Polyhedral Properties of the Induced Cluster Subgraphs

A cluster graph is a graph whose every connected component is a complete graph. Given a simple undirected graph G, a subset of vertices inducing a cluster graph is called an independent union of cliques (IUC), and the IUC polytope associated with G is defined as the convex hull of the incidence vectors of all IUCs in the graph. The Maximum IUC problem, which is to find a maximum-cardinality IUC in a graph, finds applications in network-based data analysis. In this paper, we derive several families of facet-defining valid inequalities for the IUC polytope. We also give a complete description of this polytope for some special classes of graphs. We establish computational complexity of the separation problem for most of the considered families of valid inequalities and explore the effectiveness of employing the corresponding cutting planes in an integer (linear) programming framework for the Maximum IUC problem through computational experiments.

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1 Introduction

A graph is called a cluster graph if its every connected component is a complete graph. Given a simple undirected graph , with the set of vertices and the set of edges , an independent union of cliques (IUC) is a subset of vertices inducing a cluster graph. The Maximum IUC problem is to find an IUC of maximum cardinality in . The importance of this problem stems from its application in graph clustering (partitioning the vertices of a graph into cohesive subgroups), which is a fundamental task in unsupervised data analysis; see [46]

and the references therein for a survey of graph clustering applications and methods. A cluster graph is an ideal instance from the standpoint of cluster analysis—as it is readily composed of mutually disjoint “tightly knit” subgroups—and the

Maximum IUC problem aims to identify the largest induced cluster graph contained in an input graph . Such an induced subgraph discloses the natural cores (centers) of the input graph clusters. Besides the practical importance of this problem, the definition of IUC subsumes two fundamental structures in a graph, i.e., cliques and independent sets. The Maximum Clique and Maximum Independent Set

problems are among the most popular problems of combinatorial optimization, several variants of which have been studied in the literature; see e.g. 

[32, 8, 53, 27, 40, 43]. Therefore, the study of the Maximum IUC problem, that admits both of these complementary structures as feasible solutions, is also of a particular theoretical interest.

The Maximum IUC problem has appeared under different names in the literature. Fomin et al. [22] introduced this problem as the Maximum Induced Cluster Subgraph problem, and proposed an exact -time algorithm for it, where . Ertem et al. [20] considered the Maximum IUC problem as a relaxation of the Maximum -Cluster problem with the maximum local clustering coefficient, i.e., . They proposed a graph clustering algorithm based on a given maximum IUC, referred to as disjoint 1-clusters in [20], and showed the competence of their method via experiments with real-life social networks. It was not until very recently that the term “independent union of cliques” was used by Ertem et al. [19]. In this work, the authors studied some basic properties of IUCs and analyzed complexity of the Maximum IUC problem on some restricted classes of graphs. They also performed computational experiments using a combinatorial algorithm (Russian Doll Search) and an integer (linear) programming formulation of this problem. To the best of our knowledge, this is the only work containing computational experiments on the Maximum IUC problem, and their results show that this problem is quite challenging for the exact solution methods. It is known that a graph is a cluster graph if and only if it contains no induced subgraph isomorphic to (the path graph on three vertices) [26]. In this regard, the Maximum IUC problem has also been referred to as the Maximum Induced -Free Subgraph problem [22]. In our presentation to follow, we refer to induced subgraphs isomorphic to as open triangles.

