Complexity results for equistable graphs and related classes

The class of equistable graphs is defined by the existence of a cost structure on the vertices such that the maximal stable sets are characterized by their costs. This graph class, not contained in any nontrivial hereditary class, has so far been studied mostly from a structural point of view; characterizations and polynomial time recognition algorithms have been obtained for special cases. We focus on complexity issues for equistable graphs and related classes. We describe a simple pseudo-polynomial-time dynamic programming algorithm to solve the maximum weight stable set problem along with the weighted independent domination problem in some classes of graphs, including equistable graphs. Our results are obtained within the wider context of Boolean optimization; corresponding hardness results are also provided. More specifically, we show that the above problems are APX-hard for equistable graphs and that it is co-NP-complete to determine whether a given cost function on the vertices of a graph defines an equistable cost structure of that graph. Part of the work was done while M. Milanič was at the Universität Bielefeld. Support by the group “Combinatorial Search Algorithms in Bioinformatics” funded by the Sofja Kovalevskaja Award 2004 of the Alexander von Humboldt Stiftung and the German Federal Ministry of Research and Education is gratefully acknowledged. M. Milanič ( ) FAMNIT, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia e-mail: [email protected] J. Orlin MIT Sloan School of Management, Cambridge, MA 02139, USA e-mail: [email protected] G. Rudolf Center for Biomarker Research and Personalized Medicine, Virginia Commonwealth University, Richmond, VA 23298, USA e-mail: [email protected] 360 Ann Oper Res (2011) 188:359–370