Decomposition of Trees and Paths via Correlation

TL;DR

This paper investigates the complexity of clustering trees based on pairwise costs, revealing NP-hardness in general and star trees, while providing a complete polyhedral characterization for paths, linking to combinatorial optimization.

Contribution

It characterizes the polyhedral structure of the clustering problem on trees and paths, including a complete TDI description for paths, advancing understanding of related combinatorial formulations.

Findings

NP-hardness for general and star trees

Polynomial solvability for paths

Complete TDI description of the lifted multicut polytope for paths

Abstract

We study the problem of decomposing (clustering) a tree with respect to costs attributed to pairs of nodes, so as to minimize the sum of costs for those pairs of nodes that are in the same component (cluster). For the general case and for the special case of the tree being a star, we show that the problem is NP-hard. For the special case of the tree being a path, this problem is known to be polynomial time solvable. We characterize several classes of facets of the combinatorial polytope associated with a formulation of this clustering problem in terms of lifted multicuts. In particular, our results yield a complete totally dual integral (TDI) description of the lifted multicut polytope for paths, which establishes a connection to the combinatorial properties of alternative formulations such as set partitioning.

Equations

$$\underset{y\in \{0,1{\}}^E}{\min }\sum _{\text{scalebox}0.8\left(\genfrac{}{}{0ex}{}{V}{2}\right)}{c}_{uv}\prod _{e\in P_{uv}}{y}_e$$

$$f(y)=\sum _{I\subseteq [n]}{c}_I\prod _{i\in I}{y}_i,\text{with}c:2^{[n]}\to \mathbb{R}.$$

$$\min$$

$$\sum _{S:u\in V_S}{\lambda }_S=1,\forall u\in V$$

$${\lambda }_S\ge 0,{\lambda }_S\in \mathbb{Z}.$$

$${\lambda }_S=1⟺\forall e\in S:y_e=1$$

$$\sum _{\text{scalebox}0.8\left(\genfrac{}{}{0ex}{}{V}{2}\right)}{c}_{uv}\prod _{e\in P_{uv}}{y}_e=$$

$$=$$

$$\underset{x\in X_T}{\min }\sum _{\text{scalebox}0.8\left(\genfrac{}{}{0ex}{}{V}{2}\right)}{c}_{uv}{x}_{uv},$$

$$X_T$$

$$x_{uv}\le \sum _{e\in P_{uv}}{x}_e$$

$$x_e\le x_{uv}$$

$$x_{uv}=1-\prod _{e\in P_{uv}}{y}_e$$

$$x_{uv}=0$$

$$⟺\forall e\in P_{uv}:x_e=0.$$

$$c_{uv}\prod _{e\in P_{uv}}{y}_e$$

$$=-c_{uv}{x}_{uv}+c_{uv}.$$

$$\underset{x\in X_T}{\min }\sum _{\text{scalebox}0.8\left(\genfrac{}{}{0ex}{}{V}{2}\right)}-c_{uv}{x}_{uv}+c_{uv},$$

$${\Xi }_T=\mathrm{conv}{X}_T,$$

$${\Theta }_T^1$$

$$x_{uv}\le x_{u,\vec{u}(v)}+x_{\vec{u}(v),v}$$

$$x_{\vec{u}(v),v}\le x_{uv}$$

$$x_{uv}+x_{\vec{u}(v),\vec{v}(u)}$$

$$\forall u,v\in V,\mathrm{dist}(u,v)\ge 3.$$

$$x_{01}+x_{23}\le x_{12}+x_{13}$$

$$x_{02}+x_{56}\le x_{05}+x_{06}$$

$$\displaystyle\Theta^{\mathrm{Path}}_{P}=\Big{\{}$$

$$x_{0n}\le 1,$$

$$x_{in}\le x_{i-1,n}$$

$$x_{0i}\le x_{0,i+1}$$

$$x_{i-1,i+1}\le x_{i-1,i}+x_{i,i+1}$$

$$x_{j,k}+x_{j+1,k-1}\le x_{j+1,k}+x_{j,k-1}$$

$$\displaystyle\qquad\qquad\forall j,k\in\{0,\dotsc,n\},\>j<k-2\Big{\}}.$$

$$x_{uv}+x_{\vec{u}(v),\vec{v}(u)}=x_{u,\vec{v}(u)}+x_{\vec{u}(v),v}.$$

$$x_{uw}=x_{\vec{u}(v),w}$$

$$x_{uv}+x_{u+v,v+1}$$

$$\ge x_{u+1,v}+x_{u+1,v+1}$$

$$⟹x_{uv}$$

Full text

1  Introduction

We study the problem of decomposing (clustering) a tree with respect to costs attributed to pairs of nodes, so as to minimize the sum of costs for those pairs of nodes that are in the same component (cluster). This tree partition problem is stated rigorously in Def. 1. One instance and its solution are depicted in Fig. 1.

