Stable Set Polytope Research Articles

A weak box-perfect graph theorem

A graph G is called perfect if ω(H)=χ(H) for every induced subgraph H of G, where ω(H) is the clique number of H and χ(H) its chromatic number. The Weak Perfect Graph Theorem of Lovász states that a graph G is perfect if and only if its complement G‾ is perfect. This does not hold for box-perfect graphs, which are the perfect graphs whose stable set polytope is box-totally dual integral.We prove that both G and G‾ are box-perfect if and only if G‾+ is box-perfect, where G+ is obtained by adding a universal vertex to G. Consequently, G+ is box-perfect if and only if G‾+ is box-perfect. As a corollary, we characterize when the complete join of two graphs is box-perfect.

The total matching polytope of complete bipartite graphs

The total matching polytope generalizes the stable set polytope and the matching polytope. In this paper, we first propose new facet-defining inequalities for the total matching polytope. We then give an exponential-sized, non-redundant description in the original space and a compact description in an extended space of the total matching polytope of complete bipartite graphs.

Stable set reformulations for the degree preserving spanning tree problem

Let G=(V,E) be a connected undirected graph and assume that a spanning tree is available for it. Any vertex in this tree is called degree preserving if it has the same degree in the graph and in the tree. Building upon this concept, the Degree Preserving Spanning Tree Problem (DPSTP) asks for a spanning tree of G with as many degree preserving vertices as possible. DPSTP is very much intertwined with the most important application for it, so far, i.e., the online monitoring of arc flows in a water distribution network, what serves us as an additional motivation for our investigation. We show that degree preserving vertices correspond to a stable set of a properly defined DPSTP conflict graph. This, in turn, allows us to use valid inequalities for the Stable Set Polytope (SSP) and thus strengthen the DPSTP formulations previously suggested in the literature. In doing so, the Branch-and-cut and Benders Decomposition algorithms suggested for these formulations are greatly enhanced. Additionally, as a further contribution, we also introduce a new and very challenging set of DPSTP test instances, given by graphs that closely resemble water distribution networks. For these instances, the new algorithms very clearly outperform all previous DPSTP algorithms. They not only run faster than their competitors but are also less dependent on good initial DPSTP primal bounds. For previous DPSTP test sets, the new algorithms lag behind the best existing algorithm for them, which is based on Lagrangian relaxation. However, as compared to additional DPSTP previous algorithms that do not benefit from SSP inequalities, the new ones come much closer to the Lagrangian relaxation one.

Perfectly matchable set polynomials and h⁎-polynomials for stable set polytopes of complements of graphs

A subset S of vertices of a graph G is called a perfectly matchable set of G if the subgraph induced by S contains a perfect matching. The perfectly matchable set polynomial of G, first made explicit by Ohsugi and Tsuchiya, is the (ordinary) generating function p(G;z) for the number of perfectly matchable sets of G.In this work, we provide explicit recurrences for computing p(G;z) for an arbitrary (simple) graph and use these to compute the Ehrhart h⁎-polynomials for certain lattice polytopes. Namely, we show that p(G;z) is the h⁎-polynomial for certain classes of stable set polytopes, whose vertices correspond to stable sets of G.

Chain algebras of finite distributive lattices

We introduce a family of toric algebras defined by maximal chains of a finite distributive lattice. Applying results on stable set polytopes, we conclude that every such algebra is normal and Cohen–Macaulay, and give an interpretation of its Krull dimension in terms of the combinatorics of the underlying lattice. When the lattice is planar, we show that the corresponding chain algebra is generated by a sortable set of monomials and is isomorphic to a Hibi ring of another finite distributive lattice. As a consequence, it has a defining toric ideal with a quadratic Gröbner basis, and its h-vector counts ascents in certain standard Young tableaux. If instead the lattice has dimension n>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n>2$$\\end{document}, we show that the defining ideal has minimal generators of degree at least n.

