Optimizing over the Closure of Rank Inequalities with a Small Right-Hand Side for the Maximum Stable Set Problem via Bilevel Programming

In the context of the maximum stable set problem, rank inequalities impose that the cardinality of any set of vertices contained in a stable set be, at most, as large as the stability number of the subgraph induced by such a set. Rank inequalities are very general, as they subsume many classical inequalities such as clique, hole, antihole, web, and antiweb inequalities. In spite of their generality, the exact separation of rank inequalities has never been addressed without the introduction of topological restrictions on the induced subgraph and the tightness of their closure has never been investigated systematically. In this work, we propose a methodology for optimizing over the closure of all rank inequalities with a right-hand side no larger than a small constant without imposing any restrictions on the topology of the induced subgraph. Our method relies on the exact separation of a relaxation of rank inequalities, which we call relaxed k-rank inequalities, whose closure is as tight. We investigate the corresponding separation problem, a bilevel programming problem asking for a subgraph of maximum weight with a bound on its stability number, whose study could be of independent interest. We first prove that the problem is [Formula: see text]-hard and provide some insights on its polyhedral structure. We then propose two exact methods for its solution: a branch-and-cut algorithm (which relies on a family of faced-defining inequalities which we introduce in this paper) and a purely combinatorial branch-and-bound algorithm. Our computational results show that the closure of rank inequalities with a right-hand side no larger than a small constant can yield a bound that is stronger, in some cases, than Lovász’s Theta function, and substantially stronger than bounds obtained with standard inequalities that are valid for the stable set problem, including odd-cycle inequalities and wheel inequalities. Summary of Contribution: This paper proposes two original methods for solving a challenging cut-separation problem (of bilevel type) for a large class of inequalities valid for one of the key operations research problems, namely, the max stable set problem. An extensive set of experimental results validates the proposed methods. All the source code and data sets are available online on GitHub.

QoS routing in cluster OLSR by using the artificial intelligence model MSSP in the big data environnment

<p><span id="docs-internal-guid-75ba661c-7fff-7cfb-9afe-b7c901c3fe82"><span>The most complex problems, in data science and more specifically in artificial intelligence, can be modeled as cases of the maximum stable set problem (MSSP). this article describes a new approach to solve the MSSP problem by proposing the continuous hopfield network (CHN) to build optimized link state protocol routing (OLSR) protocol cluster. our approach consists in proposing in two stages: the first acts at the level of the choice of the OLSR master cluster in order to quickly make a local minimum using the CHN, by modeling the MSSP problem. As for the second step, the objective is the improvement of the precision making a solution of efficient at the first rank of neighborhood as a linear constraint, and at the end, to find the resolution of the model using the CHN. We will show that this model determines a good solution of the MSSP problem. To test the theoretical results, we propose a comparison with a classic OLSR.</span></span></p>

Polyhedral approximations of the semidefinite cone and their application

We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (Ann Oper Res 265:155–182, 2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

Diperfect Digraphs

Let D be a digraph. A path partition P of D is a collection of paths such that {V (P ) : P 2 P } is a partition of V (D). We say D is ↵ -diperfect if for every maximum stable set S of D there exists a path partition P of D such that |S \ V (P )| = 1 for all P 2 P and this property holds for every induced subdigraph of D. A digraph C is an anti-directed odd cycle if (i) the underlying graph of C is a cycle x1x2 · · · x2k+1x1, where k 2, (ii) the longest path in C has length 2, and (iii) each of the vertices x1, x2, x3, x4, x6, x8, . . . , x2k is either a source or a sink. Berge (1982) conjectured that a digraph D is ↵ -diperfect if, and only if, D contains no induced anti-directed odd cycle. In this work, we verify this conjecture for digraphs whose underlying graph is series-parallel and for in-semicomplete digraphs.

A Set and Collection Lemma

A set $S\subseteq V(G)$ is independent if no two vertices from $S$ are adjacent. Let $\alpha\left( G\right) $ stand for the cardinality of a largest independent set.In this paper we prove that if $\Lambda$ is a nonempty collection of maximum independent sets of a graph $G$, and $S$ is an independent set, thenthere is a matching from $S-\bigcap\Lambda$ into $\bigcup\Lambda-S$, and$\left\vert S\right\vert +\alpha(G)\leq\left\vert \bigcap\Lambda\cap S\right\vert +\left\vert \bigcup\Lambda\cup S\right\vert $.Based on these findings we provide alternative proofs for a number of well-known lemmata, such as the "Maximum Stable Set Lemma" due to Claude Berge and the "Clique Collection Lemma" due to András Hajnal.

Greedoids on vertex sets of B-joins of graphs

Let Ψ(G) be the family of all local maximum stable sets of graph G, i.e., S ∈ Ψ(G) if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. It was shown that Ψ(G) is a greedoid for every forest G [15]. The cases of bipartite graphs, triangle-free graphs, and well-covered graphs, were analyzed in [16, 17, 18, 19, 20, 24]. If G1, G2 are two disjoint graphs, and B is a bipartite graph having E(B) as an edge set and bipartition {V (G1), V (G2)}, then by B-join of G1, G2 we mean the graph B (G1, G2) whose vertex set is V (G1) ∪ V (G2) and edge set is E(G1) ∪ E(G2) ∪ E (B). In this paper we present several necessary and sufficient conditions for Ψ(B (G1, G2)) to form a greedoid, an antimatroid, and a matroid, in terms of Ψ(G1), Ψ(G2) and E (B).