The Matching Problem in General Graphs is in Quasi-NC

TL;DR

This paper demonstrates that the perfect matching problem in general graphs can be solved in Quasi-NC, providing a deterministic parallel algorithm by derandomizing the Isolation Lemma, extending previous bipartite graph results.

Contribution

It introduces a deterministic parallel algorithm for perfect matching in general graphs, extending the derandomization of the Isolation Lemma beyond bipartite graphs.

Findings

Deterministic parallel algorithm for perfect matching in general graphs.

Runs in O(log^3 n) time with n^{O(log^2 n)} processors.

Relies on laminar structure of perfect matching polytope faces.

Abstract

We show that the perfect matching problem in general graphs is in Quasi-NC. That is, we give a deterministic parallel algorithm which runs in $O(\log^3 n)$ time on $n^{O(\log^2 n)}$ processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani [1987] to obtain a Randomized NC algorithm. Our proof extends the framework of Fenner, Gurjar and Thierauf [2016], who proved the analogous result in the special case of bipartite graphs. Compared to that setting, several new ingredients are needed due to the significantly more complex structure of perfect matchings in general graphs. In particular, our proof heavily relies on the laminar structure of the faces of the perfect matching polytope.

Equations

$$T(G{)}_{u,v}=\{\begin{array}{ll}{X}_{(u,v)}& \text{if (u,v)∈E and u<v},\\ -X_{(v,u)}& \text{if (u,v)∈E and u>v},\\ 0& \text{if (u,v)∈E},\end{array}$$

$$\displaystyle\begin{array}[]{rll}x(\delta(v))=&1&\quad\text{for $v\in V$},\\ x(\delta(S))\geq&1&\quad\text{for $S\subseteq V$ with $|S|$ odd,}\\ x_{e}\geq&0&\quad\text{for $e\in E$.}\end{array}$$

$$\displaystyle E(F)=\{e\in E:(\exists x\in F)\ x_{e}>0\}\quad\mbox{and}\quad\mathcal{S}(F)=\{S\subseteq V:|S|\mbox{ odd and $(\forall x\in F)\ x(\delta(S))=1$}\}\,.$$

$$\mathrm{span}(\mathcal{L})=\mathrm{span}(\mathcal{S}(F)),$$

$$F[w]:=\mathrm{argmin}\{⟨w,x⟩:x\in F\}.$$

$$\left(w\circ w^{\prime }\right)(e):=n^{21}\cdot w(e)+w^{\prime }(e).$$

$${\mathcal{W}}^k:=\{w_1\circ w_2\circ ...\circ w_k:w_1,w_2,...,w_k\in \mathcal{W}\},$$

$$⟨y_i,w^{\prime }⟩=y_i(e_{j^{\prime }})(4n^2+1{)}^{j^{\prime }}+\sum _{j<j^{\prime }}{y}_i(e_j)(4n^2+1{)}^j>(4n^2+1{)}^{j^{\prime }}+\sum _{j=-\infty }^{j^{\prime }-1}(-4n^2)(4n^2+1{)}^j=0.$$

$$\prod _{i=1}^s\left|⟨y_i,w^{\prime }⟩\right|<{\left((4n^2+1{)}^{|E|+1}\right)}^s<(4n^2+1{)}^{n^{2}s}=2^{n^{2}s\log (4n^2+1)}<2^{n^{3}s}=2^t<\mathrm{lcm}(2,...,t)$$

$$M_{12}=(M_1\setminus E(S))\cup (M_2\cap E(S))$$

$$|\delta (R)\cap M_{12}|\le |\delta (R)\cap M_1|+|\delta (R)\cap M_2|=2,$$

$$⟨y,\text{mathbbm}{1}_{\delta (S)}⟩=⟨y,\sum _{L\in \mathcal{L}}{\mu }_L\text{mathbbm}{1}_{\delta (L)}⟩=\sum _{L\in \mathcal{L}}{\mu }_L⟨y,\text{mathbbm}{1}_{\delta (L)}⟩=0.$$

$$y:=z+\sum _i\left(\text{mathbbm}{1}_{M_i^{+}}-\text{mathbbm}{1}_{M_i^{-}}\right).$$

$$⟨y,\text{mathbbm}{1}_{\delta (T)}⟩$$

$$=\sum _{i:S_i=S}⟨\text{mathbbm}{1}_{e_i^{+}}-\text{mathbbm}{1}_{e_i^{-}}+\text{mathbbm}{1}_{M_i^{+}}-\text{mathbbm}{1}_{M_i^{-}},\text{mathbbm}{1}_{\delta (T)}⟩$$

$$\{(\pm \text{mathbbm}1{)}_C:C\text{ is an alternating circuit in H of node-weight at most 2λ}\}$$

$$v_0\stackrel{e_0}{\to }{v}_1\stackrel{e_1}{\to }\cdots \stackrel{e_{k-2}}{\to }{v}_{k-1}\stackrel{e_{k-1}}{\to }{v}_0.$$

$$C_0$$

$$C_1$$

$$⋮$$

$$C_t$$

$$\underset{\ge \lambda /2}{\underbrace{\sum _{i=i_0+1}^{i_1}\mathrm{NW}(v_i)}}+\underset{\ge \lambda /2}{\underbrace{\sum _{i=i_1+1}^{i_2}\mathrm{NW}(v_i)}}+\underset{\ge \lambda /2}{\underbrace{\sum _{i=i_2+1}^{i_3}\mathrm{NW}(v_i)}}\ge \frac{3}{2}\lambda$$

$$b\stackrel{f_1}{\to }\cdots \stackrel{f_{|P_C|}}{\to }c\stackrel{g_1}{\to }\cdots \stackrel{g_{|P_D|}}{\to }b,$$

$$-(\pm \text{mathbbm}1{)}_B$$

$$=\underset{=(\pm \text{mathbbm}1{)}_{C_j}}{\underbrace{\left((-1{)}^0\text{mathbbm}{1}_{\mathrm{out}(a)}+\sum _{i=1}^{|P_C|}(-1{)}^i\text{mathbbm}{1}_{f_i}+(-1{)}^{|P_C|+1}\text{mathbbm}{1}_{\mathrm{in}(d)}\right)}}$$

$$+\underset{=-(\pm \text{mathbbm}1{)}_{D_j}}{\underbrace{\left((-1{)}^{|P_C|+2}\text{mathbbm}{1}_{\mathrm{in}(d)}+\sum _{i=1}^{|P_D|}(-1{)}^{|P_C|+2+i}\text{mathbbm}{1}_{g_i}+(-1{)}^{|P_C|+|P_D|+3}\text{mathbbm}{1}_{\mathrm{out}(a)}\right)}}.$$

$$\mathcal{Z}:=\{(\pm \text{mathbbm}1{)}_C:C\text{ is an alternating circuit in H of node-weight at most 2λ}\}$$

$$S\in {\mathcal{L}}_{\text{in}}$$

$$U_p^{(i)}$$

$$U_{p,r}^{(i)}$$

$$\displaystyle w_{t}(M_{1}\cap E(U))\neq w_{t}(M_{2}\cap E(U))\quad\mbox{for any two matchings $M_{1},M_{2}$ in $F$ that differ inside $U$.}$$

$$w_t(M_1\cap E(U))=w_t(M_2\cap E(U)).$$

Full text

Our proof extends the framework of Fenner, Gurjar and Thierauf [2016], who proved the analogous result in the special case of bipartite graphs. Compared to that setting, several new ingredients are needed due to the significantly more complex structure of perfect matchings in general graphs. In particular, our proof heavily relies on the laminar structure of the faces of the perfect matching polytope.

1 Introduction

The perfect matching problem is a fundamental question in graph theory. Work on this problem has contributed to the development of many core concepts of modern computer science, including linear-algebraic, probabilistic and parallel algorithms. Edmonds [Edm65b] was the first to give a polynomial-time algorithm for it. However, half a century later, we still do not have full understanding of the deterministic parallel complexity of the perfect matching problem. In this paper we make progress in this direction.

We consider a problem to be efficiently solvable in parallel if it has an algorithm which uses polylogarithmic time and polynomially many processors. More formally, a problem is in the class ${\sf NC}$ if it has uniform circuits of polynomial size and polylogarithmic depth. The class ${\sf RNC}$ is obtained if we also allow randomization.

We study the decision version of the problem: given an undirected simple graph, determine whether it has a perfect matching – and the search version: find and return a perfect matching if one exists. The decision version was first shown to be in ${\sf RNC}$ by Lovász [Lov79]. The search version has proved to be more difficult and it was found to be in ${\sf RNC}$ several years later by Karp, Upfal and Wigderson [KUW86] and Mulmuley, Vazirani and Vazirani [MVV87]. All these algorithms are randomized, and it remains a major open problem to determine whether randomness is required, i.e., whether either version is in ${\sf NC}$. Intuitively, one difficulty is that a graph may contain super-polynomially many perfect matchings and it is necessary to somehow coordinate the processors so that they locate the same one. Our vocabulary for this is that we want to isolate one matching.

A successful approach to the perfect matching problem has been the linear-algebraic one. It involves the Tutte matrix associated with a graph $G=(V,E)$, which is a $|V|\times|V|$ matrix defined as follows (see Figure 1 for an example):

[TABLE]

where $X_{(u,v)}$ for $(u,v)\in E$ are variables. Tutte’s theorem [Tut47] says that $\det T(G)\neq 0$ if and only if $G$ has a perfect matching. This is great news for parallelization, as computing determinants is in ${\sf NC}$ [Csa76, Ber84]. However, the matrix is defined over a ring of indeterminates, so randomness is normally used in order to test if the determinant is nonzero. One approach is to replace each indeterminate by a random value from a large field. This leads, among others, to the fastest known (single-processor) running times for dense graphs [MS04, Har09].

A second approach, adopted by Mulmuley, Vazirani and Vazirani [MVV87] for the search version, is to replace the indeterminates by randomly chosen powers of two. Namely, for each edge $(u,v)$, a random weight $w(u,v)\in\{1,2,...,2|E|\}$ is selected, and we substitute $X_{(u,v)}:=2^{w(u,v)}$. Now, let us make the crucial assumption that one perfect matching $M$ is isolated, in the sense that it is the unique minimum-weight perfect matching (minimizing $w(M)$). Then $\det T(G)$ remains nonzero after the substitution: one can show that $M$ contributes a term $\pm 2^{2w(M)}$ to $\det T(G)$, whereas all other terms are multiples of $2^{2w(M)+1}$ and thus they cannot cancel $2^{2w(M)}$ out. The determinant can still be computed in ${\sf NC}$ as all entries $2^{w(u,v)}$ of the matrix are of polynomial bit-length, and so we have a parallel algorithm for the decision version. An algorithm for the search version also follows: for every edge in parallel, test whether removing it causes this least-significant digit $2^{2w(M)}$ in the determinant to disappear; output those edges for which it does.

The fundamental claim in [MVV87] is that assigning random weights to edges does indeed isolate one matching with high probability. This is known as the Isolation Lemma and turns out to be true in the much more general setting of arbitrary set families:

Lemma 1.1** (Isolation Lemma).**

*Let $\mathcal{M}\subseteq 2^{E}$ be any nonempty family of subsets of a universe $E=\{1,2,...,|E|\}$. Suppose we define a weight function $w:E\to\{1,2,...,2|E|\}$ by selecting each $w(e)$ for $e\in E$ independently and uniformly at random. Then with probability at least $1/2$, there is a unique set $M\in\mathcal{M}$ which minimizes the weight $w(M)=\sum_{e\in M}w(e)$. *

We call such a weight function $w$ isolating. We take $\mathcal{M}$ in Lemma 1.1 to be the set of all perfect matchings.

Since Lemma 1.1 is the only randomized ingredient of the ${\sf RNC}$ algorithm, a natural approach to showing that the perfect matching problem is in ${\sf NC}$ is the derandomization of the Isolation Lemma. That is, we would like a set of polynomially many weight functions (with polynomially bounded values) which would be guaranteed to contain an isolating one. To get an ${\sf NC}$ algorithm, we should be able to generate this set efficiently in parallel; then we can try all weight functions simultaneously.

However, derandomizing the Isolation Lemma turns out to be a challenging open question. It has been done for certain classes of graphs: strongly chordal [DK98], planar bipartite [DKR10, TV12], or graphs with a small number of perfect matchings [GK87, AHT07]. More generally, there has been much interest in obtaining ${\sf NC}$ algorithms for the perfect matching problem on restricted graph classes (not necessarily using the Isolation Lemma), e.g.: regular bipartite [LPV81], $P_{4}$-tidy [Par98], dense [DHK93], convex bipartite [DS84], claw-free [CNN89], incomparability graphs [KVV85]. The general set-family setting of the Isolation Lemma is also related to circuit lower bounds and polynomial identity testing [AM08].

Recently, in a major development, Fenner, Gurjar and Thierauf [FGT16] have almost derandomized the Isolation Lemma for bipartite graphs. Namely, they define a family of weight functions which can be computed obliviously (only using the number of vertices $n$) and prove that for any bipartite graph, one of these functions is isolating. Because their family has quasi-polynomial size and the weights are quasi-polynomially large, this has placed the perfect bipartite matching problem in the class $\text{\sf quasi-}{\sf NC}$.

Nevertheless, the general-graph setting of the derandomization question (either using the Isolation Lemma or not) remained open. Even in the planar case, with ${\sf NC}$ algorithms for bipartite planar and small-genus graphs having been known for a long time [MN89, MV00], we knew no $\text{\sf quasi-}{\sf NC}$ algorithm for non-bipartite graphs. Curiously, an ${\sf NC}$ algorithm to count the number of perfect matchings in a planar graph is known [Kas67, Vaz89], which implies an algorithm for the decision version. However, it is open to give an ${\sf NC}$ algorithm for the search version. In general, the best known upper bound on the size of uniform circuits with polylogarithmic depth was exponential.

We are able to nearly bridge this gap in understanding. The main result of our paper is the following:

Theorem 1.2**.**

*For any number $n$, we can in $\text{\sf quasi-}{\sf NC}$ construct $n^{O(\log^{2}n)}$ weight functions on $\{1,2,...,{n\choose 2}\}$ with weights bounded by $n^{O(\log^{2}n)}$ such that for any graph on $n$ vertices, one of these weight functions isolates a perfect matching (if one exists). *

The results of [MVV87] and Theorem 1.2 together imply that the perfect matching problem (both the decision and the search variant) in general graphs is in $\text{\sf quasi-}{\sf NC}$. See Section 2.1 for more details on this. We remark that the implied algorithm is very simple. The complexity lies in the analysis, i.e., proving that one of the weight functions is isolating (see Theorem 4.11).

In what follows, we first give an overview of the framework in [FGT16] for bipartite graphs. We then explain how we extend the framework to general graphs. Due to the more complex structure of perfect matchings in general graphs, we need several new ideas. In particular, we exploit structural properties of the perfect matching polytope.

1.1 Isolation in bipartite graphs

In this section we shortly discuss the elegant framework introduced by Fenner, Gurjar and Thierauf [FGT16], which we extend to obtain our result.

If a weight function $w$ is not isolating, then there exist two minimum-weight perfect matchings, and their symmetric difference consists of alternating cycles. In each such cycle, the total weight of edges from the first matching must be equal to the total weight of edges from the second matching (as otherwise we could obtain another matching of lower weight). The difference between these two total weights is called the circulation of the cycle. By the above, if all cycles have nonzero circulation, then $w$ is isolating. It is known how to obtain weight functions which satisfy a polynomial number of such non-equalities (see Lemma 3.4). However, a graph may have an exponential number of cycles.

A key idea of [FGT16] is to build the weight function in $\log n$ rounds. In the first round, we find a weight function with the property that each cycle of length 4 has nonzero circulation. This is possible since there are at most $n^{4}$ such cycles. We apply this function and from now on consider only those edges which belong to a minimum-weight perfect matching. Crucially, it turns out that in the subgraph obtained this way, all cycles of length $4$ have disappeared – this follows from the simple structure of the bipartite perfect matching polytope (a face is simply the bipartite matching polytope of a subgraph) and fails to hold for general graphs. In the second round, we start from this subgraph and apply another weight function which ensures that all even cycles of length up to $8$ have nonzero circulation (one proves that there are again $\leq n^{4}$ many since the graph contains no $4$-cycles). Again, these cycles disappear from the next subgraph, and so on. After $\log n$ rounds, the current subgraph has no cycles, i.e., it is a perfect matching. The final weight function is obtained by combining the $\log n$ polynomial-sized weight functions. To get a parallel algorithm, we need to simultaneously try each such possible combination, of which there are quasi-polynomially many.

This result has later been generalized by Gurjar and Thierauf [GT17] to the linear matroid intersection problem – a natural extension of bipartite matching. From the work of Narayanan, Saran and Vazirani [NSV94], who gave an ${\sf RNC}$ algorithm for that problem (also based on computing a determinant), it again follows that derandomizing the Isolation Lemma implies a $\text{\sf quasi-}{\sf NC}$ algorithm.

1.2 Challenges of non-bipartite graphs

We find it useful to look at the method explained in the previous section from a polyhedral perspective (also used by [GT17]). We begin from the set of all perfect matchings, of which we take the convex hull: the perfect matching polytope. After applying the first weight function, we want to consider only those perfect matchings which minimize the weight; this is exactly the definition of a face of the polytope. In the bipartite case, any face was characterized by just taking a subset of edges (i.e., making certain constraints $x_{e}\geq 0$ tight), so we could simply think about recursing on a smaller subgraph. This was used to show that any cycle whose circulation has been made nonzero will not retain all of its edges in the next subgraph. The progress we made in the bipartite case could be measured by the girth (the minimum length of a cycle) of the current subgraph, which doubled as we moved from face to subface.

