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 (N ), for some > 0. We then apply our technique in conjunction with the probabilistically checkable proofs of H astad, and show that it is hard to approximate the chromatic number to within (N 1? ) for any > 0, assuming NP 6 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 ) gaps for approximating the chromatic number (such as those by Lund and Yannakakis, and by Furer) did not match the gap for independent set, and do not extend beyond (N 1=2? ).
Feige and Kilian REF proved that the chromatic number cannot be approximated within V 1 ε for any ε 0, unless NP ZPP.
Zero Knowledge and the Chromatic Number
Journal of Computer and System Sciences
Mathematics Computer Science