Coloring bipartite hypergraphs

Abstract

It is NP-Hard to find a proper 2-coloring of a given 2-colorable (bipartite) hypergraph H. We consider algorithms that will color such a hypergraph using few colors in polynomial time. The results of the paper can be summarized as follows: Let n denote the number of vertices of H and m the number of edges, (i) For bipartite hypergraphs of dimension k there is a polynomial time algorithm which produces a proper coloring using min\(\{ O(n^{1 - 1/k} ),O((m/n)^{\tfrac{1}{{k - 1}}} )\}\)colors, (ii) For 3-uniform bipartite hypergraphs, the bound is reduced to O(n 2/9). (iii) For a class of dense 3-uniform bipartite hypergraphs, we have a randomized algorithm which can color optimally. (iv) For a model of random bipartite hypergraphs with edge probability p≥ dn −2, d > O a sufficiently large constant, we can almost surely find a proper 2-coloring.KeywordsPolynomial TimeAdjacency MatrixPolynomial Time AlgorithmColor ClassMarkov InequalityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.