Maximum Cut Parameterized by Crossing Number

TL;DR

This paper introduces a fixed-parameter tractable algorithm for the NP-hard Max-Cut problem based on the crossing number of a graph's drawing, extending tractability results beyond 1-planar graphs.

Contribution

It presents the first fixed-parameter algorithm for Max-Cut parameterized by crossing number, with improved running time over previous methods restricted to 1-planar graphs.

Findings

Algorithm runs in $O(2^k \, p(n + k))$ time, where $k$ is the crossing number.

Max-Cut is fixed-parameter tractable with respect to crossing number.

Results extend to minor crossing number, broadening applicability.

Abstract

Given an edge-weighted graph $G$ on $n$ nodes, the NP-hard Max-Cut problem asks for a node bipartition such that the sum of edge weights joining the different partitions is maximized. We propose a fixed-parameter tractable algorithm parameterized by the number $k$ of crossings in a given drawing of $G$. Our algorithm achieves a running time of $O(2^k \cdot p(n + k))$, where $p$ is the polynomial running time for planar Max-Cut. The only previously known similar algorithm [8] is restricted to 1-planar graphs (i.e., at most one crossing per edge) and its dependency on $k$ is of order $3^k$ . A direct consequence of our result is that Max-Cut is fixed-parameter tractable w.r.t. the crossing number, even without a given drawing. Moreover, the results naturally carry over to the minor crossing number.

Full text

1 Introduction

Cut problems in graphs are a well-established class of problems attracting interest since the beginning of modern algorithmic research. Given an edge-weighted undirected graph, the Max-Cut problem asks for a node partition into two sets, such that the sum of the weights of the edges between the partition sets is maximized. The problem is getting increasing attention in the literature due to its applicability to various scenarios: these range from $\ell^{1}$-embeddability [11], to the layout of electronic circuits [4, 9], to solving Ising spin glass models, which are of high interest in physics [1]. Besides the theoretical merits, such models need to be solved in adiabatic quantum computation [32]. Furthermore, quadratic unconstrained binary optimization (QUBO) problems can be solved via Max-Cut. Many combinatorial optimization problems can be stated in the form of QUBO such as multicommodity-flow problems, maximum clique, vertex cover, scheduling, and many others. Also see [10, 11] for a more in-depth overview on applications.

The Max-Cut problem has been shown to be NP-hard for general graphs [24]. Papadimitriou and Yannakakis [36] have shown that the Max-Cut problem is even APX-hard, i.e., there does not exist a polynomial-time approximation scheme unless P=NP. Goemans and Williamson proposed a randomized constant-factor approximation algorithm [17], which has been derandomized by Mahajan and Ramesh [31], achieving a ratio of 0.87856. Several special cases of the problem allow polynomial algorithms: If the weights of all edges are negative we obtain a Min-Cut problem, which can be solved, e.g., via network flow. Other special cases are, e.g., graphs without long odd cycles [20] or weakly bipartite graphs [19]. The case of planar input graphs is of particular interest. Orlova and Dorfman [34] and Hadlock [21] have shown how to solve Max-Cut in polynomial time for unweighted planar graphs. Those algorithms can be extended to weighted planar graphs; the currently fastest algorithms for the weighted case have been suggested by Shih et al. [38] and by Liers and Pardella [30], and achieve a running time of $p(n)=\mathcal{O}(n^{3/2}\log n)$ on planar graphs with $n$ nodes. Barahona has shown that the planarity condition can be relaxed to graphs not contractible to $K_{5}$ [3].

Similarly, it has been shown that Max-Cut can be solved in polynomial time if the graph can be embedded on a surface of constant genus $g$ [14, 15]. However, the edge-weights have to be restricted to integers whose absolute values are bounded by a polynomial in the input. The central idea of this algorithm is to write the generating function of cuts as a linear combination of $4^{g}$ Pfaffians. This algorithm is in fact highly non-trivial to realize: In order to obtain an implementable algorithm, [15] reports on a scheme, which depends on the existence of sufficiently many prime numbers within a given interval, which cannot be guaranteed in general.

A graph is 1-planar if it allows a drawing where each edge is involved in at most one crossing. A 1-plane graph is such a graph, together with an embedding realizing this property. The Max-Cut problem on 1-plane graphs with $k$ edge crossings has recently been shown to be fixed-parameter tractable (FPT) with parameter $k$ [7]. More precisely, it was shown that such instances can be solved in $\mathcal{O}(3^{k}\cdot p(n))$ time, where $p(n)$ is the running time of a polynomial-time Max-Cut algorithm on planar graph with $n$ nodes, e.g., [30, 38]. There are no restrictions on the edge weights.

