Abstract

We investigate hierarchies of semidefinite approximations for the chromatic number χ(G) of a graph G. We introduce an operator Ψ mapping any graph parameter β(G), nested between the stability number α(G) and χ(G), to a new graph parameter Ψβ(G), nested between α(G) and χ(G); Ψβ(G) is polynomial time computable if β(G) is. As an application, there is no polynomial time computable graph parameter nested between the fractional chromatic number χ ∗ (·) and χ(·) unless P = NP. Moreover, based on the Motzkin–Straus formulation for α(G), we give (quadratically constrained) quadratic and copositive programming formulations for χ(G). Under some mild assumptions, n/β(G) ≤ Ψβ(G), but, while n/β(G) remains below χ ∗ (G), Ψβ(G) can reach χ(G) (e.g., for β(·) =α(·)). We also define new polynomial time computable lower bounds for χ(G), improving the classic Lovász theta number (and its strengthenings obtained by adding nonnegativity and triangle inequalities); experimental results on Hamming graphs, Kneser graphs, and DIMACS benchmark graphs will be given in the follow-up paper [N. Gvozdenović and M. Laurent, SIAM J. Optim., 19 (2008), pp. 592–615].

Citations