Evidently, finding a maximum IUC in is equivalent to finding a minimum number of vertices whose deletion turns into a cluster graph. The latter is the optimization version of the Cluster Vertex Deletion problem (-CVD), which asks if an input graph can be transformed into a cluster graph by deleting at most vertices, and is known to be NP-hard by Lewis and Yannakakis theorem [36]. The study of this problem started from the viewpoint of parameterized complexity with the work of Gramm et al. [26], who proposed an -time fixed-parameter tractable algorithm for -CVD, where and . Later, Hüffner et al. [30] and Boral et al. [9] improved this result to and , respectively. Le et al. [35] showed that -CVD admits a kernel with vertices, which means there exists a polynomial-time algorithm that converts an -vertex instance of -CVD to an equivalent instance with vertices. Fomin et al. [23] used the parameterized results of [9] to show that the Minimum CVD problem (hence, Maximum IUC) is solvable in time, which is better than the former result of [22]. It is also known that the Minimum CVD problem is approximable to within a constant factor. In particular, You et al. [54] proposed a -approximation algorithm for it, and Fiorini et al. [21] improved the approximation ratio to . Several other problems involving cluster subgraphs (not necessarily induced) have also been considered in the literature, including Cluster Editing [47, 18, 6, 33, 7, 24, 2], Disjoint Cliques [31, 1], -Plex Cluster Vertex Deletion [51], and -Plex Cluster Editing [28].

From a broader perspective, the Maximum IUC problem is to identify a maximum-cardinality independent set in a graph-based independence system. An independence system is a pair of a finite set together with a family of subsets of such that implies . Referring to an independence system , every is called an independent set and every , is called a dependent set. The minimal (inclusion-wise) dependent subsets of are called circuits of the independence system . In graph-theory terminology, an independent set (stable set, vertex packing) is a subset of pairwise non-adjacent vertices in a graph . Let denote the set of all independent sets (in the sense of graph theory) in . Then, is an independence system whose circuit set is given by ; hence, the two definitions coincide. Following this concept, together with the set of all IUCs in form an independence system whose set of circuits is given by the three-vertex subsets of inducing open triangles in the graph. This perspective on the Maximum IUC problem is of our interest as some of the results to follow correspond to certain properties of independence systems. It is worth noting that many well-known problems of graph theory, beyond those mentioned here, convey the notion of independence systems associated with a graph; see for example [34, 44].

This paper presents the first study of the Maximum IUC problem from the standpoint of polyhedral combinatorics. More specifically, we study the facial structure of the IUC polytope associated with a simple undirected graph , and identify several facet-defining valid inequalities for this polytope. As a result, we will be able to present the full description of the IUC polytope for some special classes of graphs. We also study the computational complexity of the separation problems for these inequalities, and perform computational experiments to examine their effectiveness when used in a branch-and-cut scheme to solve the Maximum IUC problem. The organization of this paper is as follows: we continue this section by introducing the terminology and notation used in this paper. In Section 2, the IUC polytope is defined and the strength of its fractional counterpart—obtained form relaxing the integrality of the variables in definition of the original polytope—is discussed. In Section 3, we study some facet-producing structures for the IUC polytope and present the corresponding facet-inducing inequalities, as well as convex hull characterization (when possible). In our study, we pay special attention to similarities between the facial structure of the IUC polytope and those of the independent set (vertex packing) and clique polytopes. In Section 4, we investigate the computational complexity of the separation problem for each class of the proposed valid inequalities. Section 5 contains the results of our computational experiments, and Section 6 concludes this paper.

1.1 Terminology and Notation

Throughout this paper, we consider a simple undirected graph , where is the vertex set and is the edge set. Unless otherwise stated, we assume . Given a subset of vertices , the subgraph induced by is denoted by and is defined as , where is the set of edges with both endpoints in . A clique is a subset of vertices inducing a complete graph, and an independent set (stable set, vertex packing) is a subset of vertices inducing an edgeless graph. A clique (resp. independent set) is called maximum if it is of maximum cardinality, i.e., there is no larger clique (resp. independent set) in the graph. Cardinality of a maximum clique in is called the clique number of the graph, and is denoted by . Respectively, the independence number (stability number) of is the cardinality of a maximum independent set in the graph, and is denoted by . In a similar manner, a maximum IUC in is an IUC of maximum cardinality, and the IUC number of , denoted by , is the cardinality of a maximum IUC in the graph. Since every clique as well as every independent set in is an IUC, the IUC number of the graph is bounded from below by its clique number and independence number, i.e., . The (open) neighborhood of a vertex is the set of vertices adjacent to , and is denoted by . The closed neighborhood of is defined as . The incidence vector of a subset of vertices is a binary vector such that and . Conventionally, the incidence vector of a single vertex is denoted by . We use , and to denote the vectors of all zero, one and two, respectively. The corresponding dimensions will be clear from the context.

