Conflict-Free Colouring using Maximum Independent Set and Minimum Colouring

TL;DR

This paper introduces a polynomial-time approach for conflict-free colouring of hypergraphs, linking it to maximum independent set and minimum colouring problems, and provides solutions for interval hypergraphs.

Contribution

It presents a novel reduction of conflict-free colouring to maximum independent set and characterizes the problem via co-occurrence graphs, solving an open problem for interval hypergraphs.

Findings

Polynomial-time algorithm for conflict-free colouring of interval hypergraphs.

Characterization of conflict-free colouring number through graph chromatic number.

Perfectness results for co-occurrence and conflict graphs.

Abstract

Given a hypergraph $H$, the conflict-free colouring problem is to colour vertices of $H$ using minimum colours so that each hyperedge in $H$ sees a unique colour. We present a polynomial time reduction from the conflict-free colouring problem in hypergraphs to the maximum independent set problem in a class of simple graphs, which we refer to as \textit{conflict graphs}. We also present another characterization of the conflict-free colouring number in terms of the chromatic number of graphs in an associated family of simple graphs, which we refer to as \textit{co-occurrence graphs}. We present perfectness results for co-occurrence graphs and a special case of conflict graphs. Based on these results and a linear program that returns an integer solution in polynomial time, we obtain a polynomial time algorithm to compute a minimum conflict-free colouring of interval hypergraphs, thus solving an open problem due to Cheilaris et al.\cite{CPLGARSS2014}. Finally, we use the co-occurrence graph characterization to prove that for an interval hypergraph, the conflict-free colouring number is the minimum partition of its intervals into sets such that each set has an exact hitting set (a hitting set in which each interval is hit exactly once).