## Zero Knowledge and the Chromatic Number (1996)

Venue: | Journal of Computer and System Sciences |

Citations: | 176 - 8 self |

### BibTeX

@INPROCEEDINGS{Feige96zeroknowledge,

author = {Uriel Feige and Joe Kilian},

title = {Zero Knowledge and the Chromatic Number},

booktitle = {Journal of Computer and System Sciences},

year = {1996},

pages = {278--287}

}

### Years of Citing Articles

### OpenURL

### 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 max-3-coloring and max-3-sat, showing that it is hard to approximate the chromatic number within \Omega\Gamma N ffi ), for some ffi ? 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\Gamma N 1\Gammaffl ) for any ffl ? 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 ffi ) 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...