On k-Strong Conflict–Free Multicoloring

Abstract

Let $\mathcal{H}=(\mathcal{V},\mathcal{E})$ be a hypergraph. A k-strong conflict-free coloring of $\mathcal{H}$ is an assignment of colors to the members of the vertex set $\mathcal{V}$ such that every hyperedge $E\in \mathcal{E}$, $|E|\ge k$, contains k nodes whose colors are pairwise distinct and different from the colors assigned to all the other nodes in E, whereas if $|E|<k$ all nodes in E get distinct colors. The parameter to optimize is the total number of colors. The need for such colorings originally arose as a problem of frequency assignment for cellular networks, but since then it has found applications in a variety of different areas. In this paper we consider a generalization of the above problem, where one is allowed to assign more than one color to each node. When $k=1$, our generalization reduces to the conflict-free multicoloring problem introduced by Even et al. [2003], and recently studied by Bärtschi and Grandoni [2015], and Ghaffari et al. [2017]. We motivate our generalized formulation and we point out that it includes a vast class of well known combinatorial and algorithmic problems, when the hypergraph $\mathcal{H}$ and the parameter k are properly instantiated. Our main result is an algorithm to construct a k-strong conflict-free multicolorings of an input hypergraph $\mathcal{H}$ that utilizes a total number of colors $O( \min \{(k+\log ({r}/{k}) )\log {\varGamma }+ k( k+\log ^2 ({r}/{k})), \ (k^2 + r ) \log n \})$, where n is the number of nodes, $r$ is the maximum hyperedge size, and ${\varGamma }$ is the maximum hyperedge degree; the expected number of colors per node is $O(\min \{k+\log {\varGamma }, \ (k + \log ({r}/{k})) \log n \})$. Although derived for arbitrary k, our result improves on the corresponding result by Bärtschi and Grandoni [2015], when instantiated for $k=1$. We also provide lower bounds on the number of colors needed in any k-strong conflict-free multicoloring, thus showing that our algorithm is not too far from being optimal.