The Complexity of Finding Multiple Solutions to Betweenness and Quartet Compatibility

TL;DR

This paper proves that finding multiple solutions for Betweenness and Quartet Compatibility problems is computationally hard, with implications for computational biology, by establishing their ASP-completeness and coNP-completeness.

Contribution

It demonstrates that variations of Betweenness and Quartet Compatibility are ASP-complete, revealing their inherent computational complexity and impact on related biological problems.

Findings

Betweenness variation is ASP-complete.

Quartet Compatibility is ASP-complete.

Solution uniqueness decision is coNP-complete.

Abstract

We show that two important problems that have applications in computational biology are ASP-complete, which implies that, given a solution to a problem, it is NP-complete to decide if another solution exists. We show first that a variation of Betweenness, which is the underlying problem of questions related to radiation hybrid mapping, is ASP-complete. Subsequently, we use that result to show that Quartet Compatibility, a fundamental problem in phylogenetics that asks whether a set of quartets can be represented by a parent tree, is also ASP-complete. The latter result shows that Steel's \sc Quartet Challenge, which asks whether a solution to Quartet Compatibility is unique, is coNP-complete.