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.