Unfortunately, in the non-bipartite case, the description of the perfect matching polytope is more involved (see Section 2.2). Namely, moving to a new subface may also cause new tight odd-set constraints to appear. These, also referred to as odd cut constraints, require that, for an odd set $S\subseteq V$ of vertices, exactly one edge of a matching should cross the cut defined by $S$. This complicates our task, as depicted in the left part of Figure 2 (the same example was given by [FGT16] to demonstrate the difficulty of the general-graph case). Now a face is described by not only a subset of edges, but also a family of tight odd-set constraints. Thus we can no longer guarantee that any cycle whose circulation has been made nonzero will disappear from the support of the new face, i.e., the set of edges that appear in at least one perfect matching in this face. Our idea of what it means to remove a cycle thus needs to be refined (see Section 3), as well as the measure of progress we use to prove that a single matching is isolated after $\log n$ rounds (see Section 4). We need several new ideas, which we outline in Section 1.3.

Another difficulty, of a more technical nature, concerns the counting argument used to prove that a graph with no cycles of length at most $\lambda$ contains only polynomially many cycles of length at most $2\lambda$. In the bipartite case, the symmetric difference of two even cycles contains a simple cycle, which is also even. In addition, one can show that if the two cycles share many vertices, then the symmetric difference must contain one such even cycle that is short (of length at most $\lambda$) and thus should not exist. This enables a simple checkpointing argument to bound the number of cycles of length at most $2\lambda$, assuming that no cycle of length at most $\lambda$ exists. Now, in the general case we are still only interested in removing even cycles, but the symmetric difference of two even cycles may not contain an even simple cycle (see the right part of Figure 2). This forces us to remove not only even simple cycles, but all even walks, which may contain repeated edges (we call these alternating circuits – see Definition 3.1), and to rework the counting scheme, obtaining a bound of $n^{17}$ rather than $n^{4}$ (see Lemma 5.4). Moreover, instead of simple graphs, we work on node-weighted multigraphs, which arise by contracting certain tight odd-sets.

1.3 Our approach

This section is a high-level, idealized explanation of how to deal with the main difficulty (see the left part of Figure 2); we ignore the more technical one in this description.

Removing cycles which do not cross a tight odd-set

As discussed in Section 1.2, when moving from face to subface we cannot guarantee that, for each even cycle whose circulation we make nonzero, one of its edges will be absent from the support of the new face. However, this will at least be true for cycles that do not cross any odd-set tight for the new face. This is because if there are no tight odd-set constraints, then our faces behave as in the bipartite case. So, intuitively, if we only consider those cycles which do not cross any tight set, then we can remove them using the same arguments as in that case. This implies, by the same argument as in Section 1.1, that if we apply $\log n$ weight functions in succession, then the resulting face will not contain in its support any even cycle that crosses no tight odd-set. This is less than we need, but a good first step. If, at this point, there were no tight sets, then we would be done, as we would have removed all cycles. However, in general there will still be cycles crossing tight sets, which make our task more difficult.

Decomposition into two subinstances

To deal with the tight odd-sets, we will make use of two crucial properties. The first property is easy to see: once we fix the single edge $e$ in the matching which crosses a tight set $S$, the instance breaks up into two independent subinstances. That is, every perfect matching which contains $e$ is the union of: the edge $e$, a perfect matching on the vertex set $S$ (ignoring the $S$-endpoint of $e$), and a perfect matching on the vertex set $V\setminus S$ (ignoring the other endpoint of $e$).

This will allow us to employ a divide-and-conquer strategy: to isolate a matching in the entire graph, we will take care of both subinstances and of the cut separating them. We formulate the task of dealing with such a subinstance (a subgraph induced on an odd-cardinality vertex set) as follows: we want that, once the (only) edge of a matching which lies on the boundary of the tight odd-set is fixed, the entire matching inside the set is uniquely determined. We will then call this set contractible (see Definition 4.1). This can be seen as a generalization of our isolation objective to subgraphs with an odd number of vertices. If we can get that for the tight set and for its complement, then each edge from the cut separating them induces a unique perfect matching in the graph. Therefore there are at most $n^{2}$ perfect matchings left in the current face. Now, in order to isolate the entire graph, we only need a weight function $w$ which assigns different weights to all these matchings. This demand can be written as a system of $n^{4}$ linear non-equalities on $w$, and we can generate a weight function $w$ satisfying all of them (see Lemma 3.4).

While it is not clear how to continue this scheme beyond the first level or why we could hope to have a low depth of recursion, we will soon explain how we utilize this basic strategy.

Laminarity

The second crucial property that we utilize is that the family of odd-set constraints tight for a face exhibits good structural properties. Namely, it is known that a laminar family of odd sets is enough to describe any face (see Section 2.2). Recall that a family of sets is laminar if any two sets in the family are either disjoint or one is a subset of the other (see Figure 4 for an example). This enables a scheme where we use this family to make progress in a bottom-up fashion. This is still challenging as the family does not stay fixed as we move from face to face. The good news is that it can only increase: whenever a new tight odd-set constraint appears which is not spanned by the previous ones, we can always add an odd-set to our laminar family.

Chain case

To get started, let us first discuss the special case where the laminar family of tight constraints is a chain, i.e., an increasing sequence of odd-sets $S_{1}\subsetneq S_{2}\subsetneq...\subsetneq S_{\ell}$. We remark that this will be an informal and simplified description of the proof of Lemma 5.6. For this introduction, assume $\ell=8$ as depicted in Figure 3. Denote by $U_{1},...,U_{8}$ the layers of this chain, i.e., $U_{1}=S_{1}$ and $U_{p}=S_{p}\setminus S_{p-1}$ for $p=2,3,...,8$. Suppose this chain describes the face that was obtained by applying the $\log n$ weight functions as above that remove all even cycles that do not cross a tight set. Then there is no cycle that lies inside a single layer $U_{p}$.

Notice that every layer $U_{p}$ is of even size and it touches two boundaries of tight odd-sets: $S_{p-1}$ and $S_{p}$ (that is, $\delta(U_{p})\subseteq\delta(S_{p-1})\cup\delta(S_{p})$). Any perfect matching in the current face will have one edge from $\delta(S_{p-1})$ and one edge from $\delta(S_{p})$ (possibly the same edge), therefore $U_{p}$ will have two (or zero) boundary edges in the matching. An exception is $U_{1}$, which is odd, only touches $S_{1}$ and will have one boundary edge in the matching. This motivates us to generalize our isolation objective to layers as follows: we say that a layer $U_{p}$ is contractible if choosing an edge from $\delta(S_{p-1})$ and an edge from $\delta(S_{p})$ induces a unique matching inside $U_{p}$. This definition naturally extends to layers of the form $S_{r}\setminus S_{p-1}=U_{p}\cup U_{p+1}\cup...\cup U_{r}$, which we will denote by $U_{p,r}$.

Recall that we have ensured that there is no cycle that lies inside a single layer $U_{p}=U_{p,p}$. It follows that these layers are contractible. This is because two different matchings (but with the same boundary edges) in the current face would induce an alternating cycle in their symmetric difference.

Let us say that this was the first phase of our approach (see Figure 3). In the second phase, we want to ensure contractibility for double layers: $U_{1,2}$, $U_{3,4}$, $U_{5,6}$ and $U_{7,8}$. In general, we double our progress in each phase: in the third one we deal with the quadruple layers $U_{1,4}$ and $U_{5,8}$, and in the fourth phase we deal with the octuple layer $U_{1,8}$.

Let us now describe a single phase. Take e.g. the layer $U_{5,8}$ and two boundary edges $e_{4}\in\delta(S_{4})$ and $e_{8}\in\delta(S_{8})$ (see Figure 3); we want to have only a unique matching in $U_{5,8}$ including these edges. Now we realize our divide-and-conquer approach. Note that the layers $U_{5,6}$ and $U_{7,8}$ have already been dealt with (made contractible) in the previous phase. Therefore, for each choice of boundary edge $e_{6}\in\delta(S_{6})$ for the matching, there is a unique matching inside both of these layers. Just like previously, this implies that there are only $n^{2}$ matchings using $e_{4}$ and $e_{8}$ in the layer $U_{5,8}$, and we can select a weight function that isolates one of them.We actually select only one function per phase, which works simultaneously for all layers $U_{p,r}$ in this phase (here: $U_{1,4}$ and $U_{5,8}$) and all pairs of boundary edges $e_{p-1}$ and $e_{r}$.

By generalizing this strategy (from $\ell=8$ to arbitrary $\ell$) in the natural way, we can deal with any chain in $\log\ell\leq\log n$ phases, even if it consists of $\Omega(n)$ tight sets. We remark that, in the general proof, we do not quite use a binary tree structure like in the example. Instead, in the $t$-th phase, we deal with all layers $U_{p,r}$ having $1\leq p\leq r\leq\ell$ with $r-p\leq 2^{t-1}-1$. This makes our proof simpler if $\ell$ is not a power of two.

General case

Of course, there is no reason to expect that the laminar family of tight cuts we obtain after applying the initial $\log n$ weight functions will be a chain. It also does not seem easy to directly generalize our inductive scheme from a chain to an arbitrary family. Therefore we put forth a different progress measure, which allows us to make headway even in the absence of such a favorable odd-set structure.

Since a laminar family can be represented as a tree, we might think about a bottom-up strategy based on it; however, we cannot deal with its nodes level-by-level, since it may have height $\Omega(n)$ and we can only afford $\operatorname{poly}(\log n)$ many phases. Instead, we will first deal with all tight odd-sets of size up to $4$, then up to $8$, then up to $16$ and so on, by making them contractible. At the same time, we also remove all even cycles of length up to $4$, then up to $8$ and so on.111 As discussed in Sections 1.2 and 2, the meaning of the term remove needs to be refined, as we cannot hope to always delete an edge of the cycle from the support of the current face. These two components of our progress measure, which we call $\lambda$-goodness, are mutually beneficial, as we will see below.

Making odd-sets contractible enables us not only to achieve progress, but also to simplify our setting. A contractible tight set can be, for our purposes, thought of as a single vertex – much like a blossom in Edmonds’ algorithm. This is because such a set has exactly one boundary edge in a perfect matching (as does a vertex), and choosing that edge determines the matching in the interior. As the name suggests, we will contract such sets.

Suppose that our current face is already $\lambda$-good. Roughly, this means that we have made odd-sets of size up to $\lambda$ (which we will call small) contractible and removed cycles of length up to $\lambda$. Now we want to obtain a face which is $2\lambda$-good.

The first step is to make odd-sets of size up to $2\lambda$ contractible. Let us zoom in on one such odd-set – a maximal set of size at most $2\lambda$ (see the largest dashed set in Figure 4). Once we have contracted all the small sets into single vertices, all interesting sets are now of size more than $\lambda$ but at most $2\lambda$, and any laminar family consisting of such sets must be a chain, since a set of such size cannot have two disjoint subsets of such size (see Figure 4). But this is the chain case that we have already solved!

Having made odd-sets of size up to $2\lambda$ contractible, we can contract them. The second step is now to remove cycles of length up to $2\lambda$. However, here we do not need to care about those cycles which cross an odd-set $S$ of size larger than $2\lambda$ – the reason being, intuitively, that in our technical arguments we define the length of a cycle based on the sizes of sets that it crosses, and thus such a cycle actually becomes longer than $2\lambda$. In other words, we can think about removing cycles of length up to $2\lambda$ from a version of the input graph where all small odd-sets have been contracted and all larger ones have had their boundaries erased (see Figure 4). We call this version the contraction (see Definition 4.5). Our $\lambda$-goodness progress measure (see Definition 4.7) is actually defined in terms of cycles in the contraction.

Now the second step is easy: we just need to remove all cycles of length up to $2\lambda$ from the contraction, which has no tight odd-sets and no cycles of length up to $\lambda$ – a simple scenario, already known from the bipartite case. Applying one weight function is enough to do this.

Finally, what does it mean for us to remove a cycle? When we make a cycle’s circulation nonzero, it is then eliminated from the new face in the following sense: either one of its edges disappears from the support of the face (recall that this is what always happened in the bipartite case), or a new tight odd-set appears, with the following property: the cycle crosses the set with fewer (or more) even-indexed edges than odd-indexed edges (see the example in Figure 2). In short, we say that the cycle does not respect the new face (see Section 3). This notion of removal makes sense when viewed in tandem with the contraction, because once a cycle crosses a set in the laminar family, there are two possibilities in each phase: either this set is large – then its boundary is not present in the contraction, which cancels the cycle, or it is small – then it is contracted and the cycle also disappears (for somewhat more technical reasons).

To reiterate, our strategy is to simultaneously remove cycles up to a given length and make odd-sets up to a given size contractible. We can do this in $\log n$ phases. In each such phase we need to apply a sequence of $\log n$ weight functions in order to deal with a chain of tight odd-sets (as outlined above). In all, we are able to isolate a perfect matching in the entire graph using a sequence $O(\log^{2}n)$ weight functions with polynomially bounded weights.

1.4 Future work

The most immediate open problem left by our work is to get down from $\text{\sf quasi-}{\sf NC}$ to ${\sf NC}$ for the perfect matching problem. Even for the bipartite case, this will require new insights or methods, as it is not clear how we could e.g. reduce the number of weight functions from $\log n$ to only a constant.

Proving that the search version of the perfect matching problem in planar graphs is in ${\sf NC}$ is also open. While the $\text{\sf quasi-}{\sf NC}$ result of [FGT16] gives rise to a new ${\sf NC}$ algorithm for bipartite planar graphs, which proceeds by verifying at each step whether the chosen weight function has removed the wanted cycles (it computes the girth of the support of the current face in ${\sf NC}$), our $\lambda$-goodness progress measure seems to be difficult to verify in ${\sf NC}$.

A related problem which has resisted derandomization so far is exact matching [PY82]. Here we are given a graph, whose some edges are colored red, and an integer $k$; the question is to find a perfect matching containing exactly $k$ red edges. The problem is in ${\sf RNC}$ [MVV87], but not known to even be in ${\sf P}$.

Finally, our polyhedral approach motivates the question of what other zero-one polytopes admit such a derandomization of the Isolation Lemma. One class that comes to mind are totally unimodular polyhedra. We remark that, for that class, this question has been resolved [GTV17] subsequent to the first version of this paper.

1.5 Outline

The rest of the paper is organized as follows. In Section 2 we introduce notation and define basic notions related to the perfect matching polytope and to the weight functions that we use. In Section 3 we define alternating circuits (our generalization of alternating cycles), discuss what it means for such a circuit to respect a face, and develop our tools for circuit removal. In Section 4 we introduce our measure of progress ($\lambda$-goodness), contractible sets and the contraction multigraph. We also state Theorem 4.11, which implies our main result. Finally, in Section 5 we prove our key technical theorem: that applying $\log_{2}n+1$ weight functions allows us to make progress from $\lambda$-good to $2\lambda$-good.

2 Preliminaries

Throughout the paper we consider a fixed graph $G=(V,E)$ with $n$ vertices. We remark that the isolating weight functions whose existence we prove can be generated without knowledge of the graph. For notational convenience, we assume that $\log_{2}n$ evaluates to an integer; otherwise simply replace $\log_{2}n$ by $\lceil\log_{2}n\rceil$. We also assume that $n$ is sufficiently large.

We use the following notation. For a subset $S\subseteq V$ of the vertices, let $\delta(S)=\{e\in E:|e\cap S|=1\}$ denote the edges crossing the cut $(S,V\setminus S)$ and $E(S)=\{e\in E:|e\cap S|=2\}$ denote the edges inside $S$. We shorten $\delta(\{v\})$ to $\delta(v)$ for $v\in V$. For a vector $(x_{e})_{e\in E}\in\mathbb{R}^{|E|}$, we define $x(\delta(S))=\sum_{e\in\delta(S)}x_{e}$, as well as $\operatorname{supp}(x)=\{e\in E:x_{e}>0\}$. For a subset $X\subseteq E$ we define $\mathbbm{1}_{X}$ to be the vector with $1$ on coordinates in $X$ and [math] elsewhere. We again shorten $\mathbbm{1}_{\{e\}}$ to $\mathbbm{1}_{e}$ for $e\in E$. Sometimes we identify matchings $M$ with their indicator vectors $\mathbbm{1}_{M}$.

A matching is a set of edges $M\subseteq E$ such that no two edges in $M$ share an endpoint. A matching $M$ is perfect if $|M|=\frac{n}{2}$.

2.1 Parallel complexity

The complexity class $\text{\sf quasi-}{\sf NC}$ is defined as $\text{\sf quasi-}{\sf NC}=\bigcup_{k\geq 0}\text{\sf quasi-}{\sf NC}^{k}$, where $\text{\sf quasi-}{\sf NC}^{k}$ is the class of problems having uniform circuits of quasi-polynomial size $2^{\log^{O(1)}n}$ and polylogarithmic depth $O(\log^{k}n)$ [Bar92]. Here by “uniform” we mean that the circuit can be generated in polylogarithmic space.

By the results of [MVV87], Theorem 1.2 implies that the perfect matching problem (both the decision and the search variant) in general graphs is in $\text{\sf quasi-}{\sf NC}$. The same can be said about maximum cardinality matching, as well as minimum-cost perfect matching for small costs (given in unary); see Section 5 of [MVV87].

