### Abstract

First, a simplified geometric proof is presented for the result of Lund and Yannakakis saying that for some ε > 0 it is NP-hard to approximate the chromatic number of graphs with N vertices by a factor of N^{ε}. Then, more sophisticated techniques are employed to improve the exponent. A randomized twisting method allows us to completely pack a certain space with copies of a graph without much affecting the independence number. Together with the newest results of Bellare, Goldreich and Sudan on the number of amortized free bits, it is shown that for every ε > 0 the chromatic number cannot be approximated by a factor of N^{1/5-ε} unless NP = ZPP. Finally, we get polynomial lower bounds in terms of x. Unless NP = ZPP, the performance ratio of every polynomial time algorithm approximating the chromatic number of x-colorable graphs (i.e., the chromatic number is at most x) is at least x^{1/5 - o(1)} (where the o-notation is with respect to x).

Original language | English (US) |
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Pages (from-to) | 414-421 |

Number of pages | 8 |

Journal | Annual Symposium on Foundations of Computer Science - Proceedings |

State | Published - Dec 1 1995 |

Event | Proceedings of the 1995 IEEE 36th Annual Symposium on Foundations of Computer Science - Milwaukee, WI, USA Duration: Oct 23 1995 → Oct 25 1995 |

### All Science Journal Classification (ASJC) codes

- Hardware and Architecture