Non-Gorenstein Locus and Almost Gorenstein Property of the Ehrhart Ring of the Stable Set Polytope of a Cycle Graph

Non-Gorenstein Locus and Almost Gorenstein Property of the Ehrhart Ring of the Stable Set Polytope of a Cycle Graph

Algorithms for the clique problem with multiple-choice constraints under a series–parallel dependency graph

The clique problem with multiple-choice constraints (CPMC), i.e. the problem of finding a k -clique in a k -partite graph with known partition, occurs as a substructure in many real-world applications, in particular scheduling and railway timetabling. Although CPMC is NP-complete in general, it is known to be solvable in polynomial time when the so-called dependency graph of G is a forest. In this article, we focus on the special case CPMCSP, where the dependency graph of G is series–parallel. We give a polynomial-time algorithm for CPMCSP using dynamic programming. Further, we provide some facet-inducing inequalities of the CPMCSP polytope, mainly using properties of the stable set polytope of the complement graph of G . Among these, we give a separation algorithm for the so-called embedded odd-clique-cycle inequalities using dynamic programming. If the number of vertices per subset of the k -partition is bounded, then its runtime is polynomial in the size of the dependency graph.

Perfectly contractile graphs and quadratic toric rings

AbstractPerfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Seymour, and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs, which was conjectured by Berge. We consider the class of graphs that have no odd holes, no antiholes, and no odd stretchers as induced subgraphs. In particular, every graph belonging to is perfect. Everett and Reed conjectured that a graph belongs to if and only if it is perfectly contractile. In the present paper, we discuss graphs belonging to from a viewpoint of commutative algebra. In fact, we conjecture that a perfect graph belongs to if and only if the toric ideal of the stable set polytope of is generated by quadratic binomials. Especially, we show that this conjecture is true for Meyniel graphs, perfectly orderable graphs, and clique separable graphs, which are perfectly contractile graphs.

Polyhedral results and stronger Lagrangean bounds for stable spanning trees

Given a graph G=(V,E) and a set C of unordered pairs of edges regarded as being in conflict, a stable spanning tree in G is a set of edges T inducing a spanning tree in G, such that for each left{ e_i, e_j right} in C, at most one of the edges e_i and e_j is in T. The existing work on Lagrangean algorithms to the NP-hard problem of finding minimum weight stable spanning trees is limited to relaxations with the integrality property. We exploit a new relaxation of this problem: fixed cardinality stable sets in the underlying conflict graph H =(E,C). We find interesting properties of the corresponding polytope, and determine stronger dual bounds in a Lagrangean decomposition framework, optimizing over the spanning tree polytope of G and the fixed cardinality stable set polytope of H in the subproblems. This is equivalent to dualizing exponentially many subtour elimination constraints, while limiting the number of multipliers in the dual problem to |E|. It is also a proof of concept for combining Lagrangean relaxation with the power of integer programming solvers over strongly NP-hard subproblems. We present encouraging computational results using a dual method that comprises the Volume Algorithm, initialized with multipliers determined by Lagrangean dual-ascent. In particular, the bound is within 5.5% of the optimum in 146 out of 200 benchmark instances; it actually matches the optimum in 75 cases. All of the implementation is made available in a free, open-source repository.

Three families of toric rings arising from posets or graphs with small class groups

The main objects of the present paper are (i) Hibi rings (toric rings arising from order polytopes of posets), (ii) stable set rings (toric rings arising from stable set polytopes of perfect graphs), and (iii) edge rings (toric rings arising from edge polytopes of graphs satisfying the odd cycle condition). The goal of the present paper is to analyze those three toric rings and to discuss their structures in the case where their class groups have small rank. We prove that the class groups of (i), (ii) and (iii) are torsionfree. More precisely, we give descriptions of their class groups. Moreover, we characterize the posets or graphs whose associated toric rings have rank 1 or 2. By using those characterizations, we discuss the differences of isomorphic classes of those toric rings with small class groups.

Slack matrices, [formula omitted]-products, and 2-level polytopes

In this paper, we study algorithmic questions concerning products of matrices and their consequences for recognition algorithms for polyhedra.The 1-product of matrices S1∈Rm1×n1,S2∈Rm2×n2 is a matrix in R(m1+m2)×(n1n2) whose columns are the concatenation of each column of S1 with each column of S2. The k-product generalizes the 1-product, by taking as input two matrices S1,S2 together with k−1 special rows of each of those matrices, and outputting a certain composition of S1,S2. Our first result is a polynomial time algorithm for the following problem: given a matrix S, is S a k-product of some matrices, up to permutation of rows and columns? Our algorithm is based on minimizing a symmetric submodular function that expresses mutual information from information theory.Our study is motivated by a close link between the 1-product of matrices and the Cartesian product of polytopes, and more generally between the k-product of matrices and the glued product of polytopes. These connections rely on the concept of a slack matrix, which gives an algebraic representation of classes of affinely equivalent polytopes. The slack matrix recognition problem is the problem of determining whether a given matrix is a slack matrix. This is an intriguing problem whose complexity is unknown. Our algorithm reduces the problem to instances which cannot be expressed as k-products of smaller matrices.In the second part of the paper, we give a combinatorial interpretation of k-products for two well-known classes of polytopes: 2-level matroid base polytopes and stable set polytopes of perfect graphs. We also show that the slack matrix recognition problem is polynomial-time solvable for such polytopes. Those two classes are special cases of 2-level polytopes, for which we conjecture that the slack matrix recognition problem is polynomial-time solvable.