On the one hand, the tree partition problem lacks a complexity of clustering problems for general graphs: Its feasible set, the set of all decompositions of a tree, is trivial to characterize. For example, the decompositions of any tree $T=(V,E)$ relate one-to-one to the binary edge labelings $y\in\{0,1\}^{E}$ that indicate whether neighboring nodes $\{u,v\}\in E$ are in the same component ($y_{uv}=1$) or distinct components ($y_{uv}=0$).

On the other hand, the tree partition problem exhibits a complexity not commonly found in clustering problems for general graphs: Its objective function has a multilinear form in the coordinates of $y$ (in the standard unit basis) whose polynomial degree is the length of the longest path in $T$. This form arises from the fact that any two distinct nodes $\{u,v\}\in\tbinom{V}{2}$ are in the same component iff $y_{e}=1$ for all edges $e$ on the unique path $P_{uv}$ in $T$ from $u$ to $v$, that is, iff $\prod_{e\in P_{uv}}y_{e}=1$. Hence, the tree partition problem can be stated rigorously as follows:

Definition 1**.**

For any (finite, simple, undirected) tree $T=(V,E)$ and any $c\colon\tbinom{V}{2}\to\mathbb{R}$, the optimization problem (1) is called the instance of the tree partition problem w.r.t. $T$ and $c$. If $T$ is a path, it is also called the instance of the path partition problem w.r.t. $T$ and $c$.

[TABLE]

1.1  Contribution

In Section 3, we survey several representations of the tree partition problem and discuss its computational complexity. For the general case and for the special case of the tree being a star, we show that the problem is np-hard. The path partition problem, on the other hand, is known to be polynomial time solvable (Kernighan, 1971).

In Section 4, we study the lifted multicut polytope (Horňáková et al., 2017) associated with the lifted multicut representation of the tree partition problem. We characterize several classes of facets for the general case, which in particular yield a complete totally dual integral description (TDI) of the lifted multicut polytope for paths. Our results relate the geometry of lifted multicuts to the combinatorial properties of the tree and path partition problem.

1.2  Notation

Throughout the paper, we consider $T=(V,E)$ to be a (finite, simple, undirected) tree with $n=\lvert E\rvert$ edges. Let m=\lvert\raisebox{0.43057pt}{\scalebox{0.8}{\tbinom{V}{2}}}\rvert be the number of distinct pairs of nodes in $V$. For any $\{u,v\}\in\tbinom{V}{2}$, we denote by $P_{uv}$ the unique path in $T$ connecting $u$ and $v$. Moreover, we denote by $\operatorname{dist}(u,v)$ the distance of $u$ and $v$ in $T$, i.e. the length of $P_{uv}$. To ease notation, we take the liberty of identifying any graph $G=(V_{G},E_{G})$ with its edge set $E_{G}$. Hence, we may write $e\in G$ instead of $e\in E_{G}$. For any $k\in\mathbb{N}$, we denote by $[k]:=\{1,\dotsc,k\}$ the set of integers from $1$ through $k$.

2  Related Work

2.1  Pseudo-Boolean Optimization

The optimization of pseudo-Boolean functions plays an important role in machine learning, for instance, in MAP-inference for computer vision models. The general problem can be reduced to the quadratic case (Boros and Hammer, 2002; Boros and Gruber, 2012), which is responsible for the np-hardness of the problem. The combinatorial polytope associated with the linearization of quadratic pseudo-Boolean functions was extensively studied by Padberg (1989). Computational approaches to pseudo-Boolean optimization based on the roof duality bound (Hammer et al., 1984; Kahl and Strandmark, 2012) have been quite successful in practice (Rother et al., 2007).

2.2  Correlation Clustering and Multicut Polytopes

Decompositions of graphs into an unknown number of components based on (dis-)similarities between neighboring nodes is referred to as weighted correlation clustering. It has been studied for the complete graph (Bansal et al., 2004) as well as general graphs (Demaine et al., 2006). As any decomposition of a graph is characterized by the mathematical notion of a multicut, the study of multicuts is closely related to correlation clustering. The combinatorial polytopes associated with the multicuts of a graph have been studied, among others, most notably by Grötschel and Wakabayashi (1989); Chopra and Rao (1993); Deza and Laurent (1997). The more general case where correlations between non-neighboring nodes are taken into account as well has been introduced by Horňáková et al. (2017).