We assume familiarity of the reader with fundamentals of polyhedral theory, and just briefly mention the concept of lifting [42, 52, 55], as several proofs to follow are based on lifting arguments. Let be the convex hull of the set of feasible solutions to an arbitrary system of linear inequalities with a nonnegative coefficient matrix, i.e., . Corresponding to a subset of variables indexed by , consider the polytope obtained from by setting . Let , and suppose that the inequality induces a facet of . Then, lifting a variable , into this inequality leads to an inequality , where

(1)

which is facet-defining for . In (1), denotes the coefficient vector of a variable in the system of inequalities, i.e., the -th column of . The importance of this result is due to the fact that some facets of the original polytope can be obtained from the facets of the restricted polytopes through sequential (or simultaneous) lifting procedures. In particular, some facets of the polytope associated with a graph for a certain problem may be identified through lifting the facets of the lower-dimensional polytopes corresponding to its subgraphs. Finally, note that (1) is equivalent to

(2)

which is easier to use when is defined to be the convex hull of the incidence vectors of the sets of vertices in a graph holding a certain property.

2 The IUC Polytope

The IUC polytope associated with a simple undirected graph , denoted by , is the convex hull of the incidence vectors of all IUCs in , which can be characterized by the circuit set of the corresponding independence system as follows [38]:

(3)

where denotes the set of three-vertex subsets of inducing open triangles in . Patently, , and the Maximum Weight IUC problem, which is to find an IUC with the maximum total weight in a vertex-weighted graph, can be formulated as

(4)

with denoting the weight of a vertex . We call each inequality , an open-triangle (OT) inequality. The following theorems establish the basic properties of including the strength of the OT inequalities.

Theorem 1.

is a full-dimensional polytope, and for every , the inequalities and define its facets.

Proof.

Note that . Also, every single vertex, as well as every pair of vertices in is an IUC, by definition. Full-dimensionality of is established by noting that and , are affinely independent points in belonging to this polytope. Besides, for every , 0 and , belong to , which indicates that is a facet of . Similarly, for every , the points and , are affinely independent and belong to , hence is a facet of . ∎

It follows from full-dimensionality of that every facet-defining inequality of this polytope is unique to within a positive multiple. Thus, in order to prove that a proper face of defined by a valid inequality

is a facet, it suffices to show that any supporting hyperplane of

containing is a (non-zero) scalar multiple of .

Theorem 2.

The inequality , is facet-defining for if and only if there does not exist a vertex such that induces a chordless cycle in .

Proof.

Clearly, is a proper face of . So, suppose that it is contained in a supporting hyperplane of this polytope. Observe that,

Similarly, and , which imply and . Now, consider the subgraph induced by for an arbitrary vertex . As is not a chordless cycle, it contains at most three open triangles, and those have at least one vertex in common. Let be such a vertex, then

As the interior of is nonempty, and is a (non-zero) scalar multiple of . To complete the proof, note that if induces a chordless cycle in , then is dominated by the inequality , which is facet-defining for by Theorem 3 of the next section. ∎

In Section 3 we will show that Theorem 2 is a special case of a more general statement for the so-called star inequality, defined later.

Obviously, the Maximum IUC problem can be directly solved using off-the-shelf MIP solvers through the following formulation:

(5a)
s.t. (5b)
(5c)

However, the LP relaxation of (5) is generally too weak. In fact, given the feasibility of the fractional assignment , the optimal solution value of the LP relaxation problem is at least as large as , while the computational results of Ertem et al. [19] show that the IUC number of graphs with moderate densities tends to stay in a close range of their relatively small independence and clique numbers. This is due to the fact that the fractional IUC polytope is a polyhedral relaxation of a cubically constrained region in the space of original variables. Note that the IUC property may be enforced through trilinear products of the variables in a program, i.e., by replacing the OT inequalities of (5) with . In such a cubic formulation, relaxing the integrality of variables will not change the set of optimal solutions, because given a set of variables , to satisfy the constraints, the optimality conditions necessitate . Hence, can be obtained by solving the following continuous problem:

(6)

It is known that the (convex and concave) envelopes of a trilinear monomial term when are as follows [37]:

which, along with , indicates that is a polyhedral outer-approximation of the feasible region of (6). As a result, the gap between and the optimal solution value of the LP relaxation problem of (5) is due to accumulation of the gap between the trilinear terms and their overestimating envelopes. A similar result holds between continuous and integer (linear) programming formulations of the Maximum Independent Set and Maximum Clique problems, except that the corresponding continuous formulations of those problems are quadratically constrained. The cubic nature of the IUC formulation justifies the weakness of even compared to the fractional independent set (vertex packing) and clique polytopes, which are known to be loose in general. This further reveals the importance of exploring the facial structure of the IUC polytope and applying strong cutting planes in integer (linear) programming solution methods of the Maximum IUC problem.

3 Facet-producing Structures

In this section, we present some facet-inducing valid inequalities for the IUC polytope associated with chordless cycle (and its edge complement), star (and double-star), fan and wheel graphs. We study the conditions under which these inequalities remain facet-defining for the IUC polytope of an arbitrary graph that contains the corresponding structure as an induced subgraph, and present some results with respect to lifting procedure for those that do not have this property. In some cases, we will present the full description of the IUC polytope associated with these graphs.

3.1 Chordless cycle and its complement

The following theorem concerns the facial structure of the IUC polytope associated with a (chordless) cycle graph. In an arbitrary graph , a subset of vertices , inducing a chordless cycle is called a hole. In labeling the vertices of a hole , we assume that adjacent vertices have consecutive labels with the convention .

Theorem 3.

(Hole Inequality) Let be a hole of cardinality in , where is a positive integer and . Then, the inequality

(7)

is valid for . Moreover, (i) it induces a facet of if and only if , and (ii) it is facet-defining for if .

Proof.

Consider the subgraph induced by . This subgraph contains distinct open triangles, and each vertex appears in exactly three of them. Summation of the corresponding OT inequalities leads to

After dividing both sides of the last inequality by , integrality of the left-hand side implies validity of (7) for as a Gomory-Chvátal cut.

(i) It is clear that, as a linear combination of OT inequalities, (7) is not facet-defining for if . Thus, we just need to show that is sufficient for (7) to induce a facet of this polytope. We examine the corresponding cases separately.

  • case 1:
    Consider a sequential partitioning of such that each partition contains exactly three vertices except for the last one, which is left with a single vertex. Assume a labeling of the vertices such that the first partition is given by , and let and be the incidence vectors of the following sets of vertices:

    The sets of black vertices in Figures 1 and 1 depict and , respectively.

    Figure 1: The subgraph induced by a hole , .

    Observe that, , hence is a proper face of . Suppose that is an arbitrary supporting hyperplane of containing . Then, implies that . The same argument using similar partitions can be used to show that this result holds for every two adjacent vertices in , thus , and . The proof is complete by noting that the interior of is nonempty, so .

  • case 2:
    Similar to the former case, let and be the incidence vectors of

    Then, the same argument leads to the result that the inequality induces a facet of .

(ii) To conclude the proof, we show that (7) is facet-defining for if , i.e., . By (i), defines a facet of . Let be the inequality obtained from an arbitrary sequential lifting of (to ). Observe that, for every vertex , there always exists two vertices such that , thus, . This implies , hence induces a facet of . ∎

It is easy to see the similarity between (7) and the odd-hole inequality for the vertex packing (independent set in the sense of graph theory) polytope associated with . Padberg [41] proved that, given a hole in with an odd number of vertices, the inequality is facet-defining for the vertex packing polytope associated with . Later, Nemhauser and Trotter [38] noted that the odd-hole inequality is a special case of a more general result concerning independence systems, stated as follows:

Theorem 4.

[38] Suppose is the cardinality of a maximum independent set in an independence system , and contains maximum independent sets with corresponding affinely independent (incidence) vectors . Then is facet-defining for the polytope associated with (i.e., the convex hull of the incidence vectors of all members of in ).