Our contribution.

In this paper, we improve on the latter result in several ways: Firstly, we drop the requirement of 1-planarity, i.e., we consider graphs that can be drawn with at most $k$ crossings (even if multiple such crossings lie on the same edge). We therefore handle the case of the well-established notion of the graph’s crossing number. Secondly, we reduce the runtime dependency on $k$ from $3^{k}$ to $2^{k}$. Finally, unlike the previous result, our approach can be extended to an FPT algorithm which does not even require a crossing-realizing drawing as an input; however, this increases the running time and requires a deep algorithm from the literature as a black box [18, 26]. Interestingly, we achieve these results by a simpler approach (compared to [7]). Comparing our algorithm with [15], we have no restrictions on the edge weights. Even in the restricted scenario, our algorithm is faster for graphs whose crossing number is at most twice its genus. Furthermore, we require only easily-implementable data structures and subalgorithms (if we are given a crossing-realizing drawing), compared to advanced methods from algebra.

The general idea of our algorithm is to recursively get rid of each crossing, each time resulting in two new subinstances. We end up with a set of up to $2^{k}$ planar graphs, each of which can be solved using a known polynomial-time Max-Cut algorithm for planar graphs. The maximum over all these subinstances then yields a maximum cut in the original instance.

Finally, we consider parameterizing the problem by the minor crossing number (see below for details). This measure is always at most the graph’s crossing number. While the exponential dependency on the respective parameter is identical, the running time only slightly increases in its polynomial part.

Independent work.

Kobayashi et al.[28] independently and simultaneously obtained another fixed-parameter tractable algorithm for Max-Cut parameterized by crossing number achieving the same running time. However, while we can always stay in the realm of maximum cuts when solving subinstances, they have to consider maximum weighted $b$-factor problems. Their preprint was uploaded to arXiv shortly after ours [CDJKMN19, 27].

Organization of the paper.

Section 2 recapitulates the basic definitions for cuts and crossings. In Section 3, we present our new algorithm and prove its correctness and running time. Section 4 extends the results to the minor crossing number case. We end with a conclusion and open problems in Section 5.

2 Preliminaries

Throughout this paper we consider undirected edge-weighted graphs. The input for our Max-Cut problem is a graph $G=(V,E,c)$, where $c_{e}\in\mathbb{R}$ denotes the (positive or negative) weight of edge $e$, for each edge $e\in E$. A partition of the nodes $V$ into two sets $S\subseteq V$ and $\overline{S}=V\setminus S$ defines the cut $\delta(S)=\{uv\in E\mid u\in S\Leftrightarrow v\not\in S\}$. The value $c(\delta(S))=\sum_{e\in\delta(S)}c_{e}$ of a cut is the sum of all edge weights of the edges in the cut. Given $G$, the Max-Cut problem asks for a cut with highest value. Since a graph can have multiple cuts of equal value, only the value of a maximum cut is unique, not the cut itself.

A non-degenerated drawing of a graph in the plane is a map of its nodes to distinct points in $\mathbb{R}^{2}$, and a map of its edges to curves connecting the respective endpoints, not including the points of any other node. Any point mapped to the plane either corresponds to a graph node, or is contained in at most two edge curves. A shared non-endpoint between two curves is called a crossing.

A graph is planar if it admits a drawing without any crossings. It is well known that planarity can be tested in linear time [23]. For non-planar graphs it is natural to ask for a drawing with as few crossings as possible. The smallest such number is the crossing number $\mathrm{cr}(G)$ of $G$. Not only is it NP-hard to compute $\mathrm{cr}(G)$ [16], but even the so called realizability problem turns out to be NP-hard [29]: Given a graph $G$ and a set $X$ of edge pairs, is there a drawing $\mathcal{D}$ of $G$ such that $X$ contains an edge pair if and only if the pair’s two edge curves cross in $\mathcal{D}$? The key problem in testing realizability is that it is hard to figure out whether there exist orderings of the crossings along the respective edges that allow the above properties.

Therefore, sometimes more restricted crossing variants are considered. For example, 1-planar graphs admit drawings where every edge is involved in at most one crossing. Not all graphs can be drawn in such a way, since 1-planar graphs can have at most $4|V|-8$ edges; also, the 1-planar number of crossings is in general larger than $\mathrm{cr}(G)$ [35, CKMV19].