2.3  Sequential Set Partitioning

Set partitioning problems where the elements are assumed to adhere to a linear order have been studied by Kernighan (1971), who devises a dynamic programming algorithm that essentially solves a shortest path problem in a directed acyclic graph. The corresponding integer linear programming formulation admits a totally unimodular constraint matrix (Joseph and Bryson, 1997).

3  Representations and Complexity

3.1  Sparse Pseudo-Boolean Functions

For any $n\in\mathbb{N}$, any $f\colon\{0,1\}^{n}\to\mathbb{R}$ is called an $n$-variate pseudo-Boolean function (PBF). Any $n$-variate PBF $f$ has a unique multi-linear polynomial form111In line with the literature on pseudo-Boolean optimization, we call the form (2) multi-linear despite its constant term $c_{\emptyset}$.:

[TABLE]

We observe that the objective functions of the tree partition problem form a class of sparse PBFs whose set $\{I\subseteq[n]\colon c_{I}\neq 0\}$ of non-zero coefficients is constrained by a tree. This class of functions is formalized in the following definition.

Definition 2**.**

An $n$-variate PBF is called tree-sparse w.r.t. a tree $T=(V,[n])$ iff its multi-linear polynomial form (2) is such that, for every $I\subseteq[n]$ with $c_{I}\neq 0$, the set $I$ induces a path in $T$. The function is called path-sparse w.r.t. a path $P=(V,[n])$ iff it is tree-sparse w.r.t. $P$.

3.2  Set Partitioning

For any connected subgraph (subtree) $S\subseteq T$, introduce a binary variable $\lambda_{S}\in\{0,1\}$. By defining the cost $d_{S}=\sum_{u,v\in V_{S}}c_{uv}$ for each component $S$, problem (1) can be reformulated as the set partitioning problem

[TABLE]

Here, the costs $d_{S}$ account for all pairs of nodes within component $S$ and the constraints ensure that every node is contained in exactly one component of the partition. Note that the number of variables $\lambda_{S}$ is exponential in the number of leaves of $T$.

Lemma 1**.**

The vector $y\in\{0,1\}^{E}$ is a solution of problem (1) w.r.t. the tree $T=(V,E)$ and costs $c\colon\tbinom{V}{2}\to\mathbb{R}$ iff the vector $\lambda$ defined by

[TABLE]

for every subtree $S\subseteq T$, is a solution of problem (3) w.r.t. the costs $d_{S}=\sum_{u,v\in V_{S}}c_{uv}$.

Proof.

We set $c_{uu}=0$ for all $u\in V$. Let $S_{u}$ denote the component of the partition that contains $u\in V$. The claim follows from

[TABLE]

3.3  Lifted Multicuts

In this section, we identify the tree partition problem (Def. 1) as a special case of the minimum cost lifted multicut problem (Horňáková et al., 2017) where the underlying graph is a tree. Essentially, this representation is a linearization of the tree partition problem with reversed binary encoding.

The minimum cost lifted multicut problem w.r.t. the tree $T=(V,E)$ and costs $c\colon\tbinom{V}{2}\to\mathbb{R}$ is the combinatorial optimization problem

[TABLE]

where $X_{T}$ is defined as

[TABLE]

Problem (4) is the special case of the minimum cost lifted multicut problem as presented by Horňáková et al. (2017) for the specific choice $G=T$ and G^{\prime}=(V,\raisebox{0.43057pt}{\scalebox{0.8}{\tbinom{V}{2}}}), i.e. the tree $T$ lifted to the complete graph on $V$ (cf. appendix B). Lemma 2 states that the tree partition problem can be reformulated as a minimum cost lifted multicut problem.

Lemma 2**.**

The vector $y\in\{0,1\}^{n}$ is a solution of problem (1) w.r.t. the tree $T=(V,E)$ and costs c\colon\raisebox{0.43057pt}{\scalebox{0.8}{\tbinom{V}{2}}}\to\mathbb{R} iff the unique $x\in X_{T}$ such that $x_{e}=1-y_{e}$ for all $e\in E$ is a solution of problem (4) w.r.t. $T$ and the cost function $-c$.

Proof.

