Exact Algorithms for No-Rainbow Coloring and Phylogenetic Decisiveness

TL;DR

This paper introduces exact deterministic and randomized algorithms for the NP-complete no-rainbow hypergraph coloring problem, with applications to phylogenetic decisiveness, improving computational efficiency for specific cases.

Contribution

The paper presents the first deterministic and randomized algorithms with explicit time complexities for solving the no-rainbow hypergraph coloring problem.

Findings

Deterministic algorithm runs in $O^*((r-1)^{(r-1)n/r})$ time.

Randomized algorithm runs in $O^*((r/2)^n)$ time.

Algorithms are applicable to phylogenetic decisiveness problems.

Abstract

The input to the no-rainbow hypergraph coloring problem is a hypergraph $H$ where every hyperedge has $r$ nodes. The question is whether there exists an $r$-coloring of the nodes of $H$ such that all $r$ colors are used and there is no rainbow hyperedge -- i.e., no hyperedge uses all $r$ colors. The no-rainbow hypergraph $r$-coloring problem is known to be NP-complete for $r \geq 3$. The special case of $r=4$ is the complement of the phylogenetic decisiveness problem. Here we present a deterministic algorithm that solves the no-rainbow $r$-coloring problem in $O^*((r-1)^{(r-1)n/r})$ time and a randomized algorithm that solves the problem in $O^*((\frac{r}{2})^n)$ time.