Polynomial Algorithms for Perfect Graphs

Abstract

We show that the weighted versions of the stable set problem, the clique problem, the coloring problem and the clique covering problem are solvable in polynomial time for perfect graphs. Our algorithms are based on the ellipsoid method and a polynomial time separation algorithm for a certain class of positive semidefinite matrices related to Lovász's bound θ(G) on the Shannon capacity of a graph. We show that θG) can be computed in polynomial time for all graphs G and also give a new characterization of perfect graphs in terms of this number θ(G). In addition we prove that the problem of verifying that a graph is imperfect is in NP. Moreover, we show that the computation of the stability number and the fractional stability number of a graph are unrelated with respect to hardness (if P ≠NP).