For any distinct pair of nodes $u,v\in V$, we introduce a binary variable $x_{uv}\in\{0,1\}$ via

[TABLE]

which implies

[TABLE]

Therefore, we can reformulate problem (1) in terms of the variables $x_{uv}$ by transforming the objective function according to

[TABLE]

This leads to the combinatorial optimization problem

[TABLE]

where $X_{T}\subseteq\{0,1\}^{m}$ captures the relationship (8). ∎

Note that, since the objective function of (4) is linear, we can replace $X_{T}$ by its convex hull

[TABLE]

which is called the lifted multicut polytope w.r.t. $T$. This motivates the study of the structure of $\Xi_{T}$ in Section 4. We refer to (5) and (6) as path and cut inequalities, respectively, and write $\Theta^{0}_{T}$ for the naive linear relaxation of $\Xi_{T}$, i.e. the set of vectors $x\in[0,1]^{m}$ that satisfy (5) and (6).

3.4  Complexity

We now discuss the computational complexity of the tree partition problem and the path partition problem. In Lemma 3, we show that the tree partition problem is np-hard, even if the tree is a star. On the contrary, the path partition problem is polynomial time solvable (Kernighan, 1971).

Lemma 3**.**

The tree partition problem is np-hard. It remains np-hard if $T$ is a star.

Proof.

For a tree-sparse pseudo-Boolean function defined on a star with $n$ leaves, problem (1) is equivalent to the unconstrained binary quadratic program with $n$ variables, which is well-known to be np-hard. ∎

4  Polyhedral Geometry

In this section, we establish polyhedral results for the lifted multicut polytope $\Xi_{T}$ for trees. We characterize all trivial facets and offer a tighter outer relaxation of $\Xi_{T}$. In Section 4.5, we show that our results yield a complete totally dual integral (TDI) description of the lifted multicut polytope for paths. This result relates the combinatorial properties of the sequential set partitioning problem to the geometry of the minimum cost lifted multicut problem for paths. The rather technical proofs for our claims are deferred to appendix A.

4.1  Canonical Outer Relaxation

We give another simple relaxation of $\Xi_{T}$ that is at least as tight as the naive linear relaxation $\Theta^{0}_{T}$. To this end, define $\vec{u}(v)$ to be the first node on the path $P_{uv}$ that is different from both $u$ and $v$ (cf. Fig. 2d) and consider the polytope

[TABLE]

This description is canonical in the sense that it only considers a quadratic number of node triplets, namely those which feature two neighboring nodes and an arbitrary third node. The following lemma states that $\Theta^{1}_{T}$ is indeed an outer relaxation of $\Xi_{T}$ which is at least as tight as $\Theta^{0}_{T}$.

Lemma 4**.**

It holds that $\Xi_{T}\subseteq\Theta^{1}_{T}\subseteq\Theta^{0}_{T}$.

4.2  Trivial Facets

It can be characterized exactly which inequalities of (12), (13) and $0\leq x_{uv}\leq 1$ define facets of $\Xi_{T}$. These elementary results are closely related to the more general study in (Horňáková et al., 2017).

Lemma 5**.**

The inequality (12) defines a facet of $\Xi_{T}$ if and only if $\operatorname{dist}(u,v)=2$.

Lemma 6**.**

The inequality (13) defines a facet of $\Xi_{T}$ if and only if $v$ is a leaf of $T$.

Lemma 7**.**

For any $u,v\in V,\;u\neq v$, the inequality $x_{uv}\leq 1$ defines a facet of $\Xi_{T}$ if and only if both $u$ and $v$ are leaves of $T$. Moreover, none of the inequalities $0\leq x_{uv}$ define facets of $\Xi_{T}$.

4.3  Non-Trivial Facets

We further present a large class of non-trivial facets of $\Xi_{T}$. Consider the set of inequalities given by

[TABLE]

We shall refer to (14) as square inequalities. As an example consider the graph depicted in Fig. 2c and $u=0$, $v=3$. Any square inequality is valid and facet-defining for $\Xi_{T}$.

Lemma 8**.**

The inequalities (14) are valid for $\Xi_{T}$.

Lemma 9**.**

The inequalities (14) define facets of $\Xi_{T}$.

