A Branch & Price algorithm for the minimum cost clique cover problem in max-point tolerance graphs

Abstract

A point-interval$(I_v, p_v)$ is a pair constituted by an interval $I_v$ of ${\mathbb {R}}$ and a point $p_v \in I_v$. A graph $G=(V,E)$ is a Max-Point-Tolerance (MPT) graph if each vertex $v\in V$ can be mapped to a point-interval in such a way that (u, v) is an edge of G iff $I_u \cap I_v \supseteq \{p_u, p_v\}$. MPT graphs constitute a superclass of interval graphs and naturally arise in genetic analysis as a way to represent specific relationships among DNA fragments extracted from a population of individuals. One of the most important applications of MPT graphs concerns the search for an association between major human diseases and chromosome regions from patients that exhibit loss of heterozygosity events. This task can be formulated as a minimum cost clique cover problem in a MPT graph and gives rise to a ${{\mathcal {N}}}{{\mathcal {P}}}$-hard combinatorial optimization problem known in the literature as the Parsimonious Loss of Heterozygosity Problem (PLOHP). In this article, we investigate ways to speed up the best known exact solution algorithm for the PLOHP as well as techniques to enlarge the size of the instances that can be optimally solved. In particular, we present a Branch&Price algorithm for the PLOHP and we develop a number of preprocessing techniques and decomposition strategies to dramatically reduce the size of its instances. Computational experiments show that the proposed approach is 10–30$\times$ faster than previous approaches described in the literature, and suggest new directions for the development of future exact solution approaches that may prove of fundamental assistance in practice.