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.

Optimization Problems in Multiple Subtree Graphs

“…For Maximum Independent Set, Minimum Coloring, Minimum Vertex Cover, and Maximum Clique we obtain approximation ratios of 2t, 2t, 2 − 1/t, and (t 2 − t + 1)/2, respectively, which match the approximation ratios of the corresponding algorithms for t-interval graphs in [3,6] by extending these in a natural manner. Minimum Dominating Set is different in that it is already as hard to approximate as Minimum Set Cover even in chordal graphs (i.e.…”

“…The first stage involves removing triangles from our input graph G by applying a technique originally introduced in [1] in order to remove short odd cycles. Using this technique, we obtain in polynomial-time a triangle-free subgraph G ′ of G such that any r-approximate vertex cover of G ′ can be easily transformed into a max {r, 1.5}-approximate vertex cover of G. The same triangle-cleaning phase is also performed in [6] to approximate Minimum Vertex Cover in tinterval graphs.…”

“…In [3,6], various optimization problems were considered in the class of t-interval graphs, a natural generalization of interval graphs. These are defined as intersection graphs of t-intervals, which are 1-dimensional objects formed by taking the union of t disjoint intervals.…”

“…It follows that Minimum Dominating Set and Minimum Vertex Cover are APX-hard, for t ≥ 2 [13], and that Minimum Coloring is NP-hard, for t ≥ 3 [8]. Maximum Clique was shown to be NP-hard for t ≥ 3 in [6]. Bar-Yehuda et al [3] showed that, for t ≥ 2, Maximum Independent Set in t-interval graphs is APX-hard, and that it cannot be approximated within a factor of O(t/ log t) unless P=NP.…”

Optimization Problems in Dotted Interval Graphs

“…Naturally, substantial research effort has been devoted into generalizing such algorithms to larger classes of graphs. Examples include algorithms proposed for circular arc graphs [13,15], disc graphs [11,16,19,20], rectangle graphs [1,5,9], multiple-interval graphs [4,8], and multiple-subtree graphs [14].…”