Similar to the odd-hole inequality for the vertex packing polytope, Theorem 3(i) spots a facet-inducing inequality implied by Theorem 4 for the independence system defined by the IUC property on . Note that Theorem 4 itself is of little practical value as it provides no means to identify the independence systems satisfying such properties.

Furthermore, Theorems 2 and 3(ii) correspond to maximal clique inequality of independence systems. An independence system is called -regular if every circuit of it is of cardinality . Referring to a -regular independence system , is called a clique if and all subsets of are circuits of . Clearly, this definition coincides with the graph-theoretic definition of a clique. Nemhauser and Trotter [38] also proved that the inequality is facet-inducing for the polytope associated with a -regular independence system if is a maximal (inclusion-wise) clique. A special case of this result for the vertex packing polytope associated with a graph had been originally shown by Padberg [41]. Let denote the set of all IUCs in , and be the corresponding independence system. Patently, every is a clique in , and such a clique is maximal only if it is not contained in a clique of cardinality 4. It is not hard to see that a clique of size 4 in must be a hole in . Therefore, Theorem 2 corresponds to maximality of as a clique of the corresponding independence system. Besides, by the proof of Theorem 3(ii), it became clear that cannot have a clique of cardinality greater than 4 because, for every 4-hole and every vertex , there always exist two vertices such that . Thus, a hole of cardinality 4 in is actually a maximum clique in , and Theorem 3(ii) corresponds to its maximality. This result is of a special value for the Maximum IUC problem as all facets of this type can be identified in polynomial time. Recall that the vertex packing polytope associated with may possess an exponential number of such facets.

Given this result, we extend the concept of fractional clique/independence number of a graph to the independence system . The fractional clique number of is the maximum sum of nonnegative weights that can be assigned to its vertices such that the total weight of every independent set in the graph is at most 1. Naturally, the fractional clique number of is equivalent to the fractional independence number of its complement graph, where the total weight of every clique is at most 1. We define the fractional IUC number of a graph as the maximum sum of nonnegative weights that can be assigned to its vertices such that the total weight of every clique in , i.e., the vertex set of every open triangle and 4-hole in , is at most 1. Clearly, the fractional IUC number of provides an upper bound on its IUC number, which is computable in polynomial time as opposed to the fractional clique and independence numbers of a graph which are NP-hard to compute [27].

Next, we show that if itself is a cycle graph, the inequalities identified by Theorems 2 and 3 along with the variable bounds are sufficient to describe the IUC polytope associated with . To this end, we first state a preliminary result regarding the rank of the coefficient matrix of the OT inequalities of a cycle graph.

Lemma 5.

Let denote the system of OT inequalities corresponding to a cycle graph on vertices. Then, is full-rank if and only if , for every positive integer .

Proof.

As it was pointed out in the proof of Theorem 3, contains exactly open triangles, hence is a square matrix of order . Consider the system of equations , where denotes the -th column of . Clearly, the equations corresponding to and in this system imply , given the convention ; . Now, suppose , for some positive integer . This implies

and any combination of real numbers satisfying is a solution to this system. Therefore, the columns of are linearly dependent and is not full-rank. On the other hand, if , then , leads to

which implies . Clearly, the same result holds when , which establishes linear independency of columns of and completes the proof. ∎

Theorem 6.

Let be a cycle graph on vertices. Let and be the quotient and remainder of a division of by 3, respectively (i.e., ), and for . Then,

(8)
Proof.