4.4  Facets Related to the Boolean Quadric Polytope

The argument to show Lemma 3 also implies that the lifted multicut polytope for a star with $n$ leaves is isomorphic to the Boolean Quadric Polytope (Padberg, 1989) of order $n$. Therefore, insights about the facial structure of the Boolean quadric polytope can be used to derive valid inequalities and facets for the more general polytope $\Xi_{T}$. In fact, any tree-sparse pseudo-Boolean function repeatedly admits substructures similar to the quadratic case. As this connection is beyond the scope of our paper, we only illustrate it with the following example. Consider the star graph depicted in Fig. 2b. It is known from (Padberg, 1989) that the “triangle” inequality

[TABLE]

is valid and facet-defining for $\Xi_{T}$. Observe that, similarly, for the tree in Fig. 2a

[TABLE]

holds true, but in this case $x_{02},x_{05}$ and $x_{06}$ correspond to cubic instead of quadratic terms of the associated pseudo-Boolean function.

4.5  Complete TDI Description for Paths

Suppose the node set $V=\{0,\dotsc,n\}$ is linearly ordered and consider the path $P=(V,E)$, where the edge set is given by E=\big{\{}\{i,i+1\}\mid i\in\{0,\dotsc,n-1\}\big{\}}. We show that the facets arising from (12) - (14) yield a complete description of $\Xi_{P}$. For this section, we consider only paths of length $n\geq 2$, since for $n=1$ the polytope $\Xi_{P}=[0,1]$ is trivial. Let $\Theta^{\mathrm{Path}}_{P}$ be the polytope defined by

[TABLE]

Note that the system (17) - (21) consists precisely of those trivial and square inequalities which we have shown to define facets of $\Xi_{P}$. We first prove that $\Theta^{\mathrm{Path}}_{P}$ indeed yields an outer relaxation of $\Xi_{P}$.

Lemma 10**.**

It holds that $\Xi_{P}\subseteq\Theta^{\mathrm{Path}}_{P}\subseteq\Theta^{1}_{P}$.

As our main result, we prove that $\Theta^{\mathrm{Path}}_{P}$ is in fact a complete description of $\Xi_{P}$. To derive this, we utilize the notion of total dual integrality. For an extensive reference on this subject we refer the reader to (Schrijver, 1986).

Definition 3**.**

A system of linear inequalities $Ax\leq b$ with $A\in\mathbb{Q}^{k\times m},\;b\in\mathbb{Q}^{k}$ is totally dual integral (TDI) if for any $c\in\mathbb{Z}^{m}$ such that the linear program $\max\{c^{\top}x\mid Ax\leq b\}$ is feasible and bounded, there exists an integral optimal dual solution.

Total dual integrality is an important concept in polyhedral geometry as it serves as a sufficient condition on the integrality of polyhedra according to the following fact.

Fact 1** ((Edmonds and Giles, 1977)).**

If $Ax\leq b$ is totally dual integral and $b$ is integral, then the polytope defined by $Ax\leq b$ is integral.

Theorem 1**.**

The system (17) - (21) is totally dual integral.

Corollary 1**.**

It holds that $\Xi_{P}=\Theta^{\mathrm{Path}}_{P}$.

Proof.

This is immediate from Lemma 10, Fact 1 and Theorem 1. ∎

Remark 1**.**

The constraint matrix corresponding to the system (17) - (21) is in general not totally unimodular. A minimal example is the path of length $4$.

The set partitioning representation of the path partition problem w.r.t. $P$ admits a quadratic number of variables and a linear number of constraints (opposed to a quadratic number of constraints in the description of $\Xi_{P}$). This representation corresponds to the dual program of the last problem in the proof of Theorem 1, cf. appendix A. Here, the integrality constraint need not be enforced, since the constraint matrix is totally unimodular.

5  Conclusion

We have characterized all trivial facets of the lifted multicut polytope for trees. Additionally, we provide a tighter relaxation compared to the standard linear relaxation by including additional classes of facets. The described facets provide a complete totally dual integral (TDI) description of the associated lifted multicut polytope for paths. This result relates the geometry of this problem to the combinatorial properties of alternative formulations such as the sequential set partitioning problem.

Appendix A Proofs for Section 4

Proof of Lemma 4.

We show first that $\Xi_{T}\subseteq\Theta^{1}_{T}$. For this purpose, let $x\in\Xi_{T}$. If $x$ violates (12), then $x_{uv}=1$ and $x_{u,\vec{u}(v)}=x_{\vec{u}(v),v}=0$. This contradicts the fact that $x$ satisfies all cut inequalities w.r.t. $\vec{u}(v),v$ and the path inequality w.r.t. $u,v$. If $x$ violates (13), then $x_{\vec{u}(v),v}=1$ and $x_{uv}=0$. This contradicts the fact that $x$ satisfies all cut inequalities w.r.t. $u,v$ and the path inequality w.r.t. $\vec{u}(v),v$. Hence, $x$ satisfies (12)-(13) and therefore $x\in\Theta^{1}_{T}$.

