Conflict-free coloring with respect to a subset of intervals

TL;DR

This paper presents a polynomial-time algorithm for conflict-free coloring of subhypergraphs of interval hypergraphs, with a tight approximation ratio of 2, and explores the complexity of minimal coloring with respect to the number of colors.

Contribution

It introduces a polynomial-time 2-approximation algorithm for conflict-free coloring of interval hypergraphs and analyzes its tightness and complexity.

Findings

The algorithm has an approximation ratio of 2.

The analysis of the algorithm's tightness is provided.

Deciding k-colorability is quasipolynomial-time solvable.

Abstract

Given a hypergraph H = (V, E), a coloring of its vertices is said to be conflict-free if for every hyperedge S \in E there is at least one vertex in S whose color is distinct from the colors of all other vertices in S. The discrete interval hypergraph Hn is the hypergraph with vertex set {1,...,n} and hyperedge set the family of all subsets of consecutive integers in {1,...,n}. We provide a polynomial time algorithm for conflict-free coloring any subhypergraph of Hn, we show that the algorithm has approximation ratio 2, and we prove that our analysis is tight, i.e., there is a subhypergraph for which the algorithm computes a solution which uses twice the number of colors of the optimal solution. We also show that the problem of deciding whether a given subhypergraph of Hn can be colored with at most k colors has a quasipolynomial time algorithm.