Some care is required to obtain our postulated running time, i.e., that the perfect matching problem has uniform circuits of size $n^{O(\log^{2}n)}$ and depth $O(\log^{3}n)$. We could get a $\text{\sf quasi-}{\sf NC}^{4}$ algorithm by applying the results of [MV97, Section 6.1] to compute the determinant(s). To shave off one $\log n$ factor, we use the following Chinese remaindering method, pointed out to us by Rohit Gurjar (it will also appear in the full version of [FGT16]). We first compute determinants modulo small primes; since the determinant has $2^{O(\log^{3}n)}$ bits, we need as many primes (each of $O(\log^{3}n)$ bits). For one prime this can be done in ${\sf NC}^{2}$ [Ber84]. Then we reconstruct the true value from the remainders. Doing this for an $n$-bits result would be in ${\sf NC}^{1}$ [BCH86], and thus for a result with $2^{O(\log^{3}n)}$ bits it is in $\text{\sf quasi-}{\sf NC}^{3}$.

2.2 Perfect matching polytope

Edmonds [Edm65a] showed that the following set of equalities and inequalities on the variables $(x_{e})_{e\in E}$ determines the perfect matching polytope (i.e., the convex hull of indicator vectors of all perfect matchings):

[TABLE]

Note that the constraints imply that $x_{e}\leq 1$ for any $e\in E$. We refer to the perfect matching polytope of the graph $G=(V,E)$ by $\operatorname{PM}(V,E)$ or simply by $\operatorname{PM}$. Our approach exploits the special structure of faces of the perfect matching polytope. Recall that a face of a polytope is obtained by setting a subset of the inequalities to equalities. We follow the definition of a face from the book of Schrijver [Sch03] – in particular, every face is nonempty.

Throughout the paper, we will only consider the perfect matching polytope and so the term “face” will always refer to a face of $\operatorname{PM}$. We sometimes abuse notation and say that a perfect matching $M$ is in a face $F$ if its indicator vector is in $F$. When talking about faces, we also use the following notation:

Definition 2.1**.**

For a face $F$ we define

[TABLE]

*In other words, $E(F)$ contains the edges that appear in a perfect matching in $F$ and $\mathcal{S}(F)$ contains the tight cut constraints of $F$. *

Notice that if a set is tight for a face, then it is also tight for any of its subfaces.

Standard uncrossing techniques imply that faces can be defined using laminar families of tight constraints. This is proved using Lemma 2.2 below, which is also useful in our approach.

Two subsets $S,T\subseteq V$ of vertices are said to be crossing if they intersect and none is contained in the other, i.e., $S\cap T,S\setminus T,T\setminus S\neq\emptyset$. A family $\mathcal{L}$ of subsets of vertices is laminar if no two sets $S,T\in\mathcal{L}$ are crossing. Furthermore, we say that $\mathcal{L}$ is a maximal laminar subset of a family $\mathcal{S}$ if no set in $\mathcal{S}\setminus\mathcal{L}$ can be added to $\mathcal{L}$ while maintaining laminarity.

Note that any single-vertex set is tight for any face, and therefore a maximal laminar family contains all these sets. The laminar families in our arguments will always contain all singletons.

The following lemma is known. For completeness, its proof is included in Appendix A.

Lemma 2.2**.**

Consider a face $F$. For any maximal laminar subset $\mathcal{L}$ of $\mathcal{S}(F)$ we have

[TABLE]

*where for a subset $\mathcal{T}\subseteq\mathcal{S}(F)$, $\operatorname{span}(\mathcal{T})$ denotes the linear subspace of $\mathbb{R}^{E}$ spanned by the boundaries of sets in $\mathcal{T}$, i.e., $\operatorname{span}(\mathcal{T})=\operatorname{span}\{\mathbbm{1}_{\delta(S)}:S\in\mathcal{T}\}$. *

Intuitively, Lemma 2.2 implies that a maximal laminar family $\mathcal{L}$ of $\mathcal{S}(F)$ is enough to describe a face $F$ (together with the edge set $E(F)$). Furthermore, given a subface $F^{\prime}\subseteq F$, we can extend $\mathcal{L}$ to a larger laminar family $\mathcal{L}^{\prime}\supseteq\mathcal{L}$ which describes $F^{\prime}$.

As the perfect matching polytope $\operatorname{PM}$ is defined as the convex hull of the indicator vectors of all perfect matchings, it is an integral polytope. In particular, it follows that every face of $\operatorname{PM}$ is also integral.

2.3 Weight functions

For our derandomization of the Isolation Lemma we will use families of weight functions which are possible to generate obliviously, i.e., by only using the number of vertices in $G$. We define them below.

Definition 2.3**.**

*Given $t\geq 7$, we define the family of weight functions $\mathcal{W}(t)$ as follows. Number the edge set $E=\{e_{1},...,e_{|E|}\}$ arbitrarily. Let $w_{k}:E\to\mathbb{Z}$ be given by $w_{k}(e_{j})=(4n^{2}+1)^{j}\operatorname{mod}k$ for $j=1,...,|E|$ and $k=2,...,t$. We define $\mathcal{W}(t)=\{w_{k}:k=2,...,t\}$. *

For brevity, we write $\mathcal{W}:=\mathcal{W}(n^{20})$.

In our argument we will obtain a decreasing sequence of faces. Each face arises from the previous by minimizing over a linear objective (given by a weight function).

Definition 2.4**.**

Let $F$ be a face and $w$ a weight function. The subface of $F$ minimizing $w$ will be called $F[w]$:

[TABLE]

Instead of minimizing over one weight function and then over another, we can concatenate them in such a way that minimizing over the concatenation yields the same subface. In particular, we will argue that one just needs to try all possible concatenations of $O(\log^{2}n)$ weight functions from $\mathcal{W}$ in order to find one which isolates a unique perfect matching in $G$ (i.e., it produces a single extreme point as the minimizing subface).

Definition 2.5**.**

For two weight functions $w$ and $w^{\prime}$, where $w:E\to\mathbb{Z}$ and $w^{\prime}\in\mathcal{W}$, we define their concatenation $w\circ w^{\prime}:=n^{21}w+w^{\prime}$, i.e.,

[TABLE]

We also define $\mathcal{W}^{k}$ to be the set of all concatenations of $k$ weight functions from $\mathcal{W}$, i.e.,

[TABLE]

where by $w_{1}\circ w_{2}\circ...\circ w_{k}$ we mean $\left(\left(w_{1}\circ w_{2}\right)\circ...\right)\circ w_{k}$.

Fact 2.6**.**

*We have $F[w][w^{\prime}]=F[w\circ w^{\prime}]$. *

Proof.

Both faces are integral and so we only need to show that $F[w][w^{\prime}]\cap\mathbb{Z}^{E}=F[w\circ w^{\prime}]\cap\mathbb{Z}^{E}$. The first set consists of matchings in $F$ minimizing $w$ and, among such matchings, minimizing $w^{\prime}$. The second set consists of matchings in $F$ minimizing $w\circ w^{\prime}$. These two sets are equal because for any $M$:

• •

$(w\circ w^{\prime})(M)=n^{21}\cdot w(M)+w^{\prime}(M)$,

• •

$w^{\prime}\in\mathcal{W}=\mathcal{W}(n^{20})$ implies that $0\leq w^{\prime}(M)<n^{20}\cdot\frac{n}{2}<n^{21}$ for any matching $M$,

• •

$w(M)\in\mathbb{Z}$, so that for any two matchings $M_{1}$ and $M_{2}$, $w(M_{1})>w(M_{2})$ implies $(w\circ w^{\prime})(M_{1})-(w\circ w^{\prime})(M_{2})=n^{21}\left(w(M_{1})-w(M_{2})\right)+\left(w^{\prime}(M_{1})-w^{\prime}(M_{2})\right)>0$.

Hence, the ordering given by $w\circ w^{\prime}$ is the same as the lexicographic ordering given by $(w,w^{\prime})$. ∎

3 Alternating circuits and respecting a face

In this section we introduce two notions which are vital for our approach. Before giving the formal definitions, we give an informal motivation.

Our argument is centered around removing even cycles. As discussed in Sections 1.2 and 2, the meaning of this term in the non-bipartite case needs to be more subtle than just “removing an edge of the cycle”.

In order to deal with a cycle, we find a weight function $w$ which assigns it a nonzero circulation. Formally, given an even cycle $C$ with edges numbered in order, define a vector $(\pm\mathbbm{1})_{C}\in\{-1,0,1\}^{E}$ as having $1$ on even-numbered edges of $C$, $-1$ on odd-numbered edges of $C$, and [math] elsewhere. Then, nonzero circulation means that $\left<{(\pm\mathbbm{1})_{C},w}\right>\neq 0$. Now, in the bipartite case, if such a cycle survived in the new face $F[w]$, that is, $C\subseteq E(F[w])$, then the vector $(\pm\mathbbm{1})_{C}$ could be used to obtain a point in the face $F$ with lower $w$-weight than the points in $F[w]$, a contradiction. This argument is possible because of the simple structure of the bipartite perfect matching polytope.

In the non-bipartite case, it is not enough that $C\subseteq E(F[w])$ in order to obtain such a point (and a contradiction). It is also required that, if the cycle $C$ enters a tight odd-set $S$ on an even-numbered edge, it exits it on an odd-numbered edge (and vice versa). This makes intuitive sense: if $C$ were obtained from the symmetric difference of two perfect matchings which both have exactly one edge crossing $S$, then $C$ would have this property. Formally, we require that $\left<{(\pm\mathbbm{1})_{C},\mathbbm{1}_{\delta(S)}}\right>=0$ for each $S\in\mathcal{S}(F[w])$. If $C$ satisfies these two conditions, i.e., that $C\subseteq E(F[w])$ and that $\left<{(\pm\mathbbm{1})_{C},\mathbbm{1}_{\delta(S)}}\right>=0$ for every $S\in\mathcal{S}(F[w])$, then we say that $C$ respects the face $F[w]$. The notion of respecting a face exactly formalizes what is required to obtain a contradictory point as above (see the proof of Lemma 3.3).

In other words, if we assign a nonzero circulation to a cycle, then it will not respect the new face, and this is what is now meant by removing a cycle.

To deal with the second, more technical difficulty discussed in Section 1.2, we need to remove not only simple cycles of even length, but also walks with repeated edges. However, we would run into problems if we allowed all such walks (up to a given length). Consider for example a walk $C$ of length $2$; such a walk traverses an edge back and forth. It is impossible to assign a nonzero circulation to $C$, because its vector $(\pm\mathbbm{1})_{C}$ is zero. We overcome this technicality by defining alternating circuits to be those even walks whose vector $(\pm\mathbbm{1})_{C}$ is nonzero (see Figure 5 for an example). For generality, we also formulate the definition of respect in terms of the vector $(\pm\mathbbm{1})_{C}$.

Definition 3.1**.**

Let $C=(e_{0},...,e_{k-1})$ be a nonempty cyclic walk of even length $k$.

• •

We define the alternating indicator vector $(\pm\mathbbm{1})_{C}$ of $C$ to be $(\pm\mathbbm{1})_{C}=\sum_{i=0}^{k-1}(-1)^{i}\mathbbm{1}_{e_{i}}$, where $\mathbbm{1}_{e}\in\mathbb{R}^{E}$ is the indicator vector having $1$ on position $e$ and [math] elsewhere.

• •

We say that $C$ is an alternating circuit if its alternating indicator vector is nonzero. We also refer to $C$ as an alternating (simple) cycle if it is an alternating circuit that visits every vertex at most once.

• •

When talking about a graph with node-weights, the node-weight of an alternating circuit is the sum of all node-weights of visited vertices (with multiplicities if visited multiple times).

We remark that $(\pm\mathbbm{1})_{C}$ does not need to have all entries $-1$, [math] or $1$ since edges can repeat in $C$.

Definition 3.2**.**

We say that a vector $y\in\mathbb{Z}^{E}$ respects a face $F$ if:

• •

$\operatorname{supp}(y)\subseteq E(F)$, and

• •

for each $S\in\mathcal{S}(F)$ we have $\left<{y,\mathbbm{1}_{\delta(S)}}\right>=0$.

*Furthermore, we say that an alternating circuit $C$ respects a face $F$ if its alternating indicator vector $(\pm\mathbbm{1})_{C}$ respects $F$. *

Clearly, if $F^{\prime}\subseteq F$ are faces and a vector respects $F^{\prime}$, then it also respects $F$.

Now we argue that we can remove an alternating circuit by assigning it a nonzero circulation. The proof of this lemma (which generalizes Lemma 3.2 of [FGT16]) motivates Definition 3.2.

Lemma 3.3**.**

*Let $y\in\mathbb{Z}^{E}$ be a vector and $F$ a face. If $w:E\to\mathbb{R}$ is such that $\left<{y,w}\right>\neq 0$, then $y$ does not respect the face $F^{\prime}=F[w]$. *

Proof.

Suppose towards a contradiction that $y$ respects $F^{\prime}$. Assume that $\left<{w,y}\right><0$ (otherwise use $-y$ in place of $y$). We pick $x\in F^{\prime}$ to be the average of all extreme points of $F^{\prime}$, so that the constraints of $\operatorname{PM}$ which are tight for $x$ are exactly those which are tight for $F^{\prime}$. Select $\varepsilon>0$ very small. Then $\left<{x+\varepsilon y,w}\right><\left<{x,w}\right>$, which will contradict the definition of $F^{\prime}=\operatorname{argmin}\{\left<{w,x}\right>:x\in F\}$ once we show that $x+\varepsilon y\in F$. We show that $x+\varepsilon y\in F^{\prime}\subseteq F$ by verifying:

• •

If $e\in E(F^{\prime})$ (i.e., $e$ is an edge with $x_{e}>0$), then $(x+\varepsilon y)_{e}=x_{e}+\varepsilon y_{e}\geq 0$ if $\varepsilon$ is chosen small enough.

• •

If $e\in E\setminus E(F^{\prime})$ (i.e., $e$ is an edge with $x_{e}=0$), then from $y$ respecting $F^{\prime}$ we get $e\not\in\operatorname{supp}(y)$ and so $(x+\varepsilon y)_{e}=0$.

• •

If $S\not\in\mathcal{S}(F^{\prime})$ is an odd set not tight for $F^{\prime}$, i.e., $\left<{x,\mathbbm{1}_{\delta(S)}}\right>>1$, then $\left<{x+\varepsilon y,\mathbbm{1}_{\delta(S)}}\right>=\left<{x,\mathbbm{1}_{\delta(S)}}\right>+\varepsilon\left<{y,\mathbbm{1}_{\delta(S)}}\right>\geq 1$ if $\varepsilon$ is chosen small enough.

• •

If $S\in\mathcal{S}(F^{\prime})$ is an odd set tight for $F^{\prime}$ (this includes all singleton sets), then from $y$ respecting $F^{\prime}$ we get $\left<{y,\mathbbm{1}_{\delta(S)}}\right>=0$ and thus $\left<{x+\varepsilon y,\mathbbm{1}_{\delta(S)}}\right>=\left<{x,\mathbbm{1}_{\delta(S)}}\right>=1$.

∎

The following lemma says that we can assign nonzero circulation to many vectors at once using an oblivious choice of weight function from $\mathcal{W}$. It is a minor generalization of Lemma 2.3 of [FGT16] and the proof remains similar. We give it for completeness.

Lemma 3.4**.**

*For any number $s$ and for any set of $s$ vectors $y_{1},...,y_{s}\in\mathbb{Z}^{E}\setminus\{0\}$ with the boundedness property $\|y_{i}\|_{1}\leq 4n^{2}$, there exists $w\in\mathcal{W}(n^{3}s)$ with $\left<{y_{i},w}\right>\neq 0$ for each $i=1,...,s$. *

We usually invoke Lemma 3.4 with vectors $y_{i}$ being the alternating indicator vectors of alternating circuits. Then the quantities $\left<{y_{i},w}\right>$ are the circulations of these circuits.

Proof.

Let $w^{\prime}:E\to\mathbb{Z}$ be given by $w^{\prime}(e_{j})=(4n^{2}+1)^{j}$ for $j=1,...,|E|$. Then we have $\left<{y_{i},w^{\prime}}\right>\neq 0$ for each $i$ because the highest nonzero coefficient dominates the expression. Formally, let $j^{\prime}$ be maximum index with $y_{i}(e_{j^{\prime}})\neq 0$ and suppose $y_{i}(e_{j^{\prime}})>0$ (the other case is analogous). Then, because $\|y_{i}\|_{\infty}\leq\|y_{i}\|_{1}\leq 4n^{2}$, we have

[TABLE]

Let $t=n^{3}s$. We want to show that there exists $k\in\{2,...,t\}$ such that for all $i=1,...,s$, $\left<{y_{i},w_{k}}\right>\neq 0$. Recalling the definition of $w_{k}$ (see Definition 2.3), $\left<{y_{i},w_{k}}\right>\neq 0$ is equivalent to $\left|\left<{y_{i},w^{\prime}}\right>\right|\neq 0\mod k$.