Now, we show that $\Theta^{1}_{T}\subseteq\Theta^{0}_{T}$. Let $x\in\Theta^{1}_{T}$. We need to show that $x$ satisfies all path and cut inequalities. Let $u,v\in V$ with $\operatorname{dist}(u,v)\geq 2$. We proceed by induction on $\operatorname{dist}(u,v)$. If $\operatorname{dist}(u,v)=2$, then the path inequality is directly given by (12), while the two cut inequalities are given by (13) for the two possible orderings of $u$ and $v$. If $\operatorname{dist}(u,v)>2$, then the path inequality is obtained from (12) and the induction hypothesis for the pair $\vec{u}(v),v$, since $\operatorname{dist}(\vec{u}(v),v)=\operatorname{dist}(u,v)-1$. Similarly, for any edge $e$ on the path from $u$ to $v$, we obtain the cut inequality w.r.t. $e$ by using the induction hypothesis and (13) such that (w.l.o.g.) $e$ is on the path from $\vec{u}(v)$ to $v$. Hence, $x\in\Theta^{0}_{T}$.

Proof of Lemma 5.

First, suppose $\operatorname{dist}(u,v)=2$. Then $P_{uv}$ is a path of length $2$ and thus chordless in the complete graph on $V$. Hence, facet-definingness follows directly from [Horňáková et al., 2017, Thm 4]. Now, suppose $\operatorname{dist}(u,v)>2$ and let $x\in\Xi_{T}$ be such that (12) is satisfied with equality. We show that this implies

[TABLE]

Then the face of $\Xi_{T}$ induced by (12) has dimension at most $m-2$ and hence cannot be a facet. In order to check that (22) holds, we distinguish the following three cases. If $x_{uv}=x_{u,\vec{u}(v)}=x_{\vec{u}(v),v}$, then all terms in (22) vanish. If $x_{uv}=x_{u,\vec{u}(v)}=1$ and $x_{\vec{u}(v),v}=0$, then $x_{\vec{u}(v),\vec{v}(u)}=0$ and $x_{u,\vec{v}(u)}=1$, so (22) holds. Finally, if $x_{uv}=x_{\vec{u}(v),v}=1$ and $x_{u,\vec{u}(v)}=0$, then (22) holds because $x_{\vec{u}(v),\vec{v}(u)}=x_{u,\vec{v}(u)}$ by contraction of the edge $u,\vec{u}(v)$.

Proof of Lemma 6.

First, suppose $v$ is not a leaf of $T$ and let $x\in\Xi_{T}$ be such that (13) is satisfied with equality. Since $v$ is not a leaf, there exists a neighbor $w\in V$ of $v$ such that $P_{\vec{u}(v),v}$ is a subpath of $P_{\vec{u}(v),w}$. We show that $x$ additionally satisfies the equality

[TABLE]

and thus the face of $\Xi_{T}$ induced by (13) cannot be a facet. There are two possible cases: Either $x_{uv}=x_{\vec{u}(v),v}=1$, then $x_{uw}=x_{\vec{u}(v),w}=1$ as well, or $x_{uv}=x_{\vec{u}(v),v}=0$, then $x_{uw}=x_{vw}=x_{\vec{u}(v),w}$ by contraction of the path $P_{uv}$, so (23) holds.

Now, suppose $v$ is a leaf of $T$ and let $\Sigma$ be the face of $\Xi_{T}$ induced by (13). We need to prove that $\Sigma$ has dimension $m-1$. This can be done explicitly by showing that we can construct $m-1$ distinct indicator vectors $\chi_{w,w^{\prime}}$ for $w,w^{\prime}\in V$ as linear combinations of elements from the set $S=\{x\in\Xi_{T}\mid x_{uv}=x_{\vec{u}(v),v}\}$. In fact this construction is analogous to the one used in the proof of Theorem 7 in [Horňáková et al., 2017] where the authors derive the dimension of the lifted multicut polytope $\dim(\Xi_{GG^{\prime}})=\lvert E^{\prime}\rvert$. The difference here is that the vector $\mathbbm{1}-\chi_{uv}\notin S$, so we have to distinguish between $x\in S$ with $x_{uv}=x_{\vec{u}(v),v}=0$ and $x\in S$ with $x_{uv}=x_{\vec{u}(v),v}=1$ in order to show $\chi_{u,\vec{u}(v)}\in\operatorname{lin}(S)$ and then $\chi_{e}\in\operatorname{lin}(S)$ for all other $e\neq\{u,v\},\{\vec{u}(v),v\}$.

