A (1 + ε)-approximation algorithm for partitioning hypergraphs using a new algorithmic version of the Lovász local Lemma

  • Authors:

  • Mohammad R. Salavatipour

  • Affiliations:

  • Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON N2L 3G1, Canada

  • Venue:
  • Random Structures & Algorithms
  • Year:
  • 2004

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Abstract

In his seminal result, Beck gave the first algorithmic version of the Lovász Local Lemma by giving polynomial time algorithms for 2-coloring and partitioning uniform hypergraphs. His work was later generalized by Alon, and Molloy and Reed. Recently, Czumaj and Scheideler gave an efficient algorithm for 2-coloring nonuniform hypergraphs. But the partitioning algorithm obtained based on their second paper only applies to a more limited range of hypergraphs, so much so that their work doesn't imply the result of Beck for the uniform case. Here we give an algorithmic version of the general form of the Local Lemma which captures (almost) all applications of the results of Beck and Czumaj and Scheideler, with an overall simpler proof. In particular, if H is a nonuniform hypergraph in which every edge ei intersects at most |ei|2αk other edges of size at most k, for some small constant α, then we can find a partitioning of H in expected linear time. This result implies the result of Beck for uniform hypergraphs along with a speedup in his running time.