Fixed cardinality stable sets

Abstract Given an undirected graph G = ( V , E ) and a positive integer k ∈ 1 , … , | V | , we initiate the combinatorial study of stable sets of cardinality exactly k in G . Our aim is to instigate the polyhedral investigation of the convex hull of fixed cardinality stable sets, inspired by the rich theory on the classical structure of stable sets. We introduce a large class of valid inequalities to the natural integer programming formulation of the problem. We also present simple combinatorial relaxations based on computing maximum weighted matchings, which yield dual bounds towards finding minimum-weight fixed cardinality stable sets, and particular cases which are solvable in polynomial time.

[1]  Gerhard J. Woeginger,et al.  Paths, trees and matchings under disjunctive constraints , 2011, Discret. Appl. Math..

[2]  Philippe G. H. Lehot.  An Optimal Algorithm to Detect a Line Graph and Output Its Root Graph , 1974, JACM.

[3]  John N. Tsitsiklis,et al.  Introduction to linear optimization , 1997, Athena scientific optimization and computation series.

[4]  Egon Balas,et al.  Finding a Maximum Clique in an Arbitrary Graph , 1986, SIAM J. Comput..

[5]  The Unsuitable Neighbourhood Inequalities for the Fixed Cardinality Stable Set Polytope , 2020 .

[6]  Francesco Maffioli,et al.  An annotated bibliography of combinatorial optimization problems with fixed cardinality constraints , 2006, Discret. Appl. Math..

[7]  Dimitris Bertsimas,et al.  Optimization over integers , 2005 .

[8]  Jeannette Janssen,et al.  Bounded Stable Sets: Polytopes and Colorings , 1999, SIAM J. Discret. Math..

[9]  F. McMorris,et al.  Topics in Intersection Graph Theory , 1987 .

[10]  Sebastián Urrutia,et al.  The matching relaxation for a class of generalized set partitioning problems , 2019, Discret. Appl. Math..

[11]  Ola Svensson,et al.  No Small Linear Program Approximates Vertex Cover within a Factor 2-ε , 2018, Math. Oper. Res..

[12]  Edward A. Bender,et al.  Almost all chordal graphs split , 1985, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[13]  Carlo Mannino,et al.  The stable set problem and the thinness of a graph , 2007, Oper. Res. Lett..

[14]  M. Guignard.  Lagrangean relaxation , 2003 .

[15]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

[16]  H. Whitney.  Congruent Graphs and the Connectivity of Graphs , 1932 .

[17]  Leslie E. Trotter,et al.  Properties of vertex packing and independence system polyhedra , 1974, Math. Program..

[18]  Gerhard Reinelt,et al.  A tutorial on branch and cut algorithms for the maximum stable set problem , 2012, Int. Trans. Oper. Res..

[19]  Konrad Dabrowski,et al.  Parameterized complexity of the weighted independent set problem beyond graphs of bounded clique number , 2012, J. Discrete Algorithms.

[20]  Carlo Mannino,et al.  Edge projection and the maximum cardinality stable set problem , 1993, Cliques, Coloring, and Satisfiability.

[21]  Konrad Dabrowski,et al.  Parameterized Algorithms for the Independent Set Problem in Some Hereditary Graph Classes , 2010, IWOCA.

[22]  Abilio Lucena.  Non Delayed Relax-and-Cut Algorithms , 2005, Ann. Oper. Res..

[23]  Hans Raj Tiwary,et al.  Parameterized Extension Complexity of Independent Set and Related Problems , 2018, Discret. Appl. Math..

[24]  Ulrich Pferschy,et al.  Determining a Minimum Spanning Tree with Disjunctive Constraints , 2009, ADT.

[25]  Raca Todosijevic,et al.  Mathematical programming based heuristics for the 0–1 MIP: a survey , 2017, Journal of Heuristics.

[26]  Quentin Botton.  Survivable network design with quality of service constraints : extended formulations and Benders decomposition , 2010 .

[27]  Ola Svensson,et al.  No Small Linear Program Approximates Vertex Cover within a Factor 2 -- e , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.