For a general drawing (not necessarily 1-plane), we typically encode its crossings as a crossing configuration $\mathcal{X}$. Therein, we not only store the pairs of edges that cross, but for each edge also the order of the crossings as they occur along its curve. The feasibility of a crossing configuration can be tested in time linear in $|V|+|\mathcal{X}|$ by replacing crossings with dummy nodes of degree 4, testing planarity, and checking the cyclic order around dummy nodes.111In general, a specified edge pair may not really cross but merely “touch”; this is trivial to detect after testing planarity by checking the cyclic order of the edges around the dummy node. Given such a “flawed” configuration, we trivially obtain one with less crossings by removing such crossing pairs from $\mathcal{X}$. Although we will not require this fact in the following, this also allows us to efficiently deduce a drawing that respects $\mathcal{X}$. It is well understood that we can restrict ourselves to good drawings when considering the (traditional) crossing number of graphs: adjacent edges never cross and no edge pair crosses more than once.

3 Algorithm

Our main idea for computing the maximum cut in an embedded weighted graph is to eliminate its crossings one by one. In the end, we use a Max-Cut algorithm for planar graphs. We first introduce a slight variant of Max-Cut:

Definition 1** (Partially-Fixed Maximum Cut, PF-Max-Cut)**

Given an edge weighted graph $G=(V,E,c)$ and a set of fixed edges $F\subseteq E$, find a cut of maximum value that contains all elements of $F$.

A cut is feasible if it contains $F$. A PF-Max-Cut instance is infeasible if it does not allow a feasible cut. It is easy to see that an instance is infeasible if and only if $F$ contains a cycle of odd length. We denote a maximum objective value by $\mathtt{MaxCut_{pf}}(G,F)$, and let $\mathtt{MaxCut_{pf}}(G,F)=-\infty$ for infeasible instances.

Observe that (as for Max-Cut) we do not need to consider a given crossing configuration $\mathcal{X}$ as part of the problem description (see Corollary 7). However, since having $\mathcal{X}$ allows for simplifications and a better running time, we will for now assume that we are given the graph together with a crossing configuration $\mathcal{X}$.222As noted above, we may assume that no such $\mathcal{X}$ ever specifies “touching points”; we can reduce such configurations whenever our algorithm retrieves a new crossing configuration. We will explain later how to remove this assumption in Corollary 7.

Given any edge $vw$ with weight $c_{vw}$ in a PF-Max-Cut instance, we define the operation to bisubdivide $vw$ at $v$ as follows: Subdivide $vw$ twice, i.e., replace $vw$ by a path of length $3$ with two new degree-2 nodes. We denote the new node incident to $v$ or $w$ by  \overline{\hbox{}}$$v or  \overline{\hbox{}}$$w, respectively. We consider the notation   \overline{\hbox{}}$$\ \cdot\ an operand.333Observe that per recursion step, we will bisubdivide at most one edge per incident node (recall that adjacent edges never cross in good drawings). Thus, the above simple notation is unambiguous. In the graphs of the subproblems, see below, we may assume the nodes to be named afresh, and thus we may again perform bisubdivisions without creating notational ambiguity.

The edges v\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}} and  \overline{\hbox{}}$$v \overline{\hbox{}}$$w have weight [math], \hbox to0.0pt{\hskip 1.21411pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorw}}w retains the weight $c_{vw}$. Furthermore, we add v\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}},\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}}\hbox to0.0pt{\hskip 1.21411pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorw}} to $F$, and if $vw\in F$, we replace it in $F$ by \hbox to0.0pt{\hskip 1.21411pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorw}}w. Clearly, both v\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}},\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}}\hbox to0.0pt{\hskip 1.21411pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorw}} will be in any feasible cut; node  \overline{\hbox{}}$$w will always lie in the same partition set as $v$, and  \overline{\hbox{}}$$v in the other (cf. Figure 1(b)). Most importantly this gives:

Lemma 2

The feasible cuts in an original PF-Max-Cut instance $\langle G,F\rangle$ are in 1-to-1 correspondence to feasible cuts of equal value in a bisubdivided instance $\langle G^{\prime},F^{\prime}\rangle$.

