Zero knowledge and the chromatic number

We present a new technique, inspired by zero-knowledge proof systems, for proving lower bounds on approximating the chromatic number of a graph. To illustrate this technique we present simple reductions from max-3-coloring and max-3-sat, showing that it is hard to approximate the chromatic number within Omega(N-delta) for some delta > 0. We then apply our technique in conjunction with the probabilistically checkable proofs of Hastad and show that it is hard to approximate the chromatic number to within Omega(N1-epsilon) for any epsilon > 0, assuming NP not subset of or equal to ZPP. Here, ZPP denotes the class of languages decidable by a random expected polynomial-time algorithm that makes no errors. Our result matches (up to low order terms) the known gap for approximating the size of the largest independent set. Previous O(N-delta) gaps for approximating the chromatic number (such as those by Lund and Yannakakis, and by Furer) did not match the gap for independent set nor extend beyond Omega(N1/2-epsilon). (C) 1998 Academic Press.