This will be implied if there exists $k\in\{2,...,t\}$ such that $\prod_{i}\left|\left<{y_{i},w^{\prime}}\right>\right|\neq 0\mod k$. So there should be some $k\in\{2,...,t\}$ not dividing $\prod_{i}\left|\left<{y_{i},w^{\prime}}\right>\right|$ – equivalently, $\operatorname{lcm}(2,...,t)$ should not divide $\prod_{i}\left|\left<{y_{i},w^{\prime}}\right>\right|$. Knowing that $\prod_{i}\left|\left<{y_{i},w^{\prime}}\right>\right|\neq 0$, this will follow if we have $\prod_{i}\left|\left<{y_{i},w^{\prime}}\right>\right|<\operatorname{lcm}(2,...,t)$. This is true because

[TABLE]

where we used that $\operatorname{lcm}(2,...,t)>2^{t}$ for $t\geq 7$ [Nai82]. ∎

Lemmas 3.4 and 3.3 together imply the following:

Corollary 3.5**.**

Let $F$ be a face. For any finite set of vectors $\mathcal{Y}\subseteq\mathbb{Z}^{E}\setminus\{0\}$ with the boundedness property $\|y\|_{1}\leq 4n^{2}$ for every $y\in\mathcal{Y}$, there exists $w\in\mathcal{W}(n^{3}\cdot|\mathcal{Y}|)$ such that each $y\in\mathcal{Y}$ does not respect the face $F^{\prime}=F[w]$. $\square$

4 Contractible sets and $\lambda$-goodness

We will make progress by ensuring that larger and larger parts of the graph are “isolated” in our current face $F$. By “parts of the graph” we mean sets $S$ which are tight for $F$. As discussed in Section 1.3, for such a set $S$, the following isolation property is desirable: once the (only) edge of a matching which lies on the boundary of $S$ is fixed, the entire matching inside $S$ is uniquely determined. This motivates the following definition:

Definition 4.1**.**

*Let $F$ be a face and let $S\in\mathcal{S}(F)$ be a tight set for $F$. We say that $S$ is $F$-contractible if for every $e\in\delta(S)$ there are no two perfect matchings in $F$ which both contain $e$ and are different inside $S$. *

Note that, in the above definition, there could be no such perfect matching for certain edges $e\in\delta(S)$ (this is the case if and only if $e\not\in E(F)$). Intuitively, a contractible set can be thought of as a single vertex with respect to the structure of the current face of the perfect matching polytope. The notion of contractibility enjoys the following two natural monotonicity properties:

Fact 4.2**.**

Let $F^{\prime}\subseteq F$ be two faces. If $S$ is $F$-contractible, then it is also $F^{\prime}$-contractible. $\square$

Lemma 4.3**.**

*Let $F$ be a face and $S\subseteq T$ two sets tight for $F$, i.e., $S,T\in\mathcal{S}(F)$. If $T$ is $F$-contractible, then so is $S$. *

Proof.

Let $e\in\delta(S)$. Suppose that $M_{1}$ and $M_{2}$ are two perfect matchings in $F$ which contain $e$ but are different inside $S$. We will argue that in that case there also exist two perfect matchings $M_{1}$ and $M_{12}$ in $F$ which contain $e$, are different inside $S$, and are equal outside of $S$.

Once we have that, we conclude as follows. Let $f$ be the (only) edge in $\delta(T)\cap M_{1}$ (perhaps $f=e$); then also $f\in M_{12}$. Then $M_{1}$ and $M_{12}$ are two perfect matchings in $F$ which contain $f\in\delta(T)$ but are different inside $T$, contradicting that $T$ is $F$-contractible.

To get the outstanding claim, we define

[TABLE]

to be the perfect matching that agrees with $M_{1}$ on all edges not in $E(S)$ and agrees with $M_{2}$ on all edges in $E(S)$ (see Figure 6). To see that $M_{12}$ is a perfect matching, notice that both $M_{1}$ and $M_{2}$ are in $F$ and contain $e$. Furthermore, as $e\in\delta(S)$ for the tight set $S\in\mathcal{S}(F)$, we have that $M_{1}\cap E(S)$ and $M_{2}\cap E(S)$ are both perfect matchings on the vertex set $S$ where we ignore the vertex incident to $e$. We can thus “replace” $M_{1}\cap E(S)$ by $M_{2}\cap E(S)$ to obtain the perfect matching $M_{12}$.

We now show that $M_{12}$ is in the face $F$. Suppose the contrary. Since $M_{1}$ and $M_{2}$ are both in $F$, we have $M_{12}\subseteq E(F)$. Therefore, if $M_{12}$ is not in $F$, we must have $|\delta(R)\cap M_{12}|>1$ for some tight set $R\in\mathcal{S}(F)$. Since $|R|$ is odd, also $|\delta(R)\cap M|$ is odd for any perfect matching $M$. In particular, $|\delta(R)\cap M_{12}|\geq 3$, which contradicts

[TABLE]

where the equality holds because $M_{1}$ and $M_{2}$ are perfect matchings in $F$ and $R\in\mathcal{S}(F)$ is a tight set. ∎

In our proof, we will be working with faces and laminar families which are compatible in the following sense:

Definition 4.4**.**

*Let $F$ be a face and $\mathcal{L}$ a laminar family. If $\mathcal{L}\subseteq\mathcal{S}(F)$, i.e., all sets $S\in\mathcal{L}$ are tight for $F$, then we say that $(F,\mathcal{L})$ is a face-laminar pair. *

Given a face-laminar pair $(F,\mathcal{L}$), we will often work with a multigraph obtained from $G$ by contracting all small sets, i.e., those with size being at most some parameter $\lambda$ (which is a measure of our progress). This multigraph will be called the contraction (see Figure 7 for an example).

In the contraction, we will also remove all boundaries of larger sets (i.e., those with size larger than $\lambda$). This is done to simulate working inside each such large set independently, because the contraction then decomposes into a collection of disconnected components, one per each large set. Because, in the contraction, each set in $\mathcal{L}$ has either been contracted or has had its boundary removed, our task is reduced to dealing with instances having no laminar sets.

Moreover, we only include those edges which are still in the support of the current face $F$, i.e., the set $E(F)$.

Definition 4.5**.**

Given a face-laminar pair $(F,\mathcal{L}$) and a parameter $\lambda$ (with $1\leq\lambda\leq 2n$), we define the $(F,\mathcal{L},\lambda)$-contraction of $G$ as a node-weighted multigraph as follows:

• •

the node set is the set of maximal sets of size (cardinality) at most $\lambda$ in $\mathcal{L}$,

• •

each node has a node-weight equal to the size of the corresponding set,

• •

the edge set is obtained from $E(F)\setminus\bigcup_{T\in\mathcal{L}:|T|>\lambda}\delta(T)$ by contracting each of these maximal sets. That is, an edge of $G$ maps to an edge of the contraction if it is in $E(F)$, it is not inside any of the contracted sets and it does not cross any cut defined by a set $T\in\mathcal{L}:|T|>\lambda$. Sometimes we identify edges of the contraction with their preimages in $G$.

In the $(F,\mathcal{L},\lambda)$-contractions arising in our arguments, we will always only contract sets $S\in\mathcal{L}$ which are $F$-contractible (i.e., the vertices of a contraction will always correspond to $F$-contractible sets). Then, a very useful property is that alternating circuits in the contraction can be lifted to alternating circuits in the entire graph $G$ in a canonical way. This is done in the proofs of Lemmas 5.3 and 5.8.

Finally, we need the following extension of Definition 3.2 for vectors defined on the contraction.

Definition 4.6**.**

Denote the $(F,\mathcal{L},\lambda)$-contraction of $G$ as $H$, and let $z\in\mathbb{Z}^{E(H)}$ be a vector on the edges of $H$. We say that $z$ respects a subface $F^{\prime}\subseteq F$ if 222In the following conditions we abuse notation and think of $z$ as a vector in $\mathbb{Z}^{E}$ obtained by identifying each edge of $H$ with its preimage in $G$ and letting those edges of $G$ without a preimage in $H$ have value [math].

• •

$\operatorname{supp}(z)\subseteq E(F^{\prime})$, and

• •

for each $S\in\mathcal{S}(F^{\prime})$ which is a union of sets corresponding to vertices in $V(H)$,333That is, the maximal sets of size at most $\lambda$ in $\mathcal{L}$. we have $\left<{z,\mathbbm{1}_{\delta(S)}}\right>=0$.

*As before, we say that an alternating circuit $C$ in $H$ respects a subface $F^{\prime}$ if its alternating indicator vector $(\pm\mathbbm{1})_{C}\in\mathbb{Z}^{E(H)}$ respects $F^{\prime}$. *

Now we are able to define our measure of progress. On one hand, we want to make larger and larger laminar sets contractible. On the other hand, there could very well be no laminar sets, so we also proceed as in the bipartite case: remove longer and longer alternating circuits.

Definition 4.7**.**

Let $(F,\mathcal{L})$ be a face-laminar pair and $\lambda$ a parameter (with $1\leq\lambda\leq 2n$). We say that $(F,\mathcal{L})$ is $\lambda$-good if $\mathcal{L}$ is a maximal laminar subset of $\mathcal{S}(F)$ and:

1. (i)

each $S\in\mathcal{L}$ with $|S|\leq\lambda$ is $F$-contractible, 2. (ii)

in the $(F,\mathcal{L},\lambda)$-contraction of $G$, there is no alternating circuit of node-weight at most $\lambda$.

We begin with $\lambda=1$, which is trivial, and then show that by concatenating enough weight functions we can obtain face-laminar families which are $2$-good, $4$-good, $8$-good, and so on. We are done once we have a $\lambda$-good family with $\lambda\geq n$. The components of this proof strategy are given in the following three claims. The first step is clear:

Fact 4.8**.**

Let $\mathcal{L}_{0}$ be a maximal laminar subset of $\mathcal{S}(PM)$. Then the face-laminar pair $(\operatorname{PM},\mathcal{L}_{0})$ is $1$-good. $\square$

Note that $\mathcal{L}_{0}$ contains all singleton sets, i.e., $\{\{v\}:v\in V\}\subseteq\mathcal{L}_{0}$.

We then proceed iteratively in $\log_{2}n$ rounds using the following theorem. Its proof, which constitutes the bulk of our argument, is given in Section 5.

Theorem 4.9**.**

*Let $(F,\mathcal{L})$ be a $\lambda$-good face-laminar pair. Then there exists a weight function $w\in\mathcal{W}^{\log_{2}n+1}$ and a laminar family $\mathcal{L}^{\prime}\supseteq\mathcal{L}$ such that $(F[w],\mathcal{L}^{\prime})$ is a $2\lambda$-good face-laminar pair. *

We are done once $\lambda$ exceeds $n$:

Lemma 4.10**.**

*Suppose $(F,\mathcal{L})$ is $\lambda$-good for some $\lambda\geq n$. Then $|F|=1$. *

We think that the proof of this lemma is instructive. It serves to understand and motivate Definition 4.7, and more involved versions of this argument appear in the sequel.

Proof.

Let $H$ be the $(F,\mathcal{L},\lambda)$-contraction of $G$. Also let $S_{1},S_{2},...,S_{k}$ be all maximal sets in $\mathcal{L}$. As $\mathcal{L}$ contains all singletons, their disjoint union is $V$ and we have $V(H)=\{S_{1},...,S_{k}\}$ and $E(H)=\bigcup_{i=1}^{k}\delta(S_{i})$. Since $(F,\mathcal{L})$ is $\lambda$-good with $\lambda\geq n$, each set $S\in\mathcal{L}$ is $F$-contractible, and $H$ contains no alternating circuit of node-weight at most $\lambda$ – in particular, $H$ contains no alternating simple cycle. Indeed, an upper bound on the node-weight of any alternating simple cycle is $|S_{1}|+...+|S_{k}|=n\leq\lambda$.

Now we show that there is only one perfect matching in $F$. One direction is easy: since $F$ is a face, it is nonempty by definition. For the other direction, let $M_{1}$ and $M_{2}$ be two perfect matchings in $F$. We show that $M_{1}=M_{2}$.

Because the sets $S_{1},...,S_{k}$ are tight for $F$ and no edge can possibly cross a tight odd-set of cardinality at least $\lambda\geq n$, any perfect matching (in the face $F$) in $G$ induces a perfect matching in $H$. If the matchings induced by $M_{1}$ and $M_{2}$ were different, then their symmetric difference would contain an alternating simple cycle in $H$, which is impossible. So the induced matchings must be equal, i.e., $M_{1}\cap\bigcup_{i}\delta(S_{i})=M_{2}\cap\bigcup_{i}\delta(S_{i})$. Moreover, the sets $S_{1},...,S_{k}$ are $F$-contractible, which means that, given the boundary edges, there is a unique perfect matching in $F$ inside each $S_{i}$. This yields $M_{1}=M_{2}$. ∎

Before we proceed to the proof of Theorem 4.9, let us see how 4.8, 4.9 and 4.10 together give our desired result:

Theorem 4.11**.**

*There exists an isolating weight function $w\in\mathcal{W}^{\left(\log_{2}n+1\right)\log_{2}n}$, i.e., one with $|\operatorname{PM}[w]|=1$. *

Proof.

Let $\ell=\log_{2}n$. We iteratively construct a sequence of face-laminar pairs $(F_{i},\mathcal{L}_{i})$ for $i=0,1,...,\ell$ such that $(F_{i},\mathcal{L}_{i})$ is $2^{i}$-good and $F_{i}=F_{i-1}[w_{i}]$ for some weight function $w_{i}\in\mathcal{W}^{\ell+1}$. We begin by setting $F_{0}=\operatorname{PM}$ and $\mathcal{L}_{0}$ to be a maximal laminar subset of $\mathcal{S}(PM)$. By 4.8, $(F_{0},\mathcal{L}_{0})$ is $1$-good. Then for $i=1,...,\ell$ we use Theorem 4.9 to obtain the wanted weight function $w_{i}$ along with a laminar family $\mathcal{L}_{i}\supseteq\mathcal{L}_{i-1}$. Finally, we have that $(F_{\ell},\mathcal{L}_{\ell})$ is $2^{\ell}$-good, so that by Lemma 4.10, $|F_{\ell}|=1$.

It remains to argue that $F_{\ell}=\operatorname{PM}[w]$ for some $w\in\mathcal{W}^{\left(\ell+1\right)\ell}$. To do this, we proceed as in Section 2.3: define the concatenation $w^{\prime}\bullet w^{\prime\prime}:=n^{21\left(\ell+1\right)}w^{\prime}+w^{\prime\prime}$ for two weight functions $w^{\prime}$ and $w^{\prime\prime}$, where $w^{\prime\prime}\in\mathcal{W}^{\ell+1}$. (We need to use a padding term $n^{21\left(\ell+1\right)}$ which is larger than the $n^{21}$ of Definition 2.5 because the right-hand weight functions are now from $\mathcal{W}^{\ell+1}$ rather than from $\mathcal{W}$.) By the same reasoning as for 2.6 we get that $F_{\ell}=\operatorname{PM}[w_{1}][w_{2}]...[w_{\ell}]=\operatorname{PM}[w_{1}\bullet w_{2}\bullet...\bullet w_{\ell}]$. We put $w=w_{1}\bullet w_{2}\bullet...\bullet w_{\ell}\in\mathcal{W}^{(\ell+1)\ell}$. ∎

Theorem 4.11 implies Theorem 1.2 because we have $|\mathcal{W}^{\left(\log_{2}n+1\right)\log_{2}n}|=|\mathcal{W}|^{\left(\log_{2}n+1\right)\log_{2}n}\leq n^{20\left(\log_{2}n+1\right)\log_{2}n}$, the values of any $w\in\mathcal{W}^{\left(\log_{2}n+1\right)\log_{2}n}$ are bounded by $n^{21\left(\log_{2}n+1\right)\log_{2}n}$, and the functions $w\in\mathcal{W}$ can be generated obliviously using only the number of vertices $n$.

5 Proof of the key Theorem 4.9: from $\lambda$-good to $2\lambda$-good

In this section we show how to make progress (measured by the $\lambda$ parameter of $\lambda$-goodness) by applying a new weight function to the current face. Our objective is to make larger sets contractible (by doubling the size threshold from $\lambda$ to $2\lambda$) and to ensure that in the new contracted graph, alternating circuits of an increased node-weight are not present. We do this by moving from the current face-laminar pair, which we call $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}})$, to a new face-laminar pair $(F_{\textsf{\tiny out}},\mathcal{L}_{\textsf{\tiny out}})$. Both pairs have the property that the laminar family is a maximal laminar family of sets tight for the face. The new family extends the previous, i.e., $\mathcal{L}_{\textsf{\tiny out}}\supseteq\mathcal{L}_{\textsf{\tiny in}}$.

Our main technical tools are Theorem 5.1 and Lemma 5.6. Theorem 5.1 is used to ensure that certain alternating circuits are not present in the new contraction. It says that if our current contraction has no alternating circuits of at most some node-weight, then a single weight function $w\in\mathcal{W}$ is enough to guarantee that all alternating circuits of at most twice that node-weight do not respect the new face obtained by applying $w$. We call this removing these circuits. Lemma 5.6 is used to make sure that sets in our laminar family that are of size at most $2\lambda$ become contractible. Later, new sets will be added to the laminar family in Lemma 5.8, in such a way that these properties are maintained and that the removed alternating circuits indeed do not survive in the new contraction.

