Optimization problems in multiple-interval graphs

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 clas- sical problems: Minimum Vertex Cover, Minimum Dominating Set, and Maxi- mum Clique. We describe applications for each one of these problems, and then pro- ceed to discuss approximation algorithms for them. Our results can be summarized as follows: Let t be the number of inter- vals associated with each vertex in a given multiple-interval graph. For Mini- mum Vertex Cover, we give a (2 ¡ 1=t)- approximation algorithm which equals the best known ratio for 2t¡1 bounded degree graphs. Since these graphs are known to be included in multiple-interval graphs with t intervals associated to each vertex, this ra- tio is in some sense tight. Following this, we give a t2-approximation algorithm for Min- imum Dominating Set which adapts well to more general and restricted variants of the problem. We then proceed to prove that Maximum Clique is NP-complete for the case of t = 3, and provide a (t2 ¡ t + 1)=2- approximation algorithm for the problem, using recent bounds proven for the so-called transversal number of t-interval families.