On dominating set polyhedra of circular interval graphs

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. [18] 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) := conv {x ∈ Z n + : Ax ≥ b}, 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.Fil: Bianchi, Silvia María. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; ArgentinaFil: Nasini, Graciela Leonor. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Tolomei, Paola Beatriz. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Torres, Luis Miguel. Escuela Politécnica Nacional; Ecuado

Polyhedral Combinatorics, Complexity & Algorithms for k-Clubs in Graphs

A k-club is a distance-based graph-theoretic generalization of clique, originally introduced to model cohesive subgroups in social network analysis. The k-clubs represent low diameter clusters in graphs and are suitable for various graph-based data mining applications. Unlike cliques, the k-club model is nonhereditary, meaning every subset of a k-club is not necessarily a k-club. This imposes significant challenges in developing theory and algorithms for optimization problems associated with k-clubs.We settle an open problem establishing the intractability of testing inclusion-wise maximality of k-clubs for fixed k>=2. This result is in contrast to polynomial-time verifiability of maximal cliques, and is a direct consequence of k-clubs' nonhereditary nature. A class of graphs for which this problem is polynomial-time solvable is also identified. We propose a distance coloring based upper-bounding scheme and a bounded enumeration based lower-bounding routine and employ them in a combinatorial branch-and-bound algorithm for finding a maximum k-club. Computational results on graphs with up to 200 vertices are also provided.The 2-club polytope of a graph is studied and a new family of facet inducing inequalities for this polytope is discovered. This family of facets strictly contains all known nontrivial facets of the 2-club polytope as special cases, and identifies previously unknown facets of this polytope. The separation complexity of these newly discovered facets is proved to be NP-complete and it is shown that the 2-club polytope of trees can be completely described by the collection of these facets along with the nonnegativity constraints.We also studied the maximum 2-club problem under uncertainty. Given a random graph subject to probabilistic edge failures, we are interested in finding a large "risk-averse" 2-club. Here, risk-aversion is achieved via modeling the loss in 2-club property due to edge failures, as random loss, which is a function of the decision variables and uncertain parameters. Conditional Value-at-Risk (CVaR) is used as a quantitative measure of risk that is constrained in the model. Benders' decomposition scheme is utilized to develop a new decomposition algorithm for solving the CVaR constrainedmaximum 2-club problem. A preliminary experiment is also conducted to compare the computational performance of the developed algorithm with our extension of an existing algorithm from the literature.Industrial Engineering & Managemen