Coloring 2-colorable hypergraphs with a sublinear number of colors

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 Lovasz 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 (n^1-1/dlog^1-1/dn) 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(n^2/9 log^17/8 n) colors. This result makes essential use of semidefinite programming. We further show that the semidefinite programming approach fails for larger dimensions.