On dominating set polyhedra of circular interval graphs

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Abstract

Clique-node and closed neighborhood matrices of circular interval graphs are circular matrices. The stable set polytope and the dominating set polytope on these graphs are therefore closely related to the set packing polytope and the set covering polyhedron on circular matrices. Eisenbrand et al. (2008) take advantage of this relationship to propose a complete linear description of the stable set polytope on circular interval graphs. In this paper we follow similar ideas to obtain a complete description of the dominating set polytope on the same class of graphs. As in the packing case, our results are established for a larger class of covering polyhedra of the form Q(A,b)convxZ+n:Axb, with A a circular matrix and b an integer vector. These results also provide linear descriptions of polyhedra associated with several variants of the dominating set problem on circular interval graphs.

Introduction

The well-known concept of domination in graphs was introduced by Berge [6], modeling many facility location problems in Operations Research. Given a graph G=(V,E), N[v] denotes the closed neighborhood of the node vV. A set DV is called a dominating set of G if DN[v] holds for every vV. Given a vector wRV of node weights, the Minimum-Weighted Dominating Set Problem (MWDSP for short) consists in finding a dominating set D of G that minimizes vDwv. MWDSP arises in many applications, involving the strategic placement of resources on the nodes of a network. As example, consider a computer network in which one wishes to choose a smallest set of computers that are able to transmit messages to all the remaining computers [22]. Many other interesting examples include sets of representatives, school bus routing, (r,d)-configurations, placement of radio stations, social network theory, kernels of games, etc. [21].

The MWDSP is NP-hard for general graphs and has been extensively investigated from an algorithmic point of view (see, e.g., [7], [11], [16], [19]). In particular, efficient algorithms for the problem on interval and circular arc graphs are proposed in [12].

However, only a few results about the MWDSP have been established from a polyhedral point of view. The dominating set polytope associated with a graph G is defined as the convex hull of all incidence vectors of dominating sets in G. In [10] the authors provide a complete description of the dominating set polytope of cycles. As a generalization, a description of the dominating set polytope associated with web graphs of the form Ws(2k+1)+tk, with 2s3, 0ts1, and kN, is presented in [8].

In fact, the MWDSP can be regarded as a particular case of the Minimum-Weighted Set Covering Problem (MWSCP). Given a {0,1}-matrix A of order m×n, a cover of A is a vector x{0,1}n such that Ax1, where 1Zm is the vector having all entries equal to one. The MWSCP consists in finding a cover of minimum weight with respect to a given a weight vector wRn. This problem can be formulated as the integer linear program minwTx:Ax1,xZ+n.The set Q(A)convxZ+n:Ax1 is termed as the set covering polyhedron associated with A.

The closed neighborhood matrix of a graph G is the square matrix N[G] whose rows are the incidence vectors of the sets N[v], for all vV. Observe that x is the incidence vector of a dominating set of G if and only if x is a cover of N[G]. Therefore, solving the MWSCP on N[G] is equivalent to solving the MWDSP on G. Moreover, the structure of the dominating set polytope of G can be studied by considering the set covering polyhedron associated with N[G].

The closed neighborhood matrix of a web graph is a circulant matrix. More generally, the closed neighborhood matrix of a circular interval graph is a circular matrix (both terms are explained in more detail in the next section). In this paper we are interested in studying the dominating set polytopes associated with circular interval graphs.

Another classic set optimization problem is the Set Packing Problem: given a {0,1}-matrix A of order m×n, a packing of A is a vector x{0,1}n such that Ax1. For a weight vector wRn, the Maximum-Weighted Set Packing Problem (MWSPP) can be stated as the integer linear program maxwTx:x{0,1}n,Ax1.The polytope P(A)convxZ+n:Ax1 is the set packing polytope associated with A.

Set packing polyhedra have been extensively studied because of their relationship with the stable set polytope. Indeed, given a graph G, a matrix A can be defined whose rows are incidence vectors of the maximal cliques in G. Conversely, given an arbitrary {0,1}-matrix A, the conflict graph G of A is defined as a graph having one node for each column of A and two nodes joined by an edge whenever the respective columns have scalar product distinct from zero. In both cases, stable sets in G correspond to packings of A.

In [18], [27] the authors present a complete linear description of the stable set polytope of circular interval graphs, which is equivalent to obtaining a complete linear description for the set packing polytope related to circular matrices. The authors show that if A is a circular matrix then P(A) is completely described by three classes of inequalities: non-negativity constraints, clique inequalities, and clique family inequalities introduced in [24]. Moreover, facet inducing clique family inequalities are associated with subwebs of the circular interval graph [27].

