The stable set problem: some structural properties and relaxations

This is a summary of the author’s PhD thesis supervised by Antonio Sassano and defended on June 4, 2012 at Sapienza Universita di Roma. The thesis is written in English and is available at http://padis.uniroma1.it/bitstream/10805/1598/1/ thesisCarlaMichini.pdf. The thesis deals with a polyhedral study of the fractional stable set polytope and aims mainly at establishing some new structural properties of this polytope. The fractional stable set polytope is the polytope defined by the linear relaxation of the edge formulation of the stable set problem. The edge formulation is defined by two-variable constraints, one for each edge of a graph G, expressing the simple condition that two adjacent nodes cannot belong to a stable set. Even if the fractional stable set polytope is a weak approximation of the stable set polytope, its simple geometrical structure provides deep theoretical insight as well as interesting algorithmic opportunities. Exploiting a graphic characterization of the bases, we first redefine simplex pivots in terms of simple graphic operations, that turn a given basis into an adjacent one. Among all possible pivots, we characterize degenerate and non-degenerate ones, and we differentiate those leading to an integer or to a fractional vertex. The graphic characterization of bases is crucial to prove another structural property of the fractional stable set polytope, concerning the adjacency of its vertices. In particular, we extend a necessary and sufficient condition due to Chvatal for adjacency of (integer) vertices of the stable set polytope to arbitrary (and possibly fractional) vertices of the fractional stable set polytope. These results lead us to prove that the Hirsch conjecture is true for the fractional stable set polytope, i.e. the combinatorial diameter of this fractional polytope is at most equal to the number of edges of the given graph. We actually refine