Proof 3.1**.**

Let $vw$ be the edge in $G$ that is bisubdivided at $v$ to obtain $\langle G^{\prime},F^{\prime}\rangle$. By construction, we know that both edges v\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}},  \overline{\hbox{}}$$v \overline{\hbox{}}$$w have cost [math] and are in $F^{\prime}$, and thus in any $F^{\prime}$-feasible cut. Consequently, in any $F^{\prime}$-feasible cut, $v$ and  \overline{\hbox{}}$$w will lie in a common partition set. Let $S^{\prime}\subset V(G^{\prime})$ be a node subset that induces some feasible (with respect to $F^{\prime}$) cut in $G^{\prime}$. Then, the node set S=S^{\prime}\setminus\{\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}},\hbox to0.0pt{\hskip 1.21411pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorw}}\} induces a feasible (with respect to $F$) cut in $G$. Cut $\delta(S)$ contains edge $vw$ if and only if $\delta(S^{\prime})$ contains \hbox to0.0pt{\hskip 1.21411pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorw}}w. Since both these edges have identical cost, the total costs of both cuts are equal.

Inversely, let $S\subset V(G)$ be a node subset that induces some feasible (with respect to $F$) cut in $G$. Then, consider the cut in $G^{\prime}$ induced by $S^{\prime}=S\cup\{s\}$, where s=\hbox to0.0pt{\hskip 1.21411pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorw}} if $v\in S$, and s=\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}} otherwise. Both fixed edges v\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}},  \overline{\hbox{}}$$v \overline{\hbox{}}$$w are in $\delta(S^{\prime})$ and the cut is thus feasible. Again, $\delta(S^{\prime})$ contains edge \hbox to0.0pt{\hskip 1.21411pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorw}}w if and only if $\delta(S)$ contains $vw$, and both cut values are thus equal.

When we identify two nodes $a,b$ in a graph with one another, they become a common entity that is incident to all of their former neighbors. We will only identify nodes that are neither adjacent nor share neighbors. When identifying nodes in $G$ of some PF-Max-Cut instance $\langle G,F\rangle$, the set $F$ is retained, subject to replacing the edges formerly incident to $a$ or $b$ with their new counterparts.

We are now ready to describe our algorithm. We are given a Max-Cut instance $G=(V,E,c)$, together with some crossing configuration $\mathcal{X}$ with $k$ crossings. Let $F=\emptyset$ be the set of fixed edges and consider $\langle G,F\rangle$ as a PF-Max-Cut instance. From $\langle G,F,\mathcal{X}\rangle$, we pick a crossed edge $vw$, and derive two new triplets $T_{i}=\langle G_{i},F_{i},\mathcal{X}_{i}\rangle$, for $i\in\{v,w\}$. Both derived crossing configurations $\mathcal{X}_{i}$ attain at most $k-1$ crossings and we can call our algorithm recursively on $T_{v}$ and $T_{w}$. As a base case, the derived graphs become planar and (after a preprocessing to deal with the fixed edges) we apply an efficient Max-Cut algorithm for planar graphs. The solutions of $\langle G_{i},F_{i}\rangle$, for $i\in\{v,w\}$, yield a solution of $\langle G,F\rangle$. Observe, however, that $\langle G_{i},F_{i}\rangle$ may become infeasible.

Let us describe this recursion step formally (cf. also Figures 1 and 2). We define the crossing split operation that, given a triplet $\langle G,F,\mathcal{X}\rangle$, yields the two triplets $T_{v}$ and $T_{w}$: Let $\langle G=(V,E,c),F\rangle$ be a PF-Max-Cut instance and $\mathcal{X}$ a crossing configuration of $G$. Consider a crossing $\chi\in\mathcal{X}$ with crossing edges $vw$ and $xy$. For $j\in\{v,w,x,y\}$, let $Y_{j}$ be the ordered sets of crossings in $\mathcal{X}$ between $j$ and $\chi$ (cf. the dotted edges in Figure 1: e.g., the crossings between the two dotted edges and \hbox to0.0pt{\hskip 1.21411pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorw}}w in Figure 1(b) are in $Y_{w}$ as they are between $\chi$ and $w$ in Figure 1(a)). Let the intermediate instance $\langle G^{\prime},F^{\prime}\rangle$ be obtained from $\langle G,F\rangle$ by bisubdividing $vw$ at $v$ and bisubdividing $xy$ at $x$. For $i\in\{v,w\}$, let $\langle G_{i},F_{i}\rangle$ be the PF-Max-Cut instance obtained from $\langle G^{\prime},F^{\prime}\rangle$ by identifying  \overline{\hbox{}}$$x with i$$\overline{\hbox{}} (see Figures 2(c) and 2(d)). Intuitively, the two graphs obtained by the identifications represent the two possibilities whether $x$ is on the same side of the cut as $v$ or not. We obtain a corresponding crossing configuration $\mathcal{X}_{i}$ from $\mathcal{X}$ by removing $\chi$ and placing the crossings $Y_{j}$ (retaining their order) on the edge j\hskip 0.15475pt\hbox to0.0pt{\leavevmode\hbox{\set@colorj}\hss}{\hskip-0.15475pt\leavevmode\hbox{\set@color\overline{\hbox{}}}}, for all $j\in\{v,w,x,y\}$. The triplets $T_{v}=\langle G_{v},F_{v},\mathcal{X}_{v}\rangle$ and $T_{w}=\langle G_{w},F_{w},\mathcal{X}_{w}\rangle$ are the results of the crossing split operation with respect to $\langle\chi,vw,xy\rangle$.

