Computing Semidefinite Programming Lower Bounds for the (Fractional) Chromatic Number Via Block-Diagonalization

Abstract

Recently we investigated in [SIAM J. Optim., 19 (2008), pp. 572-591] hierarchies of semidefinite approximations for the chromatic number $\chi(G)$ of a graph $G$. In particular, we introduced two hierarchies of lower bounds: the “$\psi$”-hierarchy converging to the fractional chromatic number and the “$\Psi$”-hierarchy converging to the chromatic number of a graph. In both hierarchies the first order bounds are related to the Lovasz theta number, while the second order bounds would already be too costly to compute for large graphs. As an alternative, relaxations of the second order bounds are proposed in [SIAM J. Optim., 19 (2008), pp. 572-591]. We present here our experimental results with these relaxed bounds for Hamming graphs, Kneser graphs, and DIMACS benchmark graphs. Symmetry reduction plays a crucial role as it permits us to compute the bounds by using more compact semidefinite programs. In particular, for Hamming and Kneser graphs, we use the explicit block-diagonalization of the Terwilliger algebra given by Schrijver [IEEE Trans. Inform. Theory, 51 (2005), pp. 2859-2866]. Our numerical results indicate that the new bounds can be much stronger than the Lovasz theta number. For some of the DIMACS instances we improve the best known lower bounds significantly.