Complexity Note on Mixed Hypergraphs

Abstract

A mixed hypergraph H is a triple (V, $\mathcal{C}$, $\mathcal{D}$) where V is its vertex set and $\mathcal{C}$ and $\mathcal{D}$ are families of subsets of V, $\mathcal{C}$-edges and $\mathcal{D}$-edges. The degree of a vertex is the number of edges in which it is contained. A vertex coloring of H is proper if each $\mathcal{C}$-edge contains two vertices with the same color and each $\mathcal{D}$-edge contains two vertices with different colors. The feasible set of H is the set of all k’s such that there exists a proper coloring using exactly k colors. The lower (upper) chromatic number of H is the minimum (maximum) number in the feasible set. We prove that it is NP-complete to decide whether the upper chromatic number of mixed hypergraphs with maximum degree two is at least a given k. We present polynomial time algorithms for mixed hypergraphs with maximum degree two to decide their colorability, to find a coloring using the number of colors equal to the lower chromatic number and we present a 5/3-aproximation algorithm for the upper chromatic number. We further prove that it is coNP-hard to decide whether the feasible set of a given general mixed hypergraph is an interval of integers.

The author acknowledges partial support by GAČR 201/99/0242.

The author acknowledges support by GAUK 158/99 and KONTAKT 338/99.

Supported by Ministry of Education of Czech Republic as project LN00A056