On the Approximability of Comparing Genomes with Duplicates

TL;DR

This paper investigates the computational complexity of comparing duplicated genomes using various matching models and measures, establishing hardness results and providing algorithms for specific cases in comparative genomics.

Contribution

It proves APX-hardness for optimizing similarity measures with genome matchings and offers polynomial-time solutions and approximation algorithms for certain models and measures.

Findings

Computing optimal matchings for similarity measures is APX-hard.

NP-Complete for exemplar model with no breakpoints.

Polynomial-time algorithms for maximum matching model.

Abstract

A central problem in comparative genomics consists in computing a (dis-)similarity measure between two genomes, e.g. in order to construct a phylogeny. All the existing measures are defined on genomes without duplicates. However, we know that genes can be duplicated within the same genome. One possible approach to overcome this difficulty is to establish a one-to-one correspondence (i.e. a matching) between genes of both genomes, where the correspondence is chosen in order to optimize the studied measure. In this paper, we are interested in three measures (number of breakpoints, number of common intervals and number of conserved intervals) and three models of matching (exemplar, intermediate and maximum matching models). We prove that, for each model and each measure M, computing a matching between two genomes that optimizes M is APX-hard. We also study the complexity of the following problem: is there an exemplarization (resp. an intermediate/maximum matching) that induces no breakpoint? We prove the problem to be NP-Complete in the exemplar model for a new class of instances, and we show that the problem is in P in the maximum matching model. We also focus on a fourth measure: the number of adjacencies, for which we give several approximation algorithms in the maximum matching model, in the case where genomes contain the same number of duplications of each gene.