Let denote the polytope defined by the right-hand side of (8). Note that the last inequality in the definition of is the hole inequality (7) written for . Given the validity of (7) for , we just need to show that every extreme point of is integral. We start with the case where is a multiple of 3, so (7) is redundant in the description of . Recall that is an extreme point of if and only if it lies on some linearly independent supporting hyperplanes of this polytope. By Lemma 5, the coefficient matrix of the set of OT inequalities is rank deficient, thus at least one variable is tight at its bound in every extreme point of . Suppose , for some , is such a variable in an arbitrary extreme point of , and consider the corresponding facets and . Observe that, the inequalities , , and are redundant in description of , thus at most (linearly independent) supporting hyperplanes of this facet are provided by the set of OT inequalities. Moreover, in description of , the OT inequalities involving reduce to , , and . This implies redundancy of two other OT inequalities, that is and , in description of this facet, which further indicates that OT inequalities can provide at most (linearly independent) supporting hyperplanes for . Therefore, in either case, is not the only integral component of . It is easy to verify that fixing any other variable at its (lower or upper) bound either makes more OT inequalities redundant in description of the corresponding faces, or forces another variable to be integral, which in turn reduces the number of OT inequalities in description of lower-dimensional faces of containing . Clearly, this chain continues up to the zero-dimensional face of containing , i.e., itself, where all coordinates are integral. This completes the proof for the case , for every positive integer .

Next, we examine the case where is not a multiple of . Let be an extreme point of , and assume (7) is not binding at this point. As is full-rank, the system of OT equalities has a unique fractional solution given by . Since this solution violates (7), must have an integer component, which implies is integral in its entirety by our former argument. Now, suppose (7) is binding at . In this case, exactly OT inequalities must be binding at any fractional extreme point of (if any), because all OT inequalities can not be active simultaneously with (7) and less number of them leaves at least one variable to be integer, which leads to integrality of . Let , for some positive integer and, without loss of generality, assume is the non-binding constraint. Then, the unique solution of the corresponding system of equations is given by

which violates the variable bounds. Alternatively, if , the solution of this system of equations is given by

which is integral. Consequently, such a fractional extreme point does not exist, which completes the proof. ∎

The proof of Theorem 6 also implies that the family of OT inequalities together with the variable bounds is sufficient to describe the IUC polytope associated with a path (chain) graph. A path graph on vertices is denoted by . We present this result through the following corollary.

Corollary 7.

Let . Then,

(9)
Proof.

Consider a cycle graph generated from by adding some additional vertices (and the corresponding edges) such that , for some positive integer . Let denote the set of appended vertices, and observe that the right-hand side of (9) is precisely the description of the face of defined by . Thus, each extreme point of this polytope is integral by Theorem 6 and the incidence vector of an IUC in . The proof is complete. ∎

Given the definition of IUC, it is also natural to consider the correspondence between the IUC polytope and the clique polytope associated with a graph , i.e., the convex hull of the incidence vectors of all cliques in . Patently, the odd-hole inequality for the vertex packing polytope translates to the odd-anti-hole inequality for the clique polytope associated with . An anti-hole is a subset of vertices , such that the (edge) complement of the corresponding induced subgraph is a chordless cycle, i.e., is a hole in the complement graph of . The following theorem states that, under a slightly stronger condition, the same inequality is valid for the IUC polytope associated with and possesses the same facial property on the polytope associated with the corresponding induced subgraph.

Theorem 8.

(Anti-hole Inequality) Let , be an anti-hole in . Then, the inequality

(10)

is valid for . Furthermore, it induces a facet of if and only if is odd.

Proof.

Consider a labeling of the vertices of an anti-hole , such that , and identify the pairs of non-adjacent vertices in . For a fixed sequence of labels, note that , is a hole of cardinality 4, thus the inequality , is valid for . Consider the summation of those valid inequalities. Observe that each vertex appears times, once, and twice, in the left-hand side of the resultant inequality. These partitions are illustrated by black, hatched, and gray vertices in Figure 2, respectively. The white vertices in Figure 2 are those that are not present in this inequality.

Figure 2: The subgraph induced by an anti-hole.

Since the starting vertex of the labeling sequence was arbitrary, valid inequalities of this type may be written for . Considering all those inequalities, every vertex will appear exactly twice in a black position, twice in a white, twice in a hatched, and times in a gray position. As a result, the summation of all 4-hole inequalities associated with results in

Therefore, is valid for , which further implies validity of (10) as the Gomory-Chvátal cut corresponding to this inequality. Clearly, when is even, (10) is a linear combination of the 4-hole inequalities associated with , hence not facet-defining for the corresponding polytope. Thus, it remains to show that it induces a facet of if is odd.

