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Approximation Algorithms for Intersection Graphs

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Abstract

We study three complexity parameters that, for each vertex v, are an upper bound for the number of cliques that are sufficient to cover a subset S(v) of its neighbors. We call a graph k-perfectly groupable if S(v) consists of all neighbors, k-simplicial if S(v) consists of the neighbors with a higher number after assigning distinct numbers to all vertices, and k-perfectly orientable if S(v) consists of the endpoints of all outgoing edges from v for an orientation of all edges