Proof of Lemma 7.

We apply the more general characterization given in [Horňáková et al., 2017] to our special case. The nodes $u,v\in V$ are a pair of $w$-$w^{\prime}$-cut-vertices for some vertices $w,w^{\prime}\in V$ (with at least one being different from $u$ and $v$) if and only if $u$ or $v$ is not a leaf of $V$. Thus, the claim follows from [Horňáková et al., 2017, Thm 8]. The second assertion follows from [Horňáková et al., 2017, Thm 9] and the fact that we lift to the complete graph on $V$.

Proof of Lemma 8.

Let $x\in\Xi_{T}$ and suppose that either $x_{u,\vec{v}(u)}=0$ or $x_{\vec{u}(v),v}=0$ for some $u,v\in V$ with $\operatorname{dist}(u,v)\geq 3$. Then, since $x$ satisfies all cut inequalities w.r.t. $u,\vec{v}(u)$, respectively $\vec{u}(v),v$, and the path inequality w.r.t. $\vec{u}(v),\vec{v}(u)$, it must hold that $x_{\vec{u}(v),\vec{v}(u)}=0$. Moreover, if even $x_{u,\vec{v}(u)}=0=x_{\vec{u}(v),v}$, then, by the same reasoning, we have $x_{uv}=0$ as well. Hence, $x$ satisfies (14).

Proof of Lemma 9.

Let $\Sigma$ be the face of $\Xi_{T}$ induced by (14). We need to prove that $\Sigma$ has dimension $m-1$, which can be done explicitly by showing that we can construct $m-1$ distinct indicator vectors $\chi_{w,w^{\prime}}$ for $w,w^{\prime}\in V$ as linear combinations of elements from the set $S=\{x\in\Xi_{T}\mid x_{uv}+x_{\vec{u}(v),\vec{v}(u)}=x_{u,\vec{v}(u)}+x_{\vec{u}(v),v}\}$. Again, this construction is very technical and analogous to the proof of Theorem 7 in [Horňáková et al., 2017] about the dimension of the lifted multicut polytope.

Proof of Lemma 10.

First, we show that $\Xi_{P}\subseteq\Theta^{\mathrm{Path}}_{P}$. Let $x\in\Xi_{P}$, then $x$ satisfies (17) and (20) by definition. Suppose $x$ violates (18), then $x_{in}=1$ and $x_{i-1,n}=0$. This contradicts the fact that $x$ satisfies all cut inequalities w.r.t. $i-1,n$ and the path inequality w.r.t. $i,n$. So, $x$ must satisfy (18) and, by symmetry, also (19). It follows from Lemma 8 that $x$ satisfies (21) as well and thus $x\in\Theta^{\mathrm{Path}}_{P}$.

Next, we prove that $\Theta^{\mathrm{Path}}_{P}\subseteq\Theta^{1}_{P}$. To this end, let $x\in\Theta^{\mathrm{Path}}_{P}$. We show that $x$ satisfies all inequalities (13). Let $u,v\in V$ with $u<v-1$. We need to prove that both $x_{u+1,v}\leq x_{uv}$ and $x_{u,v-1}\leq x_{uv}$ hold. For reasons of symmetry, it suffices to show only $x_{u+1,v}\leq x_{uv}$. We proceed by induction on the distance of $u$ from $n$. If $v=n$, then $x_{u+1,n}\leq x_{un}$ is given by (18). Otherwise, we use (21) for $j=u$ and $k=v+1$ and the induction hypothesis on $v+1$:

[TABLE]

It remains to show that $x$ satisfies all inequalities (12). Let $u,v\in V$ with $u<v-1$. We proceed by induction on $\operatorname{dist}(u,v)=u-v$. If $\operatorname{dist}(u,v)=2$, then (12) is given by (20). If $\operatorname{dist}(u,v)>2$, then we use (21) for $j=u$ and $k=v$ as well as the induction hypothesis on $u,v-1$, which have distance $\operatorname{dist}(u,v)-1$:

[TABLE]

Hence, $x\in\Theta^{1}_{P}$, which concludes the proof.

