Cycle killer... qu'est-ce que c'est? On the comparative approximability of hybridization number and directed feedback vertex set

TL;DR

This paper explores the computational complexity of approximating the hybridization number in phylogenetics, showing its deep connection to the well-known directed feedback vertex set problem and providing new approximation bounds.

Contribution

It establishes a polynomial-time approximation equivalence between hybridization number and directed feedback vertex set, linking their complexities and inheritance of inapproximability results.

Findings

Hybridization number approximability is tied to DFVS complexity.

Provides an O(log r log log r) approximation for hybridization number.

Hybridization number inherits inapproximability from Vertex Cover.

Abstract

We show that the problem of computing the hybridization number of two rooted binary phylogenetic trees on the same set of taxa X has a constant factor polynomial-time approximation if and only if the problem of computing a minimum-size feedback vertex set in a directed graph (DFVS) has a constant factor polynomial-time approximation. The latter problem, which asks for a minimum number of vertices to be removed from a directed graph to transform it into a directed acyclic graph, is one of the problems in Karp's seminal 1972 list of 21 NP-complete problems. However, despite considerable attention from the combinatorial optimization community it remains to this day unknown whether a constant factor polynomial-time approximation exists for DFVS. Our result thus places the (in)approximability of hybridization number in a much broader complexity context, and as a consequence we obtain that hybridization number inherits inapproximability results from the problem Vertex Cover. On the positive side, we use results from the DFVS literature to give an O(log r log log r) approximation for hybridization number, where r is the value of an optimal solution to the hybridization number problem.