The formal structure of the proof is as follows. We begin from a $\lambda$-good face-laminar pair $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}})$. Then, using our technical tools Theorems 5.1 and 5.6, we show in Theorem 5.5 the existence of a weight function $w_{\textsf{\tiny out}}\in\mathcal{W}^{\log_{2}(n)+1}$ such that the face $F_{\textsf{\tiny out}}=F_{\textsf{\tiny in}}[w_{\textsf{\tiny out}}]$ satisfies two conditions which make progress on conditions (i) and (ii) of $\lambda$-goodness:

• (i)’

For each $S\in\mathcal{L}_{\textsf{\tiny in}}$ with $|S|\leq 2\lambda$, $S$ is $F_{\textsf{\tiny out}}$-contractible.

• (ii)’

In the $(F_{\textsf{\tiny out}},\mathcal{L}_{\textsf{\tiny in}},2\lambda)$-contraction of $G$, there is no $F_{\textsf{\tiny out}}$-respecting alternating circuit of node-weight at most $2\lambda$.

This gives us the wanted face $F_{\textsf{\tiny out}}$ and weight function $w_{\textsf{\tiny out}}$. Finally, in Lemma 5.8 we show that extending the laminar family $\mathcal{L}_{\textsf{\tiny in}}$ to a maximal laminar family $\mathcal{L}_{\textsf{\tiny out}}$ (of sets tight for the new face) yields a $2\lambda$-good pair $(F_{\textsf{\tiny out}},\mathcal{L}_{\textsf{\tiny out}})$. This finishes the proof of Theorem 4.9.

5.1 Removing alternating circuits

This section is devoted to the proof of Theorem 5.1, which is a technical tool we use to remove alternating circuits of size between $\lambda$ and $2\lambda$ from the contraction.

Theorem 5.1**.**

*Consider a face-laminar pair $(F,\mathcal{L})$ such that each $S\in\mathcal{L}$ with $|S|\leq\beta$ is $F$-contractible (for a parameter $\beta$). Denote by $H$ the $(F,\mathcal{L},\beta)$-contraction of $G$. If $H$ has no alternating circuit of node-weight at most $\lambda$, then there exists $w\in\mathcal{W}$ such that $H$ has no $F[w]$-respecting alternating circuit of node-weight at most $2\lambda$.

We begin with a simple technical fact: to verify that a vector respects a face, it is enough to check this for a maximal laminar family of tight constraints.

Lemma 5.2**.**

*Consider a face $F$ and a vector $y\in\mathbb{Z}^{E}$. Let $\mathcal{L}$ be a maximal laminar subset of $\mathcal{S}(F)$. If for each $S\in\mathcal{L}$ we have $\left<{y,\mathbbm{1}_{\delta(S)}}\right>=0$, then the same holds for all $S\in\mathcal{S}(F)$. *

Proof.

As $\mathcal{L}$ is a maximal laminar subset of $\mathcal{S}(F)$, Lemma 2.2 says that $\operatorname{span}(\mathcal{L})=\operatorname{span}(\mathcal{S}(F))$. In other words, for any $S\in\mathcal{S}(F)$ we can write $\mathbbm{1}_{\delta(S)}$ as a linear combination $\sum_{L\in\mathcal{L}}\mu_{L}\mathbbm{1}_{\delta(L)}$ for some coefficients $(\mu_{L})_{L\in\mathcal{L}}$. Hence

[TABLE]

∎

Now we prove a lemma which reduces the task of removing an alternating circuit in $H$ to that of removing a vector defined on the edges of $G$, which we can do using Corollary 3.5. Throughout this section, $F$, $\mathcal{L}$ and $H$ are as in the statement of Theorem 5.1. Recall that the vertices of $H$ are elements of $\mathcal{L}$, i.e., sets of vertices, and so by $S\in V(H)$ we mean the set $S\in\mathcal{L}$ that corresponds to a vertex in $H$.

Lemma 5.3**.**

*Let $z\in\mathbb{Z}^{E(H)}$ be a nonzero vector on the edges of $H$ satisfying $\left<{z,\mathbbm{1}_{\delta(S)}}\right>=0$ for each $S\in V(H)$. Then there exists a nonzero vector $y\in\mathbb{Z}^{E}$ such that for any face $F^{\prime}\subseteq F$ we have: if $z$ respects $F^{\prime}$, then $y$ respects $F^{\prime}$. We also have $\|y\|_{1}\leq n\|z\|_{1}$. *

We remark that $y$ does not depend on $F^{\prime}$.

Proof.

We consider $z$ as a vector $z\in\mathbb{Z}^{E}$ by identifying each edge of $H$ with its preimage in $G$ and letting those edges of $G$ without a preimage in $H$ have value [math]. We may assume $\operatorname{supp}(z)\subseteq E(F)$; otherwise $z$ cannot respect $F^{\prime}$ (see Definition 4.6) and thus we are done by outputting any $y$.

The proof idea is to extend $z$ to a vector $y\in\mathbb{Z}^{E}$ which resembles an alternating indicator vector. We do this in a canonical way so that if this extension does not respect $F^{\prime}$, then it must be because $z$ itself does not respect $F^{\prime}$.

To this end, we do the following for each $S\in V(H)$: pair up the boundary edges $e\in\delta(S)$ which have $z_{e}>0$ with boundary edges $e$ which have $z_{e}<0$, respecting their multiplicities as given by $z$. For example, if we had $\delta(S)=\{e_{1},e_{2},e_{3}\}$ with $z(e_{1})=3$, $z(e_{2})=-2$ and $z(e_{3})=-1$, we would get the pairs $\{(e_{1},e_{2}),(e_{1},e_{2}),(e_{1},e_{3})\}$. Such a pairing is always possible because $\left<{z,\mathbbm{1}_{\delta(S)}}\right>=0$. Let $\left\{(e^{+}_{i},e^{-}_{i})\right\}_{i}$ be the multiset of pairs of edges obtained in this way across all $S\in V(H)$, and let $S_{i}\in V(H)$ be the set for which the pair $(e^{+}_{i},e^{-}_{i})$ has been introduced. Also denote by $v^{+}_{i},v^{-}_{i}$ the $S_{i}$-endpoints of edges $e^{+}_{i},e^{-}_{i}$.

Now, for each $i$ we have $e^{+}_{i},e^{-}_{i}\in\operatorname{supp}(z)\subseteq E(F)$, and $S$ is $F$-contractible, so there is a unique perfect matching $M_{i}^{+}$ on the vertex-induced subgraph $(S_{i}\setminus\{v^{+}_{i}\},E(S_{i}\setminus\{v^{+}_{i}\}))$ in $F$ (more precisely, $M_{i}^{+}$ is the unique perfect matching on that subgraph that extends to a matching in $F$), as well as a unique perfect matching $M_{i}^{-}$ on $(S_{i}\setminus\{v^{-}_{i}\},E(S_{i}\setminus\{v^{-}_{i}\}))$ in $F$. We let

[TABLE]

What remains now is to prove the following claim:

Claim 1**.**

*Let $F^{\prime}\subseteq F$ be such that $z$ respects $F^{\prime}$. Then $y$ respects $F^{\prime}$. *

Proof.

Since $z$ respects $F^{\prime}$ (see Definition 4.6), we have $\operatorname{supp}(z)\subseteq E(F^{\prime})$. This implies that for each $i$, $M_{i}^{+}\subseteq E(F^{\prime})$. Indeed, since $e_{i}^{+}\in\operatorname{supp}(z)\subseteq E(F^{\prime})$, there is a perfect matching on $(S_{i}\setminus\{v^{+}_{i}\},E(S_{i}\setminus\{v^{+}_{i}\}))$ in $F^{\prime}$. However, $S_{i}$ is $F$-contractible and thus $M_{i}^{+}$ is the only such matching in $F$ (thus also in $F^{\prime}$). Therefore $M_{i}^{+}\subseteq E(F^{\prime})$ and analogously $M_{i}^{-}\subseteq E(F^{\prime})$.

Now we check the conditions for $y$ to respect $F^{\prime}$ (see Definition 3.2):

• •

We have $\operatorname{supp}(y)=\operatorname{supp}(z)\cup\bigcup_{i}\left(M_{i}^{+}\cup M_{i}^{-}\right)\subseteq E(F^{\prime})$.

• •

Let $T\in\mathcal{S}(F^{\prime})$. We need to verify that $\left<{y,\mathbbm{1}_{\delta(T)}}\right>=0$. Let $\mathcal{L}^{\prime}$ be a maximal laminar subfamily of $\mathcal{S}(F^{\prime})$ extending $\mathcal{L}$, i.e., $\mathcal{L}\subseteq\mathcal{L}^{\prime}\subseteq\mathcal{S}(F^{\prime})$. By Lemma 5.2, it is enough to verify that $\left<{y,\mathbbm{1}_{\delta(T)}}\right>=0$ for $T\in\mathcal{L}^{\prime}$. For a set $T$ belonging to a laminar family which extends $\mathcal{L}$, it is not hard to see that there are two possibilities: either $T\subsetneq S$ for some $S\in V(H)$, or $T$ is a union of sets in $V(H)$ (for recall that $V(H)$ is a partitioning of $V$ that consists of sets in $\mathcal{L}$). In the latter case, $\left<{y,\mathbbm{1}_{\delta(T)}}\right>=\left<{z,\mathbbm{1}_{\delta(T)}}\right>=0$ because $y$ equals $z$ on edges crossing sets in $V(H)$ and because $z$ respects $F^{\prime}$. In the former case, we have

[TABLE]

because these are the only edges in $y$’s support that have an endpoint in $S$ (other edges cannot possibly cross $T\subseteq S$). Now it is enough to show that each summand is [math].

For this, we know that $M_{i}^{+}\cup\{e_{i}^{+}\}$ and $M_{i}^{-}\cup\{e_{i}^{-}\}$ are (partial) matchings in $F^{\prime}$ and that $T$ is tight for $F^{\prime}$. Therefore we have $|\delta(T)\cap\left(M_{i}^{+}\cup\{e_{i}^{+}\}\right)|=1$,444Formally, consider a perfect matching $M^{+}$ (on $G$) in $F^{\prime}$ which is a superset of $M_{i}^{+}\cup\{e_{i}^{+}\}$. Then we have $|\delta(T)\cap M^{+}|=1$. But $\delta(T)\cap\left(M_{i}^{+}\cup\{e_{i}^{+}\}\right)=\delta(T)\cap M^{+}$ because $T\subseteq S$. and the same holds for $M_{i}^{-}\cup\{e_{i}^{-}\}$. Therefore $\left<{\mathbbm{1}_{M_{i}^{+}\cup\{e_{i}^{+}\}},\mathbbm{1}_{\delta(T)}}\right>=1=\left<{\mathbbm{1}_{M_{i}^{-}\cup\{e_{i}^{-}\}},\mathbbm{1}_{\delta(T)}}\right>$.

∎

Regarding the norm: every edge (with multiplicity) in $z$ causes less than $n/2$ new edges (a partial matching) to appear in $y$. Therefore $\|y\|_{1}\leq(n/2+1)\|z\|_{1}\leq n\|z\|_{1}$. ∎

Our second lemma gives a bound on the number of alternating circuits we need to remove. Its proof resembles that of Lemma 3.4 in [FGT16], but it is somewhat more complex, as we are dealing with a node-weighted multigraph, as well as with alternating circuits instead of simple cycles (see Section 3). We have made no attempt to minimize the exponent $17$.

Lemma 5.4**.**

There are polynomially many alternating circuits of node-weight at most $2\lambda$ in $H$, up to identifying circuits with equal alternating indicator vectors. More precisely, the cardinality of the set

[TABLE]

*is at most $n^{17}$. *

Proof.

We will associate a small signature with each alternating circuit in $H$ of node-weight at most $2\lambda$, with the property that alternating circuits with different alternating indicator vectors are assigned different signatures. This will prove that the considered cardinality is at most the number of possible signatures, which is polynomially bounded.

Let $C=(e_{0},e_{1},...,e_{k-1})$ be an alternating circuit in $H$ of node-weight at most $2\lambda$. We want to define its signature $\sigma(C)$. To streamline notation, we let $v_{i}$ be the tail of $e_{i}$ for $i=0,\dots,k-1$. Thus $C$ is of the form

[TABLE]

(See also Figure 8 for an example.) We also let $\operatorname{NW}(v_{i})$ denote the node-weight of vertex $v_{i}$, and $\operatorname{in}(v_{i})=e_{(i-1)\operatorname{mod}k}$ and $\operatorname{out}(v_{i})=e_{i}$ be the incoming and outgoing edges of $v_{i}$ in $C$.555 The functions $\operatorname{in}(v)$ and $\operatorname{out}(v)$ are not formally well-defined since they depend on the considered alternating circuit $C$ and on which occurrence of $v$ in the circuit we are considering, but their values will be clear from the context. We now define the signature $\sigma(C)$ as the output of the following procedure:

• •

Let $i_{0}=0$ be the index of the first vertex in $C$.

• •

For $j=1,2,3$, select $i_{j}\leq k$ to be the largest index satisfying $\sum_{i=i_{j-1}+1}^{i_{j}-1}\operatorname{NW}(v_{i})\leq\lambda/2$.

• •

Let $t=\max\{j:i_{j}<k\}$ and output the signature $\sigma(C)=((-1)^{i_{j}},\operatorname{in}(v_{i_{j}}),\operatorname{out}(v_{i_{j}}))_{j=0,1,\dots,t}$. (Note that $t\leq 3$.)

The intuition of the signature is as follows (see also the left part of Figure 8). The procedure starts at the first vertex $v_{i_{0}}=v_{0}$. It then selects the farthest (according to $C$) vertex $v_{i_{1}}$ while guaranteeing that the total node-weight of the vertices visited in-between $v_{i_{0}}$ and $v_{i_{1}}$ is at most $\lambda/2$. Similarly, $v_{i_{2}}$ is selected to be the farthest vertex such that the total node-weight of the vertices $v_{i_{1}+1},\dots,v_{i_{2}-1}$ is at most $\lambda/2$, and $i_{3}$ is selected in the same fashion. The indices $i_{0},i_{1},\dots,i_{t}$ thus partition $C$ into paths

[TABLE]

so that the total node-weight of the internal vertices on each path is at most $\lambda/2$. Indeed, for $C_{j}$ with $j<3$ this follows from the selection of $i_{j}$. For $C_{3}$ (in the case $t=3$), by maximality of $i_{1}$, $i_{2}$ and $i_{3}$ we have

[TABLE]

and so the internal vertices of $C_{3}$ can have node-weight at most $\lambda/2$ (the total node-weight of $C$ being at most $2\lambda$).

We now count the number of possible signature vectors. As for each $j$ there are at most $n^{2}$ ways of choosing the incoming edge, at most $n^{2}$ ways of choosing the outgoing edge, and $i_{j}$ can have two different parities, the number of possible signatures is (summing over the choices of $t=0,1,2,3$) at most $\left(2n^{2}\cdot n^{2}\right)+\left(2n^{2}\cdot n^{2}\right)^{2}+\left(2n^{2}\cdot n^{2}\right)^{3}+\left(2n^{2}\cdot n^{2}\right)^{4}<n^{17}$.

It remains to be shown that any two alternating circuits $C$ and $D$ in $H$ of node-weight at most $2\lambda$ have different signatures if $(\pm\mathbbm{1})_{C}\neq(\pm\mathbbm{1})_{D}$. Suppose that $(\pm\mathbbm{1})_{C}\neq(\pm\mathbbm{1})_{D}$ but $\sigma(C)=\sigma(D)$. We would like to derive a contradiction with the assumption (in Theorem 5.1) that $H$ contains no alternating circuit of node-weight at most $\lambda$. This will finish the proof.

As described above, $C$ can be partitioned into disjoint paths $C_{0},\dots,C_{t}$ using its indices $i_{0},\dots,i_{t}$. Similarly we partition $D$ into $D_{0},\dots,D_{t}$. Since these are disjoint unions, $(\pm\mathbbm{1})_{C}\neq(\pm\mathbbm{1})_{D}$ implies that at least one of the four subpaths must be different between $C$ and $D$, in the sense that the part of the alternating indicator vector arising from that subpath is different. More formally, let $b_{j}$ denote the parity (i.e., the first element) of the $j$-th tuple in the signatures $\sigma(C)=\sigma(D)$. Then we have $(\pm\mathbbm{1})_{C}=\sum_{j=0}^{t}b_{j}\cdot(\pm\mathbbm{1})_{C_{j}}$ and $(\pm\mathbbm{1})_{D}=\sum_{j=0}^{t}b_{j}\cdot(\pm\mathbbm{1})_{D_{j}}$. Therefore, as $(\pm\mathbbm{1})_{C}\neq(\pm\mathbbm{1})_{D}$, there must be a $j\in\{0,\dots,t\}$ such that $(\pm\mathbbm{1})_{C_{j}}\neq(\pm\mathbbm{1})_{D_{j}}$. We will “glue” together the paths $C_{j}$ and $D_{j}$ to obtain another alternating circuit $B$.

First notice that both $C_{j}$ and $D_{j}$ are paths of the form $v_{i_{j}}=a\rightarrow b\rightarrow\dots\rightarrow c\rightarrow d=v_{i_{(j+1)\operatorname{mod}t}}$, where the segment from $b$ to $c$ differs between them.666 Here again we slightly abuse notation since $i_{j}$ might differ between $C$ and $D$; however, the vertex $v_{i_{j}}$ does not, because it is the tail of $e_{i_{j}}$, which is part of the signature $\sigma(C)=\sigma(D)$. The same applies to $v_{i_{j+1\operatorname{mod}t}}$.