Proof of Theorem 1

We rewrite system (17) - (21) more compactly in the following way. Introduce two artificial nodes $-\infty$ and $\infty$ where we associate $-\infty$ to any index less than [math] and $\infty$ to any index greater than $n$. Moreover, we introduce variables $x_{ii}$ for all $0\leq i\leq n$ as well as $x_{-\infty,i}$ and $x_{i,\infty}$ for all $1\leq i\leq n-1$ and finally $x_{-\infty,\infty}$. Now, the system (17) - (21) is equivalent to the system

[TABLE]

given the additional equality constraints

[TABLE]

Let the system defined by (24) - (28) be represented in matrix form as $Ax\leq a,\;Bx=b$. Note that $\Theta^{\mathrm{Path}}_{P}$ is non-empty and bounded. Thus, to establish TDI, we need to show that for any $c\in\mathbb{Z}^{m+3n}$ the dual program

[TABLE]

has an integral optimal solution. Here, the $y$ variables, indexed by $j,k$, correspond to the inequalities (24) and the $z$ variables, indexed by pairs of $i,-\infty$ and $\infty$, correspond to the equations (25) - (28). Then, the equation system $A^{\top}y+B^{\top}z=c$ translates to

[TABLE]

where (30) - (32) hold for all $0\leq i<i+1<\ell\leq n$ and (33), (34) hold for all $1\leq i\leq n-1$. Observe that (30) includes only $y$ variables with indices of distance $2$, (31) couples $y$ variables of distance $3$ with those of distance $2$ and (32) couples the remaining $y$ variables of any distance $d>3$ with those of distance $d-1$ and $d-2$. Hence, any choice of values for the free variables $z_{ii}$ completely determines all $y$ variables. This means we can eliminate $y$ and reformulate the dual program entirely in terms of the $z$ variables, as follows. It holds that

[TABLE]

and thus, by (32),

[TABLE]

for all $0\leq i\leq\ell\leq n$. Substituting the $y$ variables in (33) - (35) yields the following equivalent formulation of (29):

[TABLE]

The variables $z_{-\infty,i},\;z_{i,\infty}$ and $z_{-\infty,\infty}$ occur only in a single equation each. Furthermore, the matrix corresponding to the inequality constraints satisfies the “consecutive-ones” property. Therefore, the constraint matrix of the whole system is totally unimodular, which concludes the proof.

Path Partition as Sequential Set Partitioning

For each $0\leq i\leq\ell\leq n$, let

[TABLE]

then taking the dual of problem (36) and simplifying yields the problem

[TABLE]

Each variable $\lambda_{i,\ell}$ corresponds to the component containing nodes $i$ to $\ell$. Problem (37) is precisely the sequential set partitioning formulation of the path partition problem as used by Joseph and Bryson [1997].

Appendix B Lifted Multicuts of General Graphs

For any connected graph $G=(V,E)$, any supergraph $G^{\prime}=(V,E^{\prime})$ with E^{\prime}=E\mathbin{\mathchoice{\leavevmode\vtop{\halign{\hfil\m@th\displaystyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil\m@th\textstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil\m@th\scriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\cup\cr\cdot\crcr}}}}F and any $c:E^{\prime}\to\mathbb{R}$ the instance of the minimum cost lifted multicut problem w.r.t. $G$, $G^{\prime}$ and $c$ is the binary linear program

[TABLE]

with

[TABLE]

The inequalities (39), (40) and (41) are referred to as cycle, path and cut inequalities, respectively. The convex hull

[TABLE]

of $X_{GG^{\prime}}$ in $\mathbb{R}^{E^{\prime}}$ is called the lifted multicut polytope w.r.t. $G$ and $G^{\prime}$. It was shown by Horňáková et al. [2017] that $\Xi_{GG^{\prime}}$ has dimension $\lvert E^{\prime}\rvert$.

For any $x\in X_{GG^{\prime}}$, the set $M:=\{e\in E\,|\,x_{e}=1\}$ of those edges of the graph $G$ that are labeled 1 is a so-called multicut of $G$. Hence, there exists a decomposition of $G$ such that $M$ is precisely the set of those edges that span across distinct components. In addition, the set $M^{\prime}:=\{e\in E^{\prime}\,|\,x_{e}=1\}$ of those edges $\{u,v\}=e\in E^{\prime}$ of the graph $G^{\prime}$ that are labeled 1 is a multicut of $G^{\prime}$ lifted from the multicut $M$ of $G$. A multicut $M^{\prime}$ of $G^{\prime}$ lifted from $G$ makes explicit for all pairs $\{u,v\}\in E^{\prime}$ of nodes (not only those neighboring in $G$) whether $u$ and $v$ are in the same component in the decomposition of $G$.