Lemma 3**.**

Let $\langle G=(V,E,c),F\rangle$ be a PF-Max-Cut instance and $\mathcal{X}$ a crossing configuration of $G$ with $k$ crossings. Let $\chi\in\mathcal{X}$ be any crossing with some crossing edges $vw$ and $xy$, and consider the crossing split operation with respect to $\langle\chi,vw,xy\rangle$. For $i\in\{v,w\}$, let $\langle G_{i},F_{i},\mathcal{X}_{i}\rangle$ be the resulting triplets. Then we have:

for $i\in\{v,w\}$, $\mathcal{X}_{i}$ is a feasible crossing configuration for $G_{i}$ with at most $k-1$ crossings; and 2. 2.

$\mathtt{MaxCut_{pf}}(G,F)=\max_{i\in\{v,w\}}\{\ \mathtt{MaxCut_{pf}}(G_{i},F_{i})\ \}$.

Proof 3.2**.**

Consider any drawing $\mathcal{D}$ of $G$ realizing $\mathcal{X}$. By routing the new paths (v\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}},\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}}\hbox to0.0pt{\hskip 1.21411pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorw}},\hbox to0.0pt{\hskip 1.21411pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorw}}w resp. x\hbox to0.0pt{\hskip 0.35762pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorx}},\hbox to0.0pt{\hskip 0.35762pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorx}}\hbox to0.0pt{\hskip 0.13078pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colory}},\hbox to0.0pt{\hskip 0.13078pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colory}}y) along the curves of their original edges ($vw$ resp. $xy$) we obtain a drawing $\mathcal{D}^{\prime}$ of $G^{\prime}$ from $\mathcal{D}$. Thereby, for $j\in\{v,w,x,y\}$, we place the new nodes j$$\overline{\hbox{}} in a close neighborhood of $\chi$ on the curve segment between $j$ and $\chi$, so that  \overline{\hbox{}}$$x \overline{\hbox{}}$$y is only crossed by  \overline{\hbox{}}$$v \overline{\hbox{}}$$w and vice versa. Note that the number of crossings in $\mathcal{D}^{\prime}$ is equal to that of $\mathcal{D}$, since all crossings in $Y_{j}$ in $\mathcal{D}$ are transferred to the edge j\hskip 0.15475pt\hbox to0.0pt{\leavevmode\hbox{\set@colorj}\hss}{\hskip-0.15475pt\leavevmode\hbox{\set@color\overline{\hbox{}}}} in $\mathcal{D}^{\prime}$, for all $j\in\{v,w,x,y\}$, and the original crossing $\chi$ between $xy$ and $vw$ in $\mathcal{D}$ has a counterpart $\chi^{\prime}$ in $\mathcal{D}^{\prime}$ between the edges  \overline{\hbox{}}$$v \overline{\hbox{}}$$w and  \overline{\hbox{}}$$x \overline{\hbox{}}$$y. Since the edges  \overline{\hbox{}}$$v \overline{\hbox{}}$$w and  \overline{\hbox{}}$$x \overline{\hbox{}}$$y are crossing free except for $\chi^{\prime}$, we can follow (in a close neighborhood) the curves of  \overline{\hbox{}}$$v \overline{\hbox{}}$$w from any of its end points up to $\chi^{\prime}$, and onwards from there along the curve of  \overline{\hbox{}}$$v \overline{\hbox{}}$$w to any of its end points. Since these routes are crossings-free, we call them free routes. When we now identify  \overline{\hbox{}}$$x with  \overline{\hbox{}}$$v, we can locally redraw our drawing such that $\chi^{\prime}$ vanishes and no other crossings arise, see Figure 2(c). Observe that  \overline{\hbox{}}$$x has precisely two neighbors:  \overline{\hbox{}}$$y and $x$. The identification is thus such that we may remove  \overline{\hbox{}}$$x and insert edges  \overline{\hbox{}}$$y \overline{\hbox{}}$$v and x\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}} instead. The former can trivially be drawn without any crossings along the free route between  \overline{\hbox{}}$$y and  \overline{\hbox{}}$$v. The curve for the latter edge is the concatenation of the former curve of x\hbox to0.0pt{\hskip 0.35762pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorx}} and the free route between  \overline{\hbox{}}$$x and  \overline{\hbox{}}$$v. The number of crossings along the edge x\hbox to0.0pt{\hskip 0.35762pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorx}} (with now \hbox to0.0pt{\hskip 0.35762pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorx}}=\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}}) does thus not change. We can perform the analogous redrawing when identifying  \overline{\hbox{}}$$x with  \overline{\hbox{}}$$w, see Figure 2(d). This establishes claim (1).