This follows from the assumption that $\sigma(C)=\sigma(D)$. Let $P_{C}$ denote the path from $b$ to $c$ in $C_{j}$ and let $P_{D}$ denote the path from $b$ to $c$ in $D_{j}$. As the parity fields of the signatures agree, we have that $|P_{C}|+|P_{D}|$ is even. Now let $B$ be the cyclic walk of even length obtained by walking from $b$ to $c$ along the path $P_{C}$ and back from $c$ to $b$ along the path $P_{D}$ (in reverse). That is, $B$ is of the form (see also the right part of Figure 8)

[TABLE]

where we let $f_{1},\dots,f_{|P_{C}|}$ denote the edges of the path $P_{C}$ and $g_{1},\dots,g_{|P_{D}|}$ denote the edges of the reversed path $P_{D}$. To verify that $B$ is an alternating circuit we need to show that its alternating indicator vector is nonzero:

[TABLE]

The second equality is easiest to see by mentally extending $B$ from a circuit $b\to...\to c\to...\to b$ to $a\to b\to...\to c\to d\to c\to...\to b\to a$. Also recall that $|P_{C}|+|P_{D}|$ is even. Thus we get $(\pm\mathbbm{1})_{B}=-(\pm\mathbbm{1})_{C_{j}}+(\pm\mathbbm{1})_{D_{j}}$, which is nonzero by the choice of $j$. Finally, the node-weight of $B$ is at most the node-weight of the internal nodes of path $C_{j}$ plus the node-weight of the internal nodes of path $D_{j}$ and thus at most $\lambda/2+\lambda/2=\lambda$.

We have thus shown that $B$ is a nonempty cyclic walk of even length whose alternating indicator vector is nonzero – thus an alternating circuit – and whose node-weight is at most $\lambda$. This contradicts our assumption on $H$. ∎

Now we have all the tools needed to prove the main result of this section.

Proof of Theorem 5.1.

Let us fix some $w\in\mathcal{W}$. We want to articulate conditions on $w$ which will make sure that the statement is satisfied. Then we show that some $w\in\mathcal{W}$ satisfies these conditions.

Let $C$ be any alternating circuit in $H$ of node-weight at most $2\lambda$. Our condition on $w$ will be that all such circuits $C$ should not respect $F[w]$, i.e., that all vectors from the set

[TABLE]

should not respect $F[w]$. We use Lemma 5.3 to transform each $z\in\mathcal{Z}$ ($z\in\mathbb{Z}^{E(H)}$) to a vector $y=y(z)\in\mathbb{Z}^{E}$ such that if $y(z)$ does not respect $F[w]$, then $z$ does not respect $F[w]$. Let $\mathcal{Y}=\{y(z):z\in\mathcal{Z}\}$. Clearly $|\mathcal{Y}|\leq|\mathcal{Z}|$ (actually $|\mathcal{Y}|=|\mathcal{Z}|$ since the mapping $z\mapsto y(z)$ is one-to-one), and $|\mathcal{Z}|\leq n^{17}$ by Lemma 5.4. Moreover, since the alternating circuits $C$ were of node-weight at most $2\lambda\leq 4n$, we have $\|z\|_{1}\leq 4n$ for $z\in\mathcal{Z}$ and $\|y\|_{1}\leq 4n^{2}$ for $y\in\mathcal{Y}$. Now it is enough to apply Corollary 3.5 to obtain that there exists $w\in\mathcal{W}(n^{3}\cdot n^{17})=\mathcal{W}$ such that each $y\in\mathcal{Y}$ does not respect the face $F[w]$, and thus each $z\in\mathcal{Z}$ does not respect $F[w]$. ∎

5.2 The existence of a good weight function

In this section, we use Theorem 5.1 to prove the existence of a weight function defining a face $F_{\textsf{\tiny out}}$ with the desired properties (so as to be the face in our $2\lambda$-good face-laminar pair), namely:

Theorem 5.5**.**

Let $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}})$ be a $\lambda$-good face-laminar pair. Then there exists a weight function $w_{\textsf{\tiny out}}\in\mathcal{W}^{\log_{2}(n)+1}$ such that the face $F_{\textsf{\tiny out}}=F_{\textsf{\tiny in}}[w_{\textsf{\tiny out}}]$ satisfies:

• (i)’

For each $S\in\mathcal{L}_{\textsf{\tiny in}}$ with $|S|\leq 2\lambda$, $S$ is $F_{\textsf{\tiny out}}$-contractible.

• (ii)’

In the $(F_{\textsf{\tiny out}},\mathcal{L}_{\textsf{\tiny in}},2\lambda)$-contraction of $G$, there is no $F_{\textsf{\tiny out}}$-respecting alternating circuit of node-weight at most $2\lambda$.

Throughout this section, $F_{\textsf{\tiny in}}$ and $\mathcal{L}_{\textsf{\tiny in}}$ are as in the statement of Theorem 5.5. The proof of Theorem 5.5 is based on the following technical lemma.

Lemma 5.6**.**

There exists a weight function $w_{\textsf{\tiny mid}}\in\mathcal{W}^{\log_{2}n}$ such that the face $F_{\textsf{\tiny mid}}=F_{\textsf{\tiny in}}[w_{\textsf{\tiny mid}}]$ satisfies:

• (i)’

For each $S\in\mathcal{L}_{\textsf{\tiny in}}$ with $|S|\leq 2\lambda$, $S$ is $F_{\textsf{\tiny mid}}$-contractible.

Before giving the proof of Lemma 5.6 let us see how it, together with Theorem 5.1, readily implies Theorem 5.5. In short, once we have made sets of size up to $2\lambda$ contractible using Lemma 5.6, we are only left with removing alternating circuits of node-weight between $\lambda$ and $2\lambda$. One application of Theorem 5.1 is enough to achieve this.

Proof of Theorem 5.5.

Lemma 5.6 says that there is a weight function $w_{\textsf{\tiny mid}}\in\mathcal{W}^{\log_{2}(n)}$ such that the face $F_{\textsf{\tiny mid}}=F_{\textsf{\tiny in}}[w_{\textsf{\tiny mid}}]$ satisfies that every $S\in\mathcal{L}_{\textsf{\tiny in}}$ with $|S|\leq 2\lambda$ is $F_{\textsf{\tiny mid}}$-contractible. As we will obtain $F_{\textsf{\tiny out}}$ as a subface of $F_{\textsf{\tiny mid}}$, we have thus proved point $(i)^{\prime}$ of Theorem 5.5, as any set that is $F_{\textsf{\tiny mid}}$-contractible will remain contractible in any subface of $F_{\textsf{\tiny mid}}$ (by 4.2).

By the above, every vertex in the $(F_{\textsf{\tiny mid}},\mathcal{L}_{\textsf{\tiny in}},2\lambda)$-contraction of $G$ corresponds to an $F_{\textsf{\tiny mid}}$-contractible set. Moreover, by the assumption that the face-laminar pair $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}})$ is $\lambda$-good, the $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}},\lambda)$-contraction of $G$ does not have any alternating circuits of node-weight at most $\lambda$. This implies that the $(F_{\textsf{\tiny mid}},\mathcal{L}_{\textsf{\tiny in}},2\lambda)$-contraction of $G$ does not have any such alternating circuits. For suppose $C$ were one. Let $S_{1},...,S_{k}$ be maximal sets of size at most $2\lambda$ in $\mathcal{L}_{\textsf{\tiny in}}$, i.e., the vertices of the $(F_{\textsf{\tiny mid}},\mathcal{L}_{\textsf{\tiny in}},2\lambda)$-contraction of $G$. Note that $C$ cannot cross a set $S_{i}$ with $|S_{i}|>\lambda$, because then its node-weight would be larger than $\lambda$. Therefore $C$ only crosses sets $S_{i}$ with $|S_{i}|\leq\lambda$. Thus $C$ also appears in the $(F_{\textsf{\tiny mid}},\mathcal{L}_{\textsf{\tiny in}},\lambda)$-contraction of $G$, with the same node-weight, and in the $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}},\lambda)$-contraction (of which the $(F_{\textsf{\tiny mid}},\mathcal{L}_{\textsf{\tiny in}},\lambda)$-contraction is a subgraph) as well. This is a contradiction.

We can thus apply Theorem 5.1 with $\beta=2\lambda$ to the face-laminar pair $(F_{\textsf{\tiny mid}},\mathcal{L}_{\textsf{\tiny in}})$. We get a weight function $w\in\mathcal{W}$ such that the $(F_{\textsf{\tiny mid}},\mathcal{L}_{\textsf{\tiny in}},2\lambda)$-contraction has no $F_{\textsf{\tiny mid}}[w]$-respecting alternating circuit of node-weight at most $2\lambda$. Therefore, as the $(F_{\textsf{\tiny mid}}[w],\mathcal{L}_{\textsf{\tiny in}},2\lambda)$-contraction is a subgraph of the $(F_{\textsf{\tiny mid}},\mathcal{L}_{\textsf{\tiny in}},2\lambda)$-contraction, the face $F_{\textsf{\tiny out}}=F_{\textsf{\tiny mid}}[w]$ satisfies $(ii)^{\prime}$. Selecting $w_{\textsf{\tiny out}}=w_{\textsf{\tiny mid}}\circ w\in\mathcal{W}^{\log_{2}(n)+1}$ completes the proof (by 2.6). $\square$∎

The rest of this section is devoted to the proof of Lemma 5.6. Recall that we need to prove the existence of a weight function $w_{\textsf{\tiny mid}}\in\mathcal{W}^{\log_{2}(n)}$ satisfying $(i)^{\prime}$, i.e., that

[TABLE]

where $F_{\textsf{\tiny mid}}=F_{\textsf{\tiny in}}[w_{\textsf{\tiny mid}}]$. First note that the statement will be true for every $S\in\mathcal{L}_{\textsf{\tiny in}}$ with $|S|\leq\lambda$, regardless of the choice of the weight function $w_{\textsf{\tiny mid}}$. Indeed, by assumption $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}})$ is $\lambda$-good and so $S$ is $F_{\textsf{\tiny in}}$-contractible. Thus, by 4.2, $S$ remains $F_{\textsf{\tiny mid}}$-contractible for any subface $F_{\textsf{\tiny mid}}\subseteq F_{\textsf{\tiny in}}$.

It remains to deal with the sets $S\in\mathcal{L}_{\textsf{\tiny in}}$ with $\lambda<|S|\leq 2\lambda$. Let $\mathcal{L}=\{S\in\mathcal{L}_{\textsf{\tiny in}}:\lambda<|S|\leq 2\lambda\}$ be the laminar family $\mathcal{L}_{\textsf{\tiny in}}$ restricted to these sets. Notice that any set in $\mathcal{L}$ can have at most one child in $\mathcal{L}$ due to the cardinality constraints. In other words, $\mathcal{L}$ consists of disjoint chains, as depicted in Figure 9.

Notation.

We refer to the sets in the $i$-th chain of $\mathcal{L}$ by $\mathcal{L}_{i}$. Let $\ell_{i}=|\mathcal{L}_{i}|$ and index the sets of the chain $\mathcal{L}_{i}=\{S^{(i)}_{1},S^{(i)}_{2},\ldots,S^{(i)}_{\ell_{i}}\}$ so that $S^{(i)}_{1}\subseteq S^{(i)}_{2}\subseteq\cdots\subseteq S^{(i)}_{\ell_{i}}$. Let $U^{(i)}_{1}=S^{(i)}_{1}$ and $U^{(i)}_{p}=S^{(i)}_{p}\setminus S^{(i)}_{p-1}$ for $p=2,3,\ldots,\ell_{i}$. Also define $U^{(i)}_{p,r}=U^{(i)}_{p}\cup U^{(i)}_{p+1}\cup\cdots\cup U^{(i)}_{r}$.

Recall that a set $U^{(i)}_{1,r}=S^{(i)}_{r}$ is defined to be $F$-contractible if for every $e_{r}\in\delta(U^{(i)}_{1,r})$ there are no two perfect matchings in $F$ which both contain $e_{r}$ and are different inside $U^{(i)}_{1,r}$ (Definition 4.1). For the proof of Lemma 5.6, we generalize this definition to also include sets $U^{(i)}_{p,r}$ with $p\geq 2$.

Definition 5.7**.**

*Consider a face $F$. We say that a set $U^{(i)}_{p,r}$ with $2\leq p\leq r\leq\ell_{i}$ is $F$-contractible if for every $e_{p-1}\in\delta(S^{(i)}_{p-1})$ and $e_{r}\in\delta(S^{(i)}_{r})$ there are no two perfect matchings in $F$ which both contain $e_{p-1}$ and $e_{r}$ and are different inside $U^{(i)}_{p,r}$. (It is possible that $e_{p-1}=e_{r}$, in which case neither endpoint of this edge lies in $U^{(i)}_{p,r}$.) *

The intuition of this definition is similar to that of Definition 4.1. Consider the second chain in Figure 9 for an example. If we restrict our attention to perfect matchings that contain edges $e_{1}\in\delta(S^{(2)}_{1})$ and $e_{3}\in\delta(S^{(2)}_{3})$, then, as $S^{(2)}_{1}$ are $S^{(2)}_{3}$ are tight sets, the task of selecting such a matching decomposes into two independent problems: the problem of selecting a perfect matching in $U_{2,3}^{(2)}$ (ignoring the vertices incident to $e_{1}$ and $e_{3}$) and the problem of selecting a perfect matching in $V\setminus U_{2,3}^{(2)}$ (again ignoring the vertices incident to $e_{1}$ and $e_{3}$).

The proof now proceeds iteratively as follows.

• •

First we select $w_{1}\in\mathcal{W}$ such that $F_{1}=F_{\textsf{\tiny in}}[w_{1}]$ satisfies:

[TABLE]

• •

For $t=2,3,\ldots,\log_{2}(n)$ we select $w_{t}\in\mathcal{W}$ such that $F_{t}=F_{t-1}[w_{t}]$ satisfies:

[TABLE]

We remark that, having selected $w_{1},w_{2},\ldots,w_{\log_{2}(n)}$ as above, if we let $w_{\textsf{\tiny mid}}=w_{1}\circ w_{2}\circ\ldots\circ w_{\log_{2}(n)}\in\mathcal{W}^{\log_{2}(n)}$, then the face $F_{\textsf{\tiny mid}}=F_{\textsf{\tiny in}}[w_{\textsf{\tiny mid}}]$ equals $F_{\log_{2}(n)}$ (by 2.6). To see that this completes the proof of Lemma 5.6, note that $\ell_{i}<n/2$ for any chain $i$ since $|S^{(i)}_{1}|>\lambda=(2\lambda)/2\geq|S^{(i)}_{\ell_{i}}|/2$ and $|S^{(i)}_{\ell_{i}}|\leq n$. Therefore any set $S^{(i)}_{r}\in\mathcal{L}$ has $r\leq n/2$ and so, by (2), $S^{(i)}_{r}=U^{(i)}_{1,r}$ is $F_{\textsf{\tiny mid}}$-contractible.

In what follows, we complete the proof of Lemma 5.6 with a description of how to select $w_{1}$, followed by the selection of $w_{t}$, $t=2,3,\ldots,\log_{2}(n)$, in the iterative case.

5.2.1 The selection of $w_{1}$

The following claim allows us to use Theorem 5.1 to show the existence of a weight function $w_{1}$ satisfying (1).

Claim 2**.**

*If the $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}},\lambda)$-contraction of $G$ has no $F_{1}$-respecting alternating circuit of node-weight at most $2\lambda$, then every $U_{p}^{(i)}$ is $F_{1}$-contractible. *

The claim together with Theorem 5.1 completes the selection of $w_{1}$ as follows. Since $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}})$ is $\lambda$-good, we can apply Theorem 5.1 with $\beta=\lambda$ to obtain the existence of a weight function $w_{1}\in\mathcal{W}$ such that in the $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}},\lambda)$-contraction of $G$ there is no $F_{1}$-respecting alternating circuit of node-weight at most $2\lambda$, where $F_{1}=F_{\textsf{\tiny in}}[w_{1}]$. Hence, by the above claim, every $U_{p}^{(i)}$ is $F_{1}$-contractible as required.

Proof of 2.

This proof resembles that of Lemma 4.10. Fix $U_{p}^{(i)}$ and let $e_{p-1}\in\delta(S^{(i)}_{p-1})$ and $e_{p}\in\delta(S^{(i)}_{p})$.777Here and in Section 5.2.2, we abuse notation and assume $p\geq 2$. The only difference is that, given a set $U^{(i)}_{p,r}$ with $p=1$ (in this section $p=r$), we consider matchings containing one edge $e_{r}\in\delta(S^{(i)}_{r})$ instead of matchings containing two edges $e_{p-1}\in\delta(S^{(i)}_{p-1})$ and $e_{r}\in\delta(S^{(i)}_{r})$, since if $p=1$, the set $S^{(i)}_{p-1}$ is not defined. Suppose that $M_{1}$ and $M_{2}$ are two perfect matchings in $F_{1}$ that both contain $e_{p-1}$ and $e_{p}$. We want to show that $M_{1}$ and $M_{2}$ are equal inside $U_{p}^{(i)}$.

Let $S_{1},...,S_{k}$ be all maximal sets $S\in\mathcal{L}_{\textsf{\tiny in}}$ with $S\subseteq U_{p}^{(i)}$. They are those vertices of the $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}},\lambda)$-contraction which lie in $U_{p}^{(i)}$, and we have $S_{1}\cup...\cup S_{k}=U_{p}^{(i)}$. Because these sets, as well as $S_{p-1}^{(i)}$ and $S_{p}^{(i)}$, are tight for $F_{1}$, any perfect matching in $F_{1}$ containing $e_{p-1}$ and $e_{p}$ induces an almost-perfect matching on $S_{1},...,S_{k}$, that is, one where only the (up to two) sets $S_{i}$ containing endpoints of $e_{p-1}$ and $e_{p}$ are unmatched (see the left part of Figure 10).

