On the Way to Perfection: Primal Operations for Stable Sets in Graphs

Abstract

In this paper some operations are described that transform every graph into a perfect graph by replacing nodes with sets of new nodes. The transformation is done in such a way that every stable set in the perfect graph corresponds to a stable set in the original graph. These operations yield a purely combinatorial augmentation procedure for finding a maximum weighted stable set in a graph. Starting with a stable set in a given graph one defines a simplex type tableau whose associated basic feasible solution is the incidence vector of the stable set. In an iterative fashion, non-basic columns that would lead to pivoting into non-integral basic feasible solutions, are replaced by new columns that one can read off from special graph structures such as odd holes, odd antiholes, and various generalizations. Eventually, either a pivot leading to an integral basic feasible solution is performed, or the optimality of the current solution is proved.