Perfect Resolution of Strong Conflict-Free Colouring of Interval Hypergraphs

TL;DR

This paper presents an exact polynomial-time algorithm for the $k$-Strong Conflict-Free coloring problem in interval hypergraphs, solving an open problem by leveraging properties of perfect graphs and introducing new graph concepts.

Contribution

It introduces a novel polynomial-time algorithm for $k$-SCF coloring of interval hypergraphs, utilizing the perfect graph structure of associated co-occurrence and conflict graphs.

Findings

The $k$-SCF coloring problem is solvable in polynomial time for interval hypergraphs.

Co-occurrence and conflict graphs are shown to be perfect graphs in this context.

The $1$-SCF coloring number corresponds to minimal interval partitions with exact hitting sets.

Abstract

The $k$-Strong Conflict-Free ($k$-SCF, in short) colouring problem seeks to find a colouring of the vertices of a hypergraph $H$ using minimum number of colours so that in every hyperedge $e$ of $H$, there are at least $\min\{|e|,k\}$ vertices whose colour is different from that of all other vertices in $e$. In the case of interval hypergraphs, we present an exact $P$-time algorithm for the $k$-SCF problem thus solving an open problem posed by Cheilaris et al. (2014). We achieve our results by showing that for any hypergraph a $k$-SCF colouring is a proper colouring of a related simple graph which we refer to as a co-occurrence graph. We then show that a co-occurrence graph is obtained by identifying an induced subgraph of a second simple graph that we introduce, which we refer to as the conflict graph. For interval hypergraphs, we show that each co-occurrence graph and the conflict graph are perfect graphs. This property plays a crucial role in our polynomial time algorithm. Secondly, we show that for an interval hypergraph, the $1$-SCF 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).