Zero Knowledge and the Chromatic Number

  • Authors:

  • Uriel Feige;Joe Kilian

  • Affiliations:

  • -;-

  • Venue:
  • CCC '96 Proceedings of the 11th Annual IEEE Conference on Computational Complexity
  • Year:
  • 1996

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Abstract

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 {\sf max-3-coloring} and {\sf 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 Bellare, Goldreich and Sudan, and of H\a{a}stad, and show that it is hard to approximate the chromatic number to within \Omega(N^{1-\epsilon}) for any \epsilon0, assuming {\sf NP}\not\subseteq {\sf ZPP}. Here, {\sf 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, and do not extend beyond \Omega(N^{1/2 - \epsilon}).