A Note on Fractional Coloring and the Integrality gap of LP for Maximum Weight Independent Set

We prove a tight connection between two important notions in combinatorial optimization. Let G be a graph class (i.e. a subset of all graphs) and r(G) = supG∈G χf (G) ω(G) where χf (G) and ω(G) are the fractional chromatic number and clique number of G respectively. In this note, we prove that r(G) tightly captures the integrality gap of the LP relaxation with clique constraints for the Maximum Weight Independent Set (MWIS) problem. Our proof uses standard applications of multiplicative weight techniques, so it is algorithmic: Any algorithm for rounding the LP can be turned into a fractional coloring algorithm and vice versa. We discuss immediate applications of our results in approximating the fractional chromatic number of certain classes of intersection graphs.