Exact and FPT Algorithms for Max-Conflict Free Coloring in Hypergraphs

Abstract

Conflict-free coloring of hypergraphs is a very well studied question of theoretical and practical interest. For a hypergraph $H=(U, \mathcal F)$, a conflict-free coloring of H refers to a vertex coloring where every hyperedge has a vertex with a unique color, distinct from all other vertices in the hyperedge. In this paper, we initiate a study of natural maximization version of this problem, namely, Max-CFC: For a given hypergraph H and a fixed $r\ge 2$, color the vertices of U using r colors so that the number of hyperedges that are conflict-free colored is maximized. By previously known hardness results for conflict-free coloring, this maximization version is NP-hard.

We study this problem in the context of both exact and parameterized algorithms. In the parameterized setting, we study this problem with respect to the natural parameter, the solution size. In particular, we study the following question: p-CFC: For a given hypergraph, can we conflict-free color at least k hyperedges with at most r colors, the parameter being k. We show that this problem is FPT by designing an algorithm with running time $2^{\mathcal {O}(k \log \log k + k \log r)}(n+m)^{\mathcal {O}(1)}$ using a novel connection to the Unique Coverage problem and applying the method of color coding in a non-trivial manner. For the special case for hypergraphs induced by graph neighbourhoods we give a polynomial kernel. Finally, we give an exact algorithm for Max-CFC running in $\mathcal {O}(2^{n+m})$ time. All our algorithms, with minor modifications, work for a stronger version of conflict-free coloring, Unique Maximum Coloring.

The research leading to these results has received partial funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 306992.