Complexity of modification problems for best match graphs

TL;DR

This paper investigates the computational complexity of modifying best match graphs (BMGs) to adhere to their properties, showing NP-completeness of key problems and providing ILP formulations based on new characterizations.

Contribution

It introduces novel characterizations of BMGs using triples and forbidden subgraphs, and proves NP-completeness of arc modification problems for BMGs.

Findings

Arc deletion, completion, and editing problems are NP-complete.

BMGs can be characterized by triples and forbidden subgraphs.

Integer linear programs can solve these problems.

Abstract

Best match graphs (BMGs) are vertex-colored directed graphs that were introduced to model the relationships of genes (vertices) from different species (colors) given an underlying evolutionary tree that is assumed to be unknown. In real-life applications, BMGs are estimated from sequence similarity data. Measurement noise and approximation errors usually result in empirically determined graphs that in general violate characteristic properties of BMGs. The arc modification problems for BMGs aim at correcting such violations and thus provide a means to improve the initial estimates of best match data. We show here that the arc deletion, arc completion and arc editing problems for BMGs are NP-complete and that they can be formulated and solved as integer linear programs. To this end, we provide a novel characterization of BMGs in terms of triples (binary trees on three leaves) and a characterization of BMGs with two colors in terms of forbidden subgraphs.