Two nodes $v$ and $x$ can either be on the same side of a cut, or they are on opposite sides. Therefore, we create two new subproblems in which $v$ and $x$ are in the same partition set or not, respectively. In $G_{v}$ (where we identify  \overline{\hbox{}}$$x with  \overline{\hbox{}}$$v), we have a path of two edges between $v$ and $x$ (namely v\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}} and \hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}}x), both of which are in $F_{v}$. Thus, $v$ and $x$ have to be in the same partition set, see Figure 2(c). Conversely, in $G_{w}$ (where we identify  \overline{\hbox{}}$$x with  \overline{\hbox{}}$$w), we have a path of three edges between $v$ and $x$ (namely v\hbox to0.0pt{\hskip 0.103pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorv}},  \overline{\hbox{}}$$v \overline{\hbox{}}$$w, and \hbox to0.0pt{\hskip 1.21411pt\leavevmode\hbox{\set@color\overline{\hbox{}}}\hss}{\leavevmode\hbox{\set@colorw}}x), all of which are in $F_{w}$. Thus, $v$ and $x$ have to be in different partition sets, see Figure 2(d). We can see that the respective constructions do not induce any further restrictions on the set of cuts. In particular, both derived instances still allow any partition choice between $w$ and $x$, between $w$ and $y$, and between $x$ and $y$. Overall, every feasible cut in $\langle G^{\prime},F^{\prime}\rangle$ can be realized either in $\langle G_{v},F_{v}\rangle$ or in $\langle G_{w},F_{w}\rangle$.

If we know the maximum cut in instance $\langle G_{v},F_{v}\rangle$ and the maximum cut in instance $\langle G_{w},F_{w}\rangle$, we can pick the larger of these two cuts and transfer it back to $\langle G^{\prime},F^{\prime}\rangle$. By applying Lemma 2 twice (once for the bisubdivision of $vw$ at $v$ and once for the bisubdivision of $xy$ at $x$), the maximum cut in $\langle G^{\prime},F^{\prime}\rangle$ induces a maximum cut in $\langle G,F\rangle$ of the same value. Claim (2) follows.

If we are in a base case – the considered graph is planar – we can use an efficient Max-Cut algorithm for planar graphs:

Lemma 4**.**

Consider a PF-Max-Cut instance $\langle G=(V,E,c),F\rangle$ with a planar graph $G$. Let $p(|V|)$ be a polynomial upper bound on the running time of a Max-Cut algorithm on the planar graph $G$. We can compute an optimal solution to $\langle G,F\rangle$ – or decide that the instance is infeasible – in $\mathcal{O}(p(|V|))$.

Proof 3.3**.**

We transform the PF-Max-Cut instance into a traditional Max-Cut instance by attaching a large weight to the edges in $F$. Namely, we add $M$ to the weight of each edge $f\in F$, where $M=2\cdot\sum_{e\in E}|c_{e}|$. The omission of a single edge of $F$ from the solution cut (even if picking all other edges of positive weight) will already result in a worse objective value than picking all of $F$ and all edges of negative weight. The instance is infeasible if and only if the computed cut does not contain all of $F$; this can also be deduced purely by checking whether the objective value is at least $M\cdot|F|+\sum_{e\in E:c_{e}<0}c_{e}$.

We proved our lemma above for a general case (by adding $M$ to the weight of each edge in $F$), but in fact we only require a slightly weaker version, since in our algorithm $c_{f}=0$ for all $f\in F$. Thus it suffices to set $c_{f}=M$ instead of adding $M$ to $c_{f}$. Using any of the currently fastest Max-Cut algorithms for planar graphs [30, 38] leads to $\mathcal{O}(|V|^{3/2}\log|V|)$ time in the above lemma. We could speed-up infeasibility detection by checking whether $F$ contains a cycle of odd length prior to the transformation; while this only requires $\mathcal{O}(|V|)$ time via depth-first search, the overall asymptotic runtime for the lemma’s claim does of course not improve.

