We consider k-regular graphs with loops, and study the Lovász theta-numbers and Schrijver theta’-numbers of the graphs that result when the loop edges are removed. We show that the theta-number dominates a recent eigenvalue upper bound on the stability number due to Godsil and Newman [C.D. Godsil and M.W. Newman. Eigenvalue bounds for independent sets. … Read more
Adding cuts based on copositive matrices, we propose to improve Lovász’ bound on the clique number and its tightening introduced by McEliece, Rodemich, Rumsey, and Schrijver. Candidates for cheap and efficient copositivity cuts of this type are obtained from graphs with known clique number. The cost of previously established semidefinite programming bound hierarchies rapidly increases … Read more
Perfect graphs constitute a well-studied graph class with a rich structure, reflected by many characterizations w.r.t different concepts. Perfect graphs are, e.g., characterized as precisely those graphs G where the stable set polytope STAB(G) coincides with the clique constraint stable set polytope QSTAB(G). For all imperfect graphs STAB(G) \subset QSTAB(G) holds and, therefore, it is … Read more
The Lovasz theta number of a graph G can be viewed as a semidefinite programming relaxation of the stability number of G. It has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of G. We introduce a related strengthening of the Lovasz theta number toward the … Read more
Let $G=(V,E)$ be a simple graph on $n$ nodes. We show how to construct a partial subgraph $D$ of the line graph of $G$ satisfying the identity: $\overline \chi(G)+\alpha(D)=n$, where $\overline \chi(G)$ denotes the minimum number of cliques in a clique partition of $G$ and $\alpha(D)$ denotes the maximum size of a stable set of … Read more
The paper describes various combinatorial and algorithmic applications of hyperbolic (multivariate) polynomials . Section 2.2 introduces a new class of polynomials , which include as hyperbolic polynomials as well volume polynomials $Vol(x_1C_1+…+x_nC_n)$ , where $C_i$ are convex compact subsets of $R^n$. This extension leads to randomized poly-time algorithm to approximate $M(C_1,…,C_n)$ (the mixed volume) within … Read more
We consider two (0,1)-linear programming formulations of the graph (vertex-)coloring problem, in which variables are associated to stable sets of the input graph. The first one is a set covering formulation, where the set of vertices has to be covered by a minimum number of stable sets. The second is a set packing formulation, in … Read more
Perfect graphs constitute a well-studied graph class with a rich structure, reflected by many characterizations with respect to different concepts. Perfect graphs are, for instance, precisely those graphs $G$ where the stable set polytope $STAB(G)$ coincides with the fractional stable set polytope $QSTAB(G)$. For all imperfect graphs $G$ it holds that $STAB(G) \subset QSTAB(G)$. It … Read more
We consider the problem of interconnecting a set of customer sites using SONET rings of equal capacity, which can be defined as follows: Given an undirected graph G=(V,E) with nonnegative edge weight d(u,v), (u,v) in E, and two integers K and B, find a partition of the nodes of G into K subsets so that … Read more
The k-traveling salesman problem, or k-TSP is: given a graph with edge weights and an integer k, find a simple cycle of minimum weight visiting exactly k nodes. To obtain lower bounds for the traveling salesman problem the 2-matching relaxation and the 1-tree relaxation can be used. We generalize these two relaxations for the k-TSP. … Read more