If the matchings induced by $M_{1}$ and $M_{2}$ were different, then their symmetric difference would induce an alternating simple cycle $C$ in the graph obtained from $E(F_{\textsf{\tiny in}})$ by contracting $S_{1},...,S_{k}$. The cycle $C$ would also be present in the $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}},\lambda)$-contraction. This is because $S_{1},...,S_{k}$ are maximal sets of size at most $\lambda$ in $\mathcal{L}_{\textsf{\tiny in}}$, so they are indeed vertices of the contraction, and because the edges between these vertices are not on the boundary $\delta(T)$ of any $T\in\mathcal{L}_{\textsf{\tiny in}}$ with $|T|>\lambda$ (in which case they would be missing from the contraction), which in turn follows by laminarity of $\mathcal{L}_{\textsf{\tiny in}}$ and the definition of $U^{(i)}_{p}$.

Since $C$ arises from two matchings in $F_{1}$, it respects $F_{1}$. This follows by a similar argument as the proof of 1, which we repeat here for completeness. Clearly, $\operatorname{supp}(C)\subseteq M_{1}\cup M_{2}\subseteq E(F_{1})$. Let $T\in\mathcal{S}(F_{1})$ be a set tight for $F_{1}$ which is a union of the vertices of the contraction; we want to show that $\left<{(\pm\mathbbm{1})_{C},\mathbbm{1}_{\delta(T)}}\right>=0$. Because $C$ is a cycle in the contraction and $|M_{1}\cap\delta(T)|=|M_{2}\cap\delta(T)|=1$, either $C$ has no edge in $\delta(T)$ or it has two, one from $M_{1}$ and one from $M_{2}$ (and they cancel out).

Moreover, since $C$ is a simple cycle inside $U^{(i)}_{p}$, its node-weight is at most $|S_{1}|+\ldots+|S_{k}|=|U^{(i)}_{p}|\leq 2\lambda$. This contradicts our assumption.

Therefore, the induced matchings must be equal. Moreover, the sets $S_{1},...,S_{k}$ are $F_{1}$-contractible, since they are vertices of the $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}},\lambda)$-contraction, $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}})$ is $\lambda$-good, and $F_{1}\subseteq F_{\textsf{\tiny in}}$. This means that, given the boundary edges (i.e., the induced matching plus $e_{p-1}$ and $e_{p}$), there is a unique perfect matching in $F_{1}$ inside each $S_{i}$. It follows that $M_{1}$ and $M_{2}$ are equal inside $U_{p}^{(i)}$. ∎

5.2.2 The selection of $w_{t}$ for $t=2,3,\ldots,\log_{2}(n)$

In this section we show the existence of a weight function $w_{t}\in\mathcal{W}$ satisfying (2), i.e.,

[TABLE]

where $F_{t}=F_{t-1}[w_{t}]$.

The proof outline is as follows. First, in 3, we give sufficient conditions on $w_{t}$ for $U^{(i)}_{p,r}$ to be $F_{t}$-contractible. They are given as a system of linear non-equalities with coefficients in $\{-1,0,1\}$. Then, in 4, we upper-bound the number of these non-equalities by $n^{11}$. This allows us to deduce the existence of $w_{t}\in\mathcal{W}$ by applying Lemma 3.4.

The following claim gives sufficient linear non-equalities on $w_{t}$ for every $U^{(i)}_{p,r}$ to be $F_{t}$-contractible (one non-equality for each choice of $U^{(i)}_{p,r}$, $e_{p-1}$, $e_{r}$, $M_{1}$ and $M_{2}$).

Claim 3**.**

Let $U=U^{(i)}_{p,r}$ for some chain $i$ and $1\leq p\leq r\leq\ell_{i}$. Suppose that for every two edges $e_{p-1}\in\delta(S^{(i)}_{p-1})$ and $e_{r}\in\delta(S^{(i)}_{r})$ defining a face $F=\{x\in F_{t-1}:x_{e_{p-1}}=1,x_{e_{r}}=1\}$ we have:

[TABLE]

*Then $U$ is $F_{t}$-contractible. *

Proof.

We prove the contrapositive. Suppose that $U$ is not $F_{t}$-contractible. Then, by definition, there must be $e_{p-1}\in\delta(S^{(i)}_{p-1})$ and $e_{r}\in\delta(S^{(i)}_{r})$ that define a face $F^{\prime}=\{x\in F_{t}:x_{e_{p-1}}=1,x_{e_{r}}=1\}$ such that there are two matchings $M_{1}$ and $M_{2}$ in $F^{\prime}$ that differ inside $U$. Notice that $F^{\prime}\subseteq F=\{x\in F_{t-1}:x_{e_{p-1}}=1,x_{e_{r}}=1\}$. Therefore $M_{1}$ and $M_{2}$ are also two matchings in $F$ that differ inside $U$.

We complete the proof of the claim by showing that

[TABLE]

Define

[TABLE]

to be the perfect matching that agrees with $M_{1}$ on all edges not in $E(U)$ and agrees with $M_{2}$ on all edges in $E(U)$ (see the central part of Figure 10 for an example). By the same argument as in the proof of Lemma 4.3, $M_{12}$ is a perfect matching in $F^{\prime}$. It differs from $M_{1}$ inside $U$ and agrees with $M_{1}$ outside $U$.

We now use that $M_{1}$ and $M_{12}$ are perfect matchings in $F^{\prime}$ to prove (3). As $F_{t}=F_{t-1}[w_{t}]$ is the convex-hull of matchings in $F_{t-1}$ that minimize the objective function $w_{t}$, all matchings $M$ in $F_{t}$ and in its subface $F^{\prime}$ have the same weight $w_{t}(M)$. In particular,

[TABLE]

and thus $w_{t}(M_{1}\cap E(U))=w_{t}(M_{2}\cap E(U))$ as required. ∎

The above claim says that it is sufficient to write down a non-equality for each choice of $U_{p,r}^{(i)}$, $e_{p-1}$, $e_{r}$, $M_{1}$, and $M_{2}$. It is easy to upper-bound the number of ways of choosing $i$, $p$, $r$, $e_{p-1}$, and $e_{r}$. The following claim bounds the number of ways of choosing $M_{1}$ and $M_{2}$. Its proof is based on a divide-and-conquer strategy (see the right part of Figure 10). It uses the inductive assumption that $U^{(i)}_{p,r}$ is $F_{t-1}$-contractible for all chains $i$ and $1\leq p\leq r\leq\ell_{i}$ with $r-p\leq 2^{t-2}-1$.

Claim 4**.**

Let $U=U^{(i)}_{p,r}$ with $r-p\leq 2^{t-1}-1$ and define $q=\lfloor(p+r)/2\rfloor$. For any two edges $e_{p}\in\delta(S^{(i)}_{p-1})$ and $e_{r}\in\delta(S^{(i)}_{r})$ defining a face $F=\{x\in F_{t-1}:x_{e_{p-1}}=1,x_{e_{r}}=1\}$ we have

[TABLE]

We remark that the first inequality holds with equality, but we only need the inequality.

Proof.

The second inequality in the statement is trivial. We prove the first.

As we have $S^{(i)}_{q}\in\mathcal{L}_{\textsf{\tiny in}}\subseteq\mathcal{S}(F_{\textsf{\tiny in}})$ and $F\subseteq F_{t-1}\subseteq F_{\textsf{\tiny in}}$, the set $S^{(i)}_{q}$ is tight for $F$. Thus any matching $M$ in $F$ must satisfy $M\cap\delta(S^{(i)}_{q})=\{e_{q}\}$ for some edge $e_{q}\in\delta(S^{(i)}_{q})\cap E(F)$.

We prove the statement by showing that for every choice of $e_{q}$, any matching $M$ in the face $F_{e_{q}}=\{x\in F:x_{e_{q}}=1\}$ matches the nodes in $U^{(i)}_{p,r}$ in a unique way. In other words, we show that |\{M\cap E(U):M\mbox{ is a matching in F_{e_{q}}}\}|\leq 1 for every $e_{q}\in\delta(S^{(i)}_{q})\cap E(F)$, which implies

[TABLE]

To prove that |\{M\cap E(U):M\mbox{ is a matching in F_{e_{q}}}\}|\leq 1, suppose the contrary, i.e., that |\{M\cap E(U):M\mbox{ is a matching in F_{e_{q}}}\}|\geq 2. Take two such matchings $M_{1}$ and $M_{2}$ that differ inside $U$. By the definition of $F_{e_{q}}$ we have $M_{1}\cap\delta(S^{(i)}_{q})=M_{2}\cap\delta(S^{(i)}_{q})=\{e_{q}\}$ and so $M_{1}$ and $M_{2}$ must differ inside $U^{(i)}_{p,q}$ or inside $U^{(i)}_{q+1,r}$; assume the former (the argument for the other case is the same). Notice that $M_{1}$ and $M_{2}$ are two matchings in $F_{e_{q}}\subseteq F_{t-1}$ which both contain $e_{p-1}$ and $e_{q}$ but differ inside $U^{(i)}_{p,q}$, which contradicts that $U^{(i)}_{p,q}$ is $F_{t-1}$-contractible. (Note that $q-p\leq(r-p)/2\leq 2^{t-2}-1/2$, which implies that $q-p\leq 2^{t-2}-1$.) ∎

We now have all the needed tools to show the existence of a weight function $w_{t}\in\mathcal{W}$ such that the face $F_{t}=F_{t-1}[w_{t}]$ satisfies (2), i.e., that for all chains $i$ and $1\leq p\leq r\leq\ell_{i}$ with $r-p\leq 2^{t-1}-1$, $U^{(i)}_{p,r}$ is $F_{t}$-contractible. By 3, this holds if for any $U=U^{(i)}_{p,r}$ with $r-p\leq 2^{t-1}-1$ and for any $e_{p-1}\in\delta(S^{(i)}_{p-1})$ and $e_{r}\in\delta(S^{(i)}_{r})$ defining a face $F=\{x\in F_{t-1}:x_{e_{p-1}}=1,x_{e_{r}}=1\}$ we have the following:

[TABLE]

There are at most $n$ ways of choosing $i$, $n$ ways of choosing $p$, $n$ ways of choosing $r$, $n^{2}$ ways of choosing $e_{p-1}$, $n^{2}$ ways of choosing $e_{r}$, and by Claim 4 there are at most $n^{4}$ ways of choosing $M_{1}$ and $M_{2}$. In total, we can write the sufficient conditions on the weight function $w_{t}$ as a system of at most $n^{11}$ linear non-equalities with coefficients in $\{-1,0,1\}$. It follows by Lemma 3.4 that there is a weight function $w_{t}\in\mathcal{W}(n^{14})\subseteq\mathcal{W}(n^{20})=\mathcal{W}$ satisfying these conditions. This completes the selection of $w_{t}$ and the proof of Lemma 5.6.

5.3 A maximal laminar family completes the proof

In Theorem 5.5 we have demonstrated the existence of a weight function $w_{\textsf{\tiny out}}$ that defines a face $F_{\textsf{\tiny out}}$ with properties $(i)^{\prime}$ and $(ii)^{\prime}$. We now show that extending $\mathcal{L}_{\textsf{\tiny in}}$ to a maximal laminar family $\mathcal{L}_{\textsf{\tiny out}}$ of $\mathcal{S}(F_{\textsf{\tiny out}})$ yields a $2\lambda$-good face-laminar pair. Such an extension is possible because $\mathcal{L}_{\textsf{\tiny in}}$ consists of sets that are tight for $F_{\textsf{\tiny in}}$ and thus also for $F_{\textsf{\tiny out}}$. As explained in the beginning of Section 5, this will complete the proof of Theorem 4.9.

Why a maximal laminar family? Part of our argument so far was about removing certain alternating circuits $C$. In other words, we have made $C$ not respect the new face $F_{\textsf{\tiny out}}$. This means either not having some edge from $\operatorname{supp}(C)$ in the support $E(F_{\textsf{\tiny out}})$ of $F_{\textsf{\tiny out}}$, or introducing a new odd-set $S$ which is tight for $F_{\textsf{\tiny out}}$ and such that $\left<{(\pm\mathbbm{1})_{C},\mathbbm{1}_{\delta(S)}}\right>\neq 0$. In the latter case, we want to have an odd-set with this property also in the new laminar family $\mathcal{L}_{\textsf{\tiny out}}$, so that the removal of $C$ is reflected in the new contraction (which is based on $\mathcal{L}_{\textsf{\tiny out}}$). Lemma 5.2 guarantees that this will happen if we choose $\mathcal{L}_{\textsf{\tiny out}}$ to be a maximal laminar subset of $\mathcal{S}(F_{\textsf{\tiny out}})$.

Lemma 5.8**.**

*Let $(F_{\textsf{\tiny in}},\mathcal{L}_{\textsf{\tiny in}})$ be a $\lambda$-good face-laminar pair and $F_{\textsf{\tiny out}}\subseteq F_{\textsf{\tiny in}}$ be the face guaranteed by Theorem 5.5. Then $(F_{\textsf{\tiny out}},\mathcal{L}_{\textsf{\tiny out}})$ is a $2\lambda$-good face-laminar pair, where $\mathcal{L}_{\textsf{\tiny out}}$ is any maximal laminar family with $\mathcal{L}_{\textsf{\tiny in}}\subseteq\mathcal{L}_{\textsf{\tiny out}}\subseteq\mathcal{S}(F_{\textsf{\tiny out}})$. *

Proof of Lemma 5.8.

Recall that Theorem 5.5 guarantees that:

• (i)’

each $S\in{\mathcal{L}_{\textsf{\tiny in}}}$ with $|S|\leq 2\lambda$ is $F_{\textsf{\tiny out}}$-contractible,

• (ii)’

in the $(F_{\textsf{\tiny out}},{\mathcal{L}_{\textsf{\tiny in}}},2\lambda)$-contraction of $G$, there is no alternating circuit of node-weight at most $2\lambda$ which respects $F_{\textsf{\tiny out}}$.

We want to show that the pair $(F_{\textsf{\tiny out}},\mathcal{L}_{\textsf{\tiny out}})$ satisfies Definition 4.7, that is,

1. (i)

each $S\in{\mathcal{L}_{\textsf{\tiny out}}}$ with $|S|\leq 2\lambda$ is $F_{\textsf{\tiny out}}$-contractible, 2. (ii)

in the $(F_{\textsf{\tiny out}},{\mathcal{L}_{\textsf{\tiny out}}},2\lambda)$-contraction of $G$, there is no alternating circuit of node-weight at most $2\lambda$.

Property (i).

Fix a set $S\in\mathcal{L}_{\textsf{\tiny out}}$ with $|S|\leq 2\lambda$. Let $S_{1},...,S_{k}$ be all maximal subsets of $S$ in $\mathcal{L}_{\textsf{\tiny in}}$ (we have $S=S_{1}\cup...\cup S_{k}$). If $S$ is contained in a set from $\mathcal{L}_{\textsf{\tiny in}}$ of size at most $2\lambda$, then that set is $F_{\textsf{\tiny out}}$-contractible by $(i)^{\prime}$, and thus $S$ is $F_{\textsf{\tiny out}}$-contractible by Lemma 4.3. So assume that is not the case; therefore, by laminarity, each $S_{i}$ is a maximal set of size at most $2\lambda$ in $\mathcal{L}_{\textsf{\tiny in}}$, that is, a vertex of the $(F_{\textsf{\tiny out}},\mathcal{L}_{\textsf{\tiny in}},2\lambda)$-contraction of $G$. By $(i)^{\prime}$, each $S_{i}$ is $F_{\textsf{\tiny out}}$-contractible.

Now the proof proceeds as in 2. We present it for completeness. Let $M_{1}$ and $M_{2}$ be two perfect matchings in $F_{\textsf{\tiny out}}$ which both contain an edge $e\in\delta(S)$. We want to show that $M_{1}$ and $M_{2}$ are equal inside $S$. Because $S$ and $S_{1},...,S_{k}$ are tight for $F_{\textsf{\tiny out}}$, any perfect matching (on $G$) in $F_{\textsf{\tiny out}}$ containing $e$ induces an almost-perfect matching on $S_{1},...,S_{k}$, that is, one where only the set $S_{i}$ containing the $S$-endpoint of $e$ is unmatched.

If the matchings induced by $M_{1}$ and $M_{2}$ were different, then their symmetric difference would contain an alternating simple cycle in the $(F_{\textsf{\tiny out}},\mathcal{L}_{\textsf{\tiny in}},2\lambda)$-contraction. Since this cycle arises from two matchings in $F_{\textsf{\tiny out}}$, it respects $F_{\textsf{\tiny out}}$. Moreover, since it is a simple cycle inside $S$, its node-weight is at most $|S_{1}|+\ldots+|S_{k}|=|S|\leq 2\lambda$. This would contradict our assumption (ii)’.

Therefore the induced matchings must be equal. Moreover, the sets $S_{1},...,S_{k}$ are $F_{\textsf{\tiny out}}$-contractible, which means that, given the boundary edges (i.e., the induced matching plus $e$), there is a unique perfect matching in $F_{\textsf{\tiny out}}$ inside each $S_{i}$. It follows that $M_{1}$ and $M_{2}$ are equal inside $S$.