Theorem 5**.**

Let $G=(V,E,c)$ be an edge-weighted graph and $\mathcal{X}$ a crossing configuration of $G$ with $k$ crossings. Let $p(n)$ be a polynomial upper bound on the running time of a Max-Cut algorithm on planar graphs with $n$ nodes. We can compute a maximum cut in $G$ in $\mathcal{O}(2^{k}\cdot p(|V|+k))$ time.

Proof 3.4**.**

As described above, we solve the instance by considering the PF-Max-Cut instance $\langle G,F=\emptyset\rangle$ together with $\mathcal{X}$. Thus the triplet $\langle G,F,\mathcal{X}\rangle$ forms the initial input of our recursive algorithm $\mathcal{R}$.

Algorithm $\mathcal{R}$ proceeds as follows on a given triplet: If the triplet’s graph is planar, we solve $\langle G,F\rangle$ via Lemma 4. Otherwise, we use Lemma 3 to obtain two new input triples $T_{v},T_{w}$, for each of which we call $\mathcal{R}$ recursively. Their returned solutions (i.p., their solution values) induce the optimum solution for the current input triplet. However, while the number of crossings decreases by (at least) one per recursion step, the graph’s size increases by three nodes.

The runtime complexity follows from the fact that we consider two choices per crossing in the given $\mathcal{X}$, and thus construct $2^{k}$ graphs. For each such graph, which has $|V|+3k$ nodes, we run the planar Max-Cut algorithm.

Above, we trivially have $k\in\mathcal{O}(|V|^{4})$ and thus $|V|+k\in\mathcal{O}(\mathrm{poly}(|V|))$.

Corollary 6**.**

The above algorithm is an FPT algorithm with parameter $k$, provided that a crossing configuration $\mathcal{X}$ with $k$ crossings is part of the input. Moreover, the attained running time is polynomial for any $k\in\mathcal{O}(\log|V|)$. Using the currently fastest Max-Cut algorithm for planar graphs [30, 38], our algorithm yields a running time of $\mathcal{O}(2^{k}\cdot(|V|+k)^{3/2}\log(|V|+k))$.

Quite sophisticated results by Grohe [18] and Karabayashi and Reed [26] show that the problem to compute the crossing number of a graph is in FPT (even in linear time) with respect to its natural parameterization: Given a graph $G$ and a number $k\in\mathbb{N}$, we can answer the question “$\mathrm{cr}(G)\leq k$ ?” in time $\mathcal{O}(f(k)\cdot n)$. In case of a yes-instance, we obtain a corresponding crossing configuration $\mathcal{X}$ as a witness. The computable function $f(k)$ is purely dependent on $k$. However, the dependency $f(k)$ is double exponential, and the algorithm far from being practical. Still, these results formally allow us to get rid of the requirement that $\mathcal{X}$ is part of the input:

Corollary 7**.**

Given an edge-weighted undirected graph $G$. Computing a maximum cut in $G$ is FPT with parameter $\mathrm{cr}(G)$.

4 Minor Crossing Number

We say $G$ is a minor of $H$, denoted by $G\preceq H$, if $G$ can be obtained from $H$ by deletion and contraction of edges. The minor crossing number of $G$ is given by $\mathrm{mcr}(G)=\min\{\mathrm{cr}(H)\mid G\preceq H\}$. A realization of $\mathrm{mcr}(G)$ is a pair $(H,\mathcal{X})$ with $G\preceq H$ and $\mathcal{X}$ being a crossing configuration of $H$ with $\mathrm{mcr}(G)$ crossings. It is easy to see that for graphs $G^{\prime}$ of maximum degree $3$ we have $\mathrm{cr}(G^{\prime})=\mathrm{mcr}(G^{\prime})$. Similarly, any graph $G$ allows a realizing graph $H$ ($\mathrm{cr}(H)=\mathrm{mcr}(G)$) of maximum degree $3$ where vertices of $G$ are replaced by disjoint cubic trees.

By definition we always have $\mathrm{mcr}(G)\leq\mathrm{cr}(G)$; as such $\mathrm{mcr}(G)$ can be a stronger FPT-parameter. Also, in contrast to crossing number, the minor crossing number is monotone with respect to graph minors, i.e., the family $\{G\mid\mathrm{mcr}(G)\leq k\}$ is minor closed. Thus, by [37], we can (theoretically) check whether $\mathrm{mcr}(G)\leq k$ in $\mathcal{O}(|V(G)|^{3})$ time for fixed $k\in\mathbb{N}$.

