Exact and Fixed Parameter Tractable Algorithms for Max-Conflict-Free Coloring in Hypergraphs
SIAM Journal on Discrete Mathematics
Conflict-free coloring of hypergraphs is a very well studied question of theoretical and practical interest. For a hypergraph H = (U, 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 a natural maximization version of this problem, namely, Max-CFC: For a given hypergraph H and a fixed r ≥ 2, color the vertices of U using r colors so that
... e number of hyperedges that are conflict-free colored is maximized. By previously known hardness results for conflict-free coloring, this maximization version is NPhard. We study this problem in the context of both exact and parameterized algorithms. In the parameterized setting, we study this problem with respect to a natural parameter-the solution size. In particular, the question we study is the following: p-CFC: For a given hypergraph, can we conflict-free color at least k hyperedges with at most r colors, the parameter being the solution size k. We show that this problem is fixed parameter tractable by designing an algorithm with running time 2 O(k log log k+k log r) (n + m) O(1) using a novel connection to the Unique Coverage problem and applying the method of color coding in a nontrivial manner. For the special case for hypergraphs induced by graph neighborhoods we give a polynomial kernel. Finally, we give an exact algorithm for Max-CFC running in O(2 n+m ) time. All our algorithms, with minor modifications, work for a stronger version of conflict-free coloring, Unique Maximum Coloring.