Combinatorial differential algebra of xp

We link n-jets of the affine monomial scheme defined by xp to the stable set polytope of some perfect graph. We prove that, as p varies, the dimension of the coordinate ring of a certain subscheme of the scheme of n-jets as a C-vector space is a polynomial of degree n+1, namely the Ehrhart polynomial of the stable set polytope of that graph. One main ingredient for our proof is a result of Zobnin who determined a differential Gröbner basis of the differential ideal generated by xp. We generalize Zobnin's result to the bivariate case. We study (m,n)-jets, a higher-dimensional analog of jets, and relate them to regular unimodular triangulations.

On the facets of stable set polytopes of circular interval graphs

A linear description of the stable set polytope STAB(G) of a quasi-line graph G is given in Eisenbrand et al. (Combinatorica 28(1):45–67, 2008), where the so called Ben Rebea Theorem (Oriolo in Discrete Appl Math 132(3):185–201, 2003) is proved. Such a theorem establishes that, for quasi-line graphs, STAB(G) is completely described by non-negativity constraints, clique inequalities, and clique family inequalities (CFIs). As quasi-line graphs are a superclass of line graphs, Ben Rebea Theorem can be seen as a generalization of Edmonds’ characterization of the matching polytope (Edmonds in J Res Natl Bureau Stand B 69:125–130, 1965), showing that the matching polytope can be described by non-negativity constraints, degree constraints and odd-set inequalities. Unfortunately, the description given by the Ben Rebea Theorem is not minimal, i.e., it is not known which are the (non-rank) clique family inequalities that are facet defining for STAB(G). To the contrary, it would be highly desirable to have a minimal description of STAB(G), pairing that of Edmonds and Pulleyblank (in: Berge, Chuadhuri (eds) Hypergraph seminar, pp 214–242, 1974) for the matching polytope. In this paper, we start the investigation of a minimal linear description for the stable set polytope of quasi-line graphs. We focus on circular interval graphs, a subclass of quasi-line graphs that is central in the proof of the Ben Rebea Theorem. For this class of graphs, we move an important step forward, showing some strong sufficient conditions for a CFI to induce a facet of STAB(G). In particular, such conditions come out to be related to the existence of certain proper circulant graphs as subgraphs of G. These results allows us to settle two conjectures on the structure of facet defining inequalities of the stable set polytope of circulant graphs (Pêcher and Wagler in Math Program 105:311–328, 2006) and of (fuzzy) circular graphs (Oriolo and Stauffer in Math Program 115:291–317, 2008), and to slightly refine the Ben Rebea Theorem itself.

Total Coloring and Total Matching: Polyhedra and Facets

A total coloring of a graph G=(V,E) is an assignment of colors to vertices and edges such that neither two adjacent vertices nor two incident edges get the same color, and, for each edge, the end-points and the edge itself receive different colors. Any valid total coloring induces a partition of the elements of G into total matchings, which are defined as subsets of vertices and edges that can take the same color. In this paper, we propose Integer Linear Programming models for both the Total Coloring and the Total Matching problems, and we study the strength of the corresponding Linear Programming relaxations. The total coloring is formulated as the problem of finding the minimum number of total matchings that cover all the graph elements. This covering formulation can be solved by a Column Generation algorithm, where the pricing subproblem corresponds to the Weighted Total Matching Problem. Hence, we study the Total Matching Polytope. We introduce three families of nontrivial valid inequalities: vertex-clique inequalities based on standard clique inequalities of the Stable Set Polytope, congruent-2k3 cycle inequalities based on the parity of the vertex set induced by the cycle, and even-clique inequalities induced by complete subgraphs of even order. We prove that congruent-2k3 cycle inequalities are facet-defining only when k=4, while the vertex-clique and even-cliques are always facet-defining. Finally, we present preliminary computational results of a Column Generation algorithm for the Total Coloring Problem and a Cutting Plane algorithm for the Total Matching Problem.

