The potential to improve the choice: list conflict-free coloring for geometric hypergraphs

TL;DR

This paper introduces a new list-coloring algorithm for conflict-free coloring of geometric hypergraphs, providing bounds on list sizes and an efficient method to compute such colorings, with applications in wireless networks.

Contribution

It presents a novel potential-based list coloring algorithm that guarantees unique-maximum colorings in geometric hypergraphs, improving bounds and computational methods.

Findings

Algorithm produces stronger unique-maximum colorings.

Provides asymptotically sharp bounds on list sizes.

Offers an efficient coloring computation method.

Abstract

Given a geometric hypergraph (or a range-space) $H=(V,\cal E)$, a coloring of its vertices is said to be conflict-free if for every hyperedge $S \in \cal E$ there is at least one vertex in $S$ whose color is distinct from the colors of all other vertices in $S$. The study of this notion is motivated by frequency assignment problems in wireless networks. We study the list-coloring (or choice) version of this notion. In this version, each vertex is associated with a set of (admissible) colors and it is allowed to be colored only with colors from its set. List coloring arises naturally in the context of wireless networks. Our main result is a list coloring algorithm based on a new potential method. The algorithm produces a stronger unique-maximum coloring, in which colors are positive integers and the maximum color in every hyperedge occurs uniquely. As a corollary, we provide asymptotically sharp bounds on the size of the lists required to assure the existence of such unique-maximum colorings for many geometric hypergraphs (e.g., discs or pseudo-discs in the plane or points with respect to discs). Moreover, we provide an algorithm, such that, given a family of lists with the appropriate sizes, computes such a coloring from these lists.