Property (ii).

Let $H_{\textsf{\tiny out}}$ be the $(F_{\textsf{\tiny out}},\mathcal{L}_{\textsf{\tiny out}},2\lambda)$-contraction of $G$ and let $H_{\textsf{\tiny in}}$ be the $(F_{\textsf{\tiny out}},\mathcal{L}_{\textsf{\tiny in}},2\lambda)$-contraction of $G$. Thus $H_{\textsf{\tiny out}}$ can also be obtained by further contracting $H_{\textsf{\tiny in}}$, as well as removing all edges that are in the boundaries of sets $S\in\mathcal{L}_{\textsf{\tiny out}}\setminus\mathcal{L}_{\textsf{\tiny in}}$ with $|S|>2\lambda$. This will be our perspective. Suppose towards a contradiction that there is an alternating circuit $C_{\textsf{\tiny out}}$ in $H_{\textsf{\tiny out}}$ of node-weight at most $2\lambda$.

To obtain a contradiction, we are going to lift $C_{\textsf{\tiny out}}$ back to an $F_{\textsf{\tiny out}}$-respecting alternating circuit $C_{\textsf{\tiny in}}$ in $H_{\textsf{\tiny in}}$, which should not exist by $(ii)^{\prime}$. (This is in the same spirit as the proof of Lemma 5.3.) Namely, whenever $C_{\textsf{\tiny out}}$ visits a vertex $S\in V(H_{\textsf{\tiny out}})$, we connect up the dangling endpoints of this visit inside $S$ to obtain a walk in $H_{\textsf{\tiny in}}$. More precisely, let $e_{1}$ and $e_{2}$ be two consecutive edges of $C_{\textsf{\tiny out}}$, whose common endpoint in $H_{\textsf{\tiny out}}$ is $S$. Between them, we insert a simple path $P_{e_{1}e_{2}}$ inside the image of $S$ in $H_{\textsf{\tiny in}}$, which is constructed as follows.

Since $e_{1},e_{2}\in\operatorname{supp}(C_{\textsf{\tiny out}})\subseteq E(F_{\textsf{\tiny out}})$, there exist matchings $M_{1}$ and $M_{2}$ (on $G$) in $F_{\textsf{\tiny out}}$ containing $e_{1}$ and $e_{2}$, respectively. Let $S_{1},...,S_{k}$ be all maximal subsets of $S$ in $\mathcal{L}_{\textsf{\tiny in}}$ ($S_{i}$ are vertices of $H_{\textsf{\tiny in}}$ and we have $S=S_{1}\cup...\cup S_{k}$). Denote by $S_{e_{1}}$ and $S_{e_{2}}$ the sets $S_{i}$ which contain the $S$-endpoint of $e_{1}$ and $e_{2}$, respectively. The sets $S$ and $S_{1},...,S_{k}$ are tight for $F_{\textsf{\tiny out}}$, so $M_{1}$ induces a perfect matching on $\{S_{1},...,S_{k}\}\setminus\{S_{e_{1}}\}$ (and similarly for $M_{2}$ and $e_{2}$). The symmetric difference of these two induced matchings contains a simple path $P_{e_{1}e_{2}}$ from $S_{e_{1}}$ to $S_{e_{2}}$ in $H_{\textsf{\tiny in}}$ which has even length (possibly [math]). For an example, see Figure 11. We obtain $C_{\textsf{\tiny in}}$ by inserting such a path $P_{e_{1}e_{2}}$ between each two consecutive edges $e_{1},e_{2}$ in $C_{\textsf{\tiny out}}$.

To obtain a contradiction, we need to prove that $C_{\textsf{\tiny in}}$ is an alternating circuit of node-weight at most $2\lambda$ which respects $F_{\textsf{\tiny out}}$.

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That $C_{\textsf{\tiny in}}$ is an alternating circuit follows by construction because each path $P_{e_{1}e_{2}}$ is of even length and the alternating indicator vector of $C_{\textsf{\tiny in}}$ is nonzero since it contains the alternating indicator vector of $C_{\textsf{\tiny out}}$ (via the natural mapping of edges).

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For the node-weight, note that in $C_{\textsf{\tiny out}}$, the visit to $S$ (on the edge $e_{1}$) incurs a node-weight increase of $|S|$, whereas in $C_{\textsf{\tiny in}}$, the visit to a certain subset of $\{S_{1},...,S_{k}\}$ (on $e_{1}$ and $P_{e_{1}e_{2}}$) incurs an increase of at most $|S_{1}|+\ldots+|S_{k}|=|S|$ because $P_{e_{1}e_{2}}$ is a simple path. Therefore the node-weight of $C_{\textsf{\tiny in}}$ is at most that of $C_{\textsf{\tiny out}}$ – at most $2\lambda$.

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To see that $C_{\textsf{\tiny in}}$ respects $F_{\textsf{\tiny out}}$, we use the assumption that $\mathcal{L}_{\textsf{\tiny out}}$ is a maximal laminar subset of $\mathcal{S}(F_{\textsf{\tiny out}})$. This is in similar spirit as the proof of 1. To work around the fact that we are dealing with the contraction, we use the following version of Lemma 5.2:

Claim 5**.**

*Consider a vector $z\in\mathbb{Z}^{E(H_{\textsf{\tiny in}})}$. If for each $T\in\mathcal{L}_{\textsf{\tiny out}}$ which is a union of sets in $V(H_{\textsf{\tiny in}})$ we have $\left<{z,\mathbbm{1}_{\delta(T)}}\right>=0$, then the same holds for each $T\in\mathcal{S}(F_{\textsf{\tiny out}})$ which is a union of sets in $V(H_{\textsf{\tiny in}})$. *

Let us defer the proof of 5 and first use it to show that $C_{\textsf{\tiny in}}$ respects $F_{\textsf{\tiny out}}$. We verify the conditions of Definition 4.6. First, note that $\operatorname{supp}(C_{\textsf{\tiny in}})\subseteq E(F_{\textsf{\tiny out}})$ by construction. Second, let $T\in\mathcal{S}(F_{\textsf{\tiny out}})$ be a union of vertices of $H_{\textsf{\tiny in}}$; we need to show that $\left<{(\pm\mathbbm{1})_{C_{\textsf{\tiny in}}},\mathbbm{1}_{\delta(T)}}\right>=0$. By 5, we may assume that $T\in\mathcal{L}_{\textsf{\tiny out}}$. We consider two cases:

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If $|T|>2\lambda$, then all boundary edges of $T$ are absent from $H_{\textsf{\tiny out}}$ (see Definition 4.5), so $\operatorname{supp}(C_{\textsf{\tiny out}})\cap\delta(T)=\emptyset$. In this case $T$ is a union of vertices of $H_{\textsf{\tiny out}}$ and so no path $P_{e_{1}e_{2}}$ contains any edges from $\delta(T)$ either. Hence $\operatorname{supp}(C_{\textsf{\tiny in}})\cap\delta(T)=\emptyset$.888Here we take advantage of the fact that the boundaries of large sets are erased in the definition of the contraction. If they were not erased, we would be unable to proceed, as there would be no reason for $C_{\textsf{\tiny out}}$ (and thus $C_{\textsf{\tiny in}}$) to respect $F_{\textsf{\tiny out}}$.

• –

If $|T|\leq 2\lambda$, then $T$ must be contained in a single set $S\in V(H_{\textsf{\tiny out}})$ (as depicted in Figure 11). This is because $T\in\mathcal{L}_{\textsf{\tiny out}}$ and the sets $S\in V(H_{\textsf{\tiny out}})$ are maximal sets $S\in\mathcal{L}_{\textsf{\tiny out}}$ with $|S|\leq 2\lambda$. For every path $P_{e_{1}e_{2}}$ inside $S$, the path $e_{1},P_{e_{1}e_{2}},e_{2}$ is a path from outside of $S$ to outside of $S$ which is part of the symmetric difference of two matchings in $F_{\textsf{\tiny out}}$. If this path enters $T$, it must also leave $T$. Suppose it entered $T$ on an edge of the first matching. Then it must exit $T$ on an edge of the second matching, since $T\in\mathcal{L}_{\textsf{\tiny out}}\subseteq\mathcal{S}(F_{\textsf{\tiny out}})$ is tight for $F_{\textsf{\tiny out}}$ and both matchings are in $F_{\textsf{\tiny out}}$, and so the corresponding $\pm 1$ terms cancel out. Abusing notation, we have $\left<{(\pm\mathbbm{1})_{e_{1},P_{e_{1}e_{2}},e_{2}},\mathbbm{1}_{\delta(T)}}\right>=0$. Since this holds for every path $P_{e_{1}e_{2}}$ inside $S$, we get $\left<{(\pm\mathbbm{1})_{C_{\textsf{\tiny in}}},\mathbbm{1}_{\delta(T)}}\right>=0$ as required.

Thus $C_{\textsf{\tiny in}}$ is an alternating circuit of node-weight at most $2\lambda$ which respects $F_{\textsf{\tiny out}}$. Its existence contradicts $(ii)^{\prime}$. We conclude with the proof of 5:

Proof of 5.

We wish to reduce our setting to that of Lemma 5.2. Define $H_{\textsf{\tiny in}}^{\prime}$ to be the graph obtained from $(V,E(F_{\textsf{\tiny out}}))$ by contracting all maximal sets $S\in\mathcal{L}_{\textsf{\tiny in}}$ with $|S|\leq 2\lambda$, but not erasing the boundaries of sets $S\in\mathcal{L}_{\textsf{\tiny in}}$ with $|S|>2\lambda$. (If we did erase them, we would obtain $H_{\textsf{\tiny in}}$; instead, we have $V(H_{\textsf{\tiny in}})=V(H_{\textsf{\tiny in}}^{\prime})$ and $E(H_{\textsf{\tiny in}})\subseteq E(H_{\textsf{\tiny in}}^{\prime})$.)

Next, we define $F_{\textsf{\tiny out}}^{\prime}$ to be the image of $F_{\textsf{\tiny out}}$ in $H_{\textsf{\tiny in}}^{\prime}$. More precisely, since the contracted sets $S\in\mathcal{L}_{\textsf{\tiny in}}$ with $|S|\leq 2\lambda$ are tight for $F_{\textsf{\tiny out}}$, each perfect matching on $G$ in $F_{\textsf{\tiny out}}$ induces a perfect matching on $H_{\textsf{\tiny in}}^{\prime}$. We let $F_{\textsf{\tiny out}}^{\prime}$ be the convex hull of the indicator vectors of these induced matchings. Note that there is a one-to-one correspondence between subsets of $V(H_{\textsf{\tiny in}})$ that are tight for $F_{\textsf{\tiny out}}^{\prime}$ and subsets of $V$ that are tight for $F_{\textsf{\tiny out}}$ and are unions of sets in $V(H_{\textsf{\tiny in}})$.

Finally, $\mathcal{L}_{\textsf{\tiny out}}$ also naturally maps to a laminar family $\mathcal{L}_{\textsf{\tiny out}}^{\prime}$ of subsets of vertices of $H_{\textsf{\tiny in}}^{\prime}$ since $\mathcal{L}_{\textsf{\tiny in}}\subseteq\mathcal{L}_{\textsf{\tiny out}}$. Specifically, there is a set in $\mathcal{L}_{\textsf{\tiny out}}^{\prime}$ corresponding to each set in $\mathcal{L}_{\textsf{\tiny out}}$ that is a union of sets in $V(H_{\textsf{\tiny in}}^{\prime})$. Note that each set in $\mathcal{L}_{\textsf{\tiny out}}^{\prime}$ is still tight for $F_{\textsf{\tiny out}}^{\prime}$, and that $\mathcal{L}_{\textsf{\tiny out}}^{\prime}$ is still a maximal subset of $\mathcal{S}(F_{\textsf{\tiny out}}^{\prime})$. Indeed, if it were possible to add any set in $\mathcal{S}(F_{\textsf{\tiny out}}^{\prime})$ to $\mathcal{L}_{\textsf{\tiny out}}^{\prime}$ while maintaining laminarity, then that set could be mapped back to a set in $\mathcal{S}(F_{\textsf{\tiny out}})$ and used to enlarge $\mathcal{L}_{\textsf{\tiny out}}$.

Now, by assumption, for each $T\in\mathcal{L}_{\textsf{\tiny out}}$ which is a union of sets in $V(H_{\textsf{\tiny in}})$ we have $\left<{z,\mathbbm{1}_{\delta(T)}}\right>=0$. This is equivalent to saying that $\left<{z,\mathbbm{1}_{\delta(T^{\prime})}}\right>=0$ for each $T^{\prime}\in\mathcal{L}_{\textsf{\tiny out}}^{\prime}$. By Lemma 5.2 (applied to $H_{\textsf{\tiny in}}^{\prime}$, $F_{\textsf{\tiny out}}^{\prime}$, and $\mathcal{L}_{\textsf{\tiny out}}^{\prime}$), we have the same for all $T^{\prime}\in\mathcal{S}(F_{\textsf{\tiny out}}^{\prime})$. Finally, that is equivalent to having $\left<{z,\mathbbm{1}_{\delta(T)}}\right>=0$ for each $T\in\mathcal{S}(F_{\textsf{\tiny out}})$ which is a union of sets in $V(H_{\textsf{\tiny in}})$. ∎

∎

Acknowledgment

We thank the anonymous FOCS 2017 reviewers for their exceptionally detailed and insightful comments.

Appendix A Proof of Lemma 2.2

The proof proceeds via the primal uncrossing technique; it is adapted from [LRS11]. Assume without loss of generality that $E=E(F)$. We can do this since including the constraint $x_{e}=0$ yields the same face as removing the edge $e$ from $G$. We begin with an uncrossing lemma.

Lemma A.1** (uncrossing).**

Let $S,T\in\mathcal{S}(F)$ be two sets which are crossing (i.e., $S\cap T,S\setminus T,T\setminus S\neq\emptyset$). Then:

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if $|S\cap T|$ is odd: then $S\cap T,S\cup T\in\mathcal{S}(F)$ and $\mathbbm{1}_{\delta(S)}+\mathbbm{1}_{\delta(T)}=\mathbbm{1}_{\delta(S\cap T)}+\mathbbm{1}_{\delta(S\cup T)}$,

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otherwise: $S\setminus T,T\setminus S\in\mathcal{S}(F)$ and $\mathbbm{1}_{\delta(S)}+\mathbbm{1}_{\delta(T)}=\mathbbm{1}_{\delta(S\setminus T)}+\mathbbm{1}_{\delta(T\setminus S)}$.

Proof.

Case $|S\cap T|$ odd. Note that we have

[TABLE]

For any $x\in F$, since $S,T\in\mathcal{S}(F)$ and because $S\cap T$, $S\cup T$ are nonempty odd sets, we have

[TABLE]

where the inequality must be an equality, and thus $x(\delta(S\cap T))=1$, $x(\delta(S\cup T))=1$ (implying $S\cap T,S\cup T\in\mathcal{S}(F)$) and $x(\delta(S\setminus T,T\setminus S))=0$ for all $x\in F$ (which, given that $E=E(F)$, implies that $\delta(S\setminus T,T\setminus S)=\emptyset$ and thus $\mathbbm{1}_{\delta(S\setminus T,T\setminus S)}=0$).

Case $|S\cap T|$ even. Now we have

[TABLE]

The sets $S\setminus T$ and $T\setminus S$ are odd and nonempty, and we proceed as above. ∎

Define $\mathrm{cross}(S,\mathcal{L})$ to be the number of sets in $\mathcal{L}$ that cross $S$.

Proposition A.2**.**

*If $S\not\in\mathcal{L}$ and $T\in\mathcal{L}$ are crossing, then all four numbers $\mathrm{cross}(S\cap T,\mathcal{L})$, $\mathrm{cross}(S\cup T,\mathcal{L})$, $\mathrm{cross}(S\setminus T,\mathcal{L})$ and $\mathrm{cross}(T\setminus S,\mathcal{L})$ are smaller than $\mathrm{cross}(S,\mathcal{L})$. *

Proof.

See Claim 9.1.6 in [LRS11]. ∎

Now we can prove Lemma 2.2. Towards a contradiction suppose that $\operatorname{span}(\mathcal{L})\subsetneq\operatorname{span}(\mathcal{S}(F))$. Then there exists $S\in\mathcal{S}(F)$ with $\mathbbm{1}_{\delta(S)}\not\in\operatorname{span}(\mathcal{L})$. Pick such a set with minimum $\mathrm{cross}(S,\mathcal{L})$. Clearly $\mathrm{cross}(S,\mathcal{L})\geq 1$, for otherwise $\mathcal{L}\cup\{S\}$ would be laminar, contradicting maximality of $\mathcal{L}$. Let $T\in\mathcal{L}$ be a set crossing $S$. Assume that $|S\cap T|$ is odd; the other case is analogous. Then by Lemma A.1, $S\cap T,S\cup T\in\mathcal{S}(F)$ and

[TABLE]

By Proposition A.2 and our choice of $S$ we have $\mathbbm{1}_{\delta(S\cap T)},\mathbbm{1}_{\delta(S\cup T)}\in\operatorname{span}(\mathcal{L})$, and of course also $\mathbbm{1}_{\delta(T)}\in\operatorname{span}(\mathcal{L})$. This and (4) implies that $\mathbbm{1}_{\delta(S)}\in\operatorname{span}(\mathcal{L})$, a contradiction. $\square$