Optimization problems in multiple-interval graphs

Abstract

Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multiple-interval graphs by considering three classical problems: Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. We describe applications for each one of these problems, and then proceed to discuss approximation algorithms for them. Our results can be summarized as follows: Let t be the number of intervals associated with each vertex in a given multiple-interval graph. For Minimum Vertex Cover, we give a (2−1/ t )-approximation algorithm which also works when a t -interval representation of our given graph is absent. Following this, we give a t 2 -approximation algorithm for Minimum Dominating Set which adapts well to more general variants of the problem. We then proceed to prove that Maximum Clique is NP -hard already for 3-interval graphs, and provide a ( t 2 − t +1)/2-approximation algorithm for general values of t ≥ 2, using bounds proven for the so-called transversal number of t -interval families.