Given an odd anti-hole , consider . Associated with a fixed labeling of as described above, let be the incidence vector of the set of even vertices, i.e., . Observe that is a clique in and , which implies is a proper face of . Suppose that is contained in a supporting hyperplane of . Let and note that is also a clique, hence . Then, implies that . Since the start vertex of the labeling was arbitrary, this results holds for all pairs of vertices with consecutive labels. That is, , and . The proof is complete noting that the interior of is nonempty, thus . ∎

Theorem 8 is closely related to a basic property of IUCs. Ertem et al. [19] showed that, in an arbitrary graph, a (maximal) clique is a maximal IUC if and only if it is a dominating set. In an anti-cycle graph , each maximal clique is maximum, and in case , it is also a maximum IUC by Theorem 8. It is easy to verify that the anti-cycle graphs on 4 and 5 vertices are the only ones in which a maximal clique is not a dominating set.

In the subsequent sections, we will show that the IUC polytope associated with an anti-cycle graph has many other facets than those induced by the 4-hole inequalities and (10), due to the fan substructures that appear in this graph. We postpone our discussion in this regard to Section 3.3.

Finally, it should be mentioned that the odd-hole and odd-anti-hole inequalities for the vertex packing polytope have been generalized by Trotter [50] via introducing a general class of facet-producing subgraphs, called webs, that subsumes cycle and anti-cycle graphs as special cases. Given the connection between the vertex packing and IUC polytopes, webs may lead to further facet-inducing inequalities for the IUC polytope as well. We leave this topic for future research and will not pursue it in this paper.

Lifting hole and anti-hole inequalities

Let , be an induced chordless cycle in , where and . Then by Theorem 3, inequality (7) induces a facet of , but it is not necessarily facet-defining for . Consider lifting a variable , into (7) to generate a facet-defining inequality for of the form . The value of in such an inequality depends on cardinality of as well as distances (lengths of the shortest paths) among the vertices of in . As a case in point, consider the graphs of Figure 3, and lifting into the inequality corresponding to . In both graphs, and the difference is due to the distance between the vertices 4 and 7 (equivalently, 1 and 7) in the left-hand-side graph, and the vertices 4 and 6 (equivalently, 1 and 6) in the right-hand-side graph. The sets of black vertices illustrate the maximum size IUCs containing vertex 8 in these graphs. Clearly, the coefficient of vanishes in lifting for the left-hand-side graph, indicating that this inequality is facet-defining for . On the other hand, for the IUC polytope associated with the right-hand-side graph, this inequality is dominated by the lifted (facet-defining) inequality .

Figure 3: Example graphs for lifting the hole inequality.

Obviously, the structure of in becomes irrelevant at extreme values of . This result can be slightly strengthened by showing that the lengths of the paths among the neighbors of in are redundant when is restricted to 3 and 2 for and , respectively. This leads to the following theorem concerning the lifting procedure of the hole inequality to higher-dimensional spaces.

Theorem 9.

Consider the hole inequality (7) where . In every sequential lifting of (7), the coefficient of a variable , vanishes under the following conditions: (i) if , (ii) if .

Proof.

Note that, in an arbitrary sequential lifting of the hole inequality (7), even if some variables not-satisfying the theorem conditions have already been lifted into (7) before , existence of vertices in forming an IUC with is sufficient for the coefficient of to vanish. Thus, we just need to establish the existence of these vertices under the theorem conditions. Such structures for all possible distributions of neighbors of in are provided in Appendix A. ∎

In Section 3.3, we will discuss implication of this theorem about the facial structure of the IUC polytope associated with a defective wheel graph.

We conclude this section by stating a similar result concerning the anti-hole inequality. By construction of a maximum IUC in an anti-cycle graph, presented it the proof of Theorem 8, it is easy to verify that the coefficient of a variable , vanishes in lifting the odd-anti-hole inequality (10) to higher-dimensional spaces if