In fact, their results are stated for a more general packing polyhedron P(A,b), defined as the convex hull of non-negative integer solutions of the system Axb, with bZ+m and A a circular matrix. In the covering case, a similar polyhedron Q(A,b) can be defined as the convex hull of the integer points in Q(A,b)={xR+n:Axb}, with bZ+m. When A is the closed neighborhood matrix of a graph, the extreme points of Q(A,b) correspond to some variants of dominating sets in graphs. In particular, if b=k1, they correspond to {k}- dominating functions [4] and, in the general case, they are related to L-dominating functions [23]. Considering the symmetry in the definition of P(A,b) and Q(A,b), it is natural to ask if the ideas proposed in [18], [27] can be applied in the covering context.

In this paper we present a complete linear description of Q(A,b) for any circular matrix A and any vector bZ+n. This yields a complete description of the polyhedron associated with L-dominating functions of circular interval graphs. The linear inequalities have a particular structure when b=k1, which includes the case of {k}-dominating functions. Finally, if k=1, facet defining inequalities of Q(A) provide a characterization of facets of the dominating set polytope on circular interval graphs. These inequalities are related to circulant minors of A.

In the light of previous results obtained by Chudnovsky and Seymour [13], the linear description presented in [18], [27] actually provided the final piece for establishing a complete linear description of the stable set polytope for the much broader class of quasi-line graphs. The fact that the dominating set problem is known to be NP-hard already for the particular subclass of line-graphs [30], discourages seeking for an analogous result regarding domination on quasi-line graphs. Nonetheless, we present here some positive results for a prominent subclass of them.

Some results presented in this paper appeared without proofs in [29].

Section snippets

Preliminaries

A circular-arc graph is the intersection graph of a set of arcs on the circle, i.e., G=(V,E) is a circular-arc graph if each node vV can be associated with an arc C(v) on the circle in such a way that uvE if and only C(u) intersects C(v). If additionally the family C(v):vV can be defined in such a way that no arc properly contains another, then G is a proper circular-arc graph. Proper circular-arc graphs are also termed as circular interval graphs in [14] and defined in a different, but

Following the ideas of the packing case

As we have mentioned, the study of the covering polyhedra of circular matrices closely follows the ideas proposed in [18], [27] for the corresponding packing polytopes. Some of these ideas can be straightforwardly translated to the covering case. In this section we review them, including the corresponding proofs for the sake of completeness.

Given a circular matrix A, bZ+m and βN, the slice of Q(A,b) defined by β is the polyhedron: Qβ(A,b)Q(A,b)xRn:1Tx=β.Remind that Ã=AI and M is the

A complete linear description of Q(A,b)

Consider a circular matrix A and the directed graph D(A) defined in Section 2. Recall from Definition 2.1 that for any arc a in D(A), l(a) denotes its (oriented) length.

Given a closed directed (not necessarily simple) path Γ=(V(Γ),E(Γ)) in D(A), its winding number p(Γ) is defined by: p(Γ)=1naE(Γ)l(a).Observe that p(Γ) is an integer, as Γ is a closed path in D(A) and hence aE(Γ)l(a) is a multiple of n.

For every i[m], let Pi+ (resp. Pi) be the path of short forward (resp. reverse) arcs in D(

The case of homogeneous right-hand side

In this section, we consider polyhedra Q(A,b) for a circular matrix A and b=α1 with αN. Observe that for these class of polyhedra dominating rows of the matrix A are associated with redundant constraints. For this reason, in the remaining of this article we always assume that A has no dominating rows.

We will prove that in this case, relevant circuit inequalities are induced by circuits in D(A) without reverse row arcs.

Remind that, for every i[m], Pi+ (resp. Pi) is the path of short forward

Circuits without reverse row arcs and their inequalities

Given a circular matrix A, let us call F(A) the digraph with nodes in [n] and all arcs in D(A) except for reverse row arcs. We are interested in the circuits in F(A) whose Γ- inequalities are relevant in the description of Q(A,α1). Therefore, from now on, circuits Γ in F(A) verify p(Γ)2 (see Theorem 5.3).

Let Γ be a circuit in F(A). Keeping the same notation introduced in [27], we consider the partition of the nodes of F(A) into the following three classes:

  • (i)

    circles (Γ)j[n]:(j1,j)E(Γ),

  • (ii)

Set covering polyhedron of circular matrices and circulant minors

Throughout this section, we restrict our attention to the set covering polyhedron Q(A)=Q(A,1) of circular matrices A. Recall that we are considering matrices without dominating rows.

Given a circuit Γ in F(A), we say that a bullet is an essential bullet if it is reached by Γ. We will see that essential bullets induce a partition of V(Γ) into blocks.

Let Γ be a circuit of F(A) with winding number p and s essential bullets {bj:j=1,,s}, with 1b1<b2<<bsn.

For j{1,,s}, define the block Bj as

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We thank Gianpaolo Oriolo and Gautier Stauffer for earlier discussions that motivated our work on this topic.

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    Partially supported by PIP-CONICET 277, PID-UNR 416, PICT-ANPCyT 0586, EPN PIJ-16-06, and MathAmSud 15MATH06 PACK-COVER .

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