Coloring 2-colorable hypergraphs with a sublinear number of colors

  • Authors:

  • Noga Alon;Pierre Kelsen;Sanjeev Mahajan;Hariharan Ramesh

  • Affiliations:

  • School of Mathematical Sciences, The Raymond and Beverly Sacker Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel;Max-Planck Institut für Informatik, D-66123 Saarbrücken, Germany;Max-Planck Institut für Informatik, D-66123 Saarbrücken, Germany;Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India

  • Venue:
  • Nordic Journal of Computing
  • Year:
  • 1996

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Abstract

A coloring of a hypergraph is a mapping of vertices to colors such that no hyperedge is monochromatic. We are interested in the problem of coloring 2-colorable hypergraphs. For the special case of graphs (hypergraphs of dimension 2) this can easily be done in linear time. The problem for general hypergraphs is much more difficult since a result of Lovász implies that the problem is NP-hard even if all hyperedges have size three.In this paper we develop approximation algorithms for this problem. Our first result is an algorithm that colors any 2-colorable hypergraph on n vertices and dimension d with O(n1-1/dlog1-1/d n) colors. This is the first algorithm that achieves a sublinear number of colors in polynomial time. This algorithm is based on a new technique for reducing degrees in a hypergraph that should be of independent interest. For the special case of hypergraphs of dimension three we improve on the previous result by obtaining an algorithm that uses only O(n2/9 log17/8 n) colors. This result makes essential use of semidefinite programming. We further show that the semidefinite programming approach fails for larger dimensions.