Towards optimal lower bounds for clique and chromatic number

  • Authors:

  • Lars Engebretsen;Jonas Holmerin

  • Affiliations:

  • Department of Numerical Analysis and Computer Science, Royal Institute of Technolooy, SE-100 44 Stockholm, Sweden;Department of Numerical Analysis and Computer Science, Royal Institute of Technolooy, SE-100 44 Stockholm, Sweden

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2003

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Abstract

It was previously known that neither Max Clique nor Min Chromatic Number can be approximated in polynomial time within n1-ε, for any constant ε 0, unless NP = ZPP. In this paper, we extend the reductions used to prove these results and combine the extended reductions with a recent result of Samorodnitsky and Trevisan to show that unless NP ⊆ ZPTIME(2O(log n(log log n)3/2)), neither Max Clique nor Min Chromatic Number can be approximated in polynomial time within n1-ε(n) where ε ∈ O((log log n)-1/2). Since there exists polynomial time algorithms approximating both problems within n1-ε(n) where ε(n) ∈ Ω(log log n/log n), our result shows that the best possible ratio we can hope for is of the form n1-o(1), for some--yet unknown--value of o(1) between O((log log n)-1/2) and Ω(log log n/log n).