Given a connected graph $G$ with $\mathrm{mcr}(G)=k$, we can obtain a graph $H$ from $G$ realizing $\mathrm{mcr}(G)$ in polynomial time as follows: Choose a node $v$ of degree at least $4$. Try different pairs of neighbors $w_{1},w_{2}\in N(v)$ until finding the first with $\mathrm{mcr}(\tilde{G})\leq k$, where $\tilde{G}$ is obtained from $G$ by splitting $v$ into two nodes $v_{1}$ and $v_{2}$ with $N(v_{1})=\{v_{2},w_{1},w_{2}\}$, $N(v_{2})=(N(v)\cup\{v_{1}\})\setminus\{w_{1},w_{2}\}$444Observe that in general this splitting operation may increase $\mathrm{mcr}$; we search for a split (which has to exists) for which it does not increase. Since the split is an inverse minor operation, $\mathrm{mcr}$ can never decrease.. We call the edge $v_{1}v_{2}$ a split edge. Iterating this for each high degree node, yields a graph $H$ of maximum degree $3$ realizing $\mathrm{mcr}(G)=\mathrm{cr}(H)$. Note that $H$ has at most $\mathcal{O}(|E(G)|)$ nodes.

Let $N=-3\cdot\sum_{e\in E(G)}|c_{e}|$. Attaching the weight $N$ to each split edge, we can make sure that these edges are not in any maximum cut of $H$. Clearly, the cuts in $H$ not containing any split edges are in one-to-one correspondence with cuts in $G$. Using 5, we obtain an algorithm computing a maximum cut on $G$ parameterized by the $\mathrm{mcr}(G)$. Similarly to Corollary 7 we do not require an explicit realization as part of the input (using the above construction method for $H$).

Corollary 8**.**

(i) Let $G=(V,E,c)$ be an edge-weighted undirected graph with $\mathrm{mcr}(G)=k$, $(H,\mathcal{X})$ a realization of $\mathrm{mcr}(G)$, and $p(n)$ be a polynomial upper bound on the running time of a Max-Cut algorithm on planar graphs with $n$ nodes. We can compute a maximum cut in $G$ in $\mathcal{O}(2^{k}\cdot p(|E(G)|+k))$ time.

(ii) Given an edge-weighted undirected graph $G$, computing a maximum cut in $G$ is FPT with parameter $\mathrm{mcr}(G)$.

5 Conclusion and Open Problems

Given a graph together with a feasible crossing configuration with $k$ crossings, we previously only knew that Max-Cut is polynomial time solvable if $k$ is constant and the graph is 1-planar, i.e., each edge is involved in at most one crossing. The runtime dependency on $k$ has been to the order of $3^{k}$ [7].

Herein, we improved on this in several ways: Firstly, we decreased the dependency on $k$ to the order of $2^{k}$. Secondly, we extended the applicability to any graph with (at most) $k$ crossings: our parameter becomes the true crossing number of the graph, without any 1-planarity restriction. This shows that Max-Cut is in FPT with respect to the graph’s crossing number. Moreover, we achieve these improvements by introducing simpler ideas than those proposed for the former result, yielding an overall surprisingly simple algorithm. Compared to the result of Kobayashi et al.[28], we are able to stay within the realm of Max-Cut. Finally, our result naturally carries over to the minor crossing number.

The skewness of a graph is the minimum number of edges to remove such that the graph becomes planar. The genus of a graph is the minimum oriented genus of a surface onto which the graph can be embedded without crossings. In FPT research, there are many algorithmic approaches that consider graphs with bounded genus $g$, see, e.g. [5, 6, 12, 13]. However, the obtained FPT algorithms are typically parameterized by the objective value $z$, or by the combined parameter $(z,g)$. There are much fewer results that obtain FPT algorithms parameterized purely with $g$. Notable examples are the graph genus problem itself [33] (where $z$ and $g$ coincide by definition), and the graph isomorphism problem [25] (which generalizes the linear-time algorithm for the problem on planar graphs). There are even fewer parameterized results with respect to skewness; the probably best known example is that maximum flow can be solved in the running time of planar graphs, if the graph’s skewness is fixed [22]. Our above algorithm seems to be the first time that the crossing number has been proposed as an efficient non-trivial FPT parameter for any widely known problem.

Besides the weight-restricted case of [15] (briefly described in the introduction), it is unclear whether Max-Cut could be FPT with respect to either skewness or genus. We deem this an interesting question for further research.