Improved formulations and branch-and-cut algorithms for the angular constrained minimum spanning tree problem

The Angular Constrained Minimum Spanning Tree Problem (\(\alpha \)-MSTP) is defined in terms of a complete undirected graph \(G=(V,E)\) and an angle \(\alpha \in (0,2\pi ]\). Vertices of G define points in the Euclidean plane while edges, the line segments connecting them, are weighted by the Euclidean distance between their endpoints. A spanning tree is an \(\alpha \)-spanning tree (\(\alpha \)-ST) of G if, for any \(i \in V\), the smallest angle that encloses all line segments corresponding to its i-incident edges does not exceed \(\alpha \). \(\alpha \)-MSTP consists in finding an \(\alpha \)-ST with the least weight. In this work, we discuss families of \(\alpha \)-MSTP valid inequalities. One of them is a lifting of existing angular constraints found in the literature and the others come from the Stable Set polytope, a structure behind \(\alpha \)-STs disclosed here. We show that despite being already satisfied by the previously strongest known formulation, \({\mathcal {F}}_{xy}\), these lifted angular constraints are capable of strengthening another existing \(\alpha \)-MSTP model so that both become equally strong, at least for the instances tested here. Inequalities from the Stable Set polytope improve the best known Linear Programming Relaxation (LPRs) bounds by about 1.6%, on average, for the hardest instances of the problem. Additionally, we indicate how formulation \({\mathcal {F}}_{xy}\) can be more effectively used in Branch-and-cut (BC) algorithms, by reducing the number of variables explicitly enforced to be integer constrained and by eliminating constraints that do not change the quality of its LPR bounds. Extensive computational experiments conducted here suggest that the combination of the ideas above allows us to redefine the best performing \(\alpha \)-MSTP algorithms, for almost the entire spectrum of \(\alpha \) values, the exception being the easy instances, those with \(\alpha \ge \frac{2\pi }{3}\). In particular, for the hardest ones (corresponding to \(\alpha \in \{\frac{\pi }{2}, \frac{\pi }{3},\frac{2\pi }{5}\}\)) that could be solved to proven optimality, the best BC algorithm suggested here improves on average CPU times by factors of up to 5, on average.

ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH

In this paper, we give a criterion of the Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ is Gorenstein if and only if (1) sizes of maximal cliques are constant (say $n$) and (2) (a) $n=1$, (b) $n=2$ and there is no odd cycle without chord and length at least 7 or (c) $n\geq 3$ and there is no odd cycle without chord and length at least 5.

Strengthened clique-family inequalities for the stable set polytope

The stable set polytope is a fundamental object in combinatorial optimization. Among the many valid inequalities that are known for it, the clique-family inequalities play an important role. Pêcher and Wagler showed that the clique-family inequalities can be strengthened under certain conditions. We show that they can be strengthened even further, using a surprisingly simple mixed-integer rounding argument.

Extended formulations for stable set polytopes of graphs without two disjoint odd cycles

Let $G$ be an $n$-node graph without two disjoint odd cycles. The algorithm of Artmann, Weismantel and Zenklusen (STOC'17) for bimodular integer programs can be used to find a maximum weight stable set in $G$ in strongly polynomial time. Building on structural results characterizing sufficiently connected graphs without two disjoint odd cycles, we construct a size-$O(n^2)$ extended formulation for the stable set polytope of $G$.

Bounds on the Number of 2-Level Polytopes, Cones, and Configurations

We prove an upper bound of the form \(2^{O(d^2 \mathop {\mathrm {polylog}}d)}\) on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones, and d-configurations. This in particular answers positively a conjecture of Bohn et al. on 2-level polytopes. We obtain our upper bound by relating affine (resp. linear) equivalence classes of 2-level d-polytopes, d-cones, and d-configurations to faces of the correlation cone. We complement this with a \(2^{\varOmega (d^2)}\) lower bound, by estimating the number of nonequivalent stable set polytopes of bipartite graphs.

Persistency of linear programming relaxations for the stable set problem

The Nemhauser–Trotter theorem states that the standard linear programming (LP) formulation for the stable set problem has a remarkable property, also known as (weak) persistency: for every optimal LP solution that assigns integer values to some variables, there exists an optimal integer solution in which these variables retain the same values. While the standard LP is defined by only non-negativity and edge constraints, a variety of other LP formulations have been studied and one may wonder whether any of them has this property as well. We show that any other formulation that satisfies mild conditions cannot have the persistency property on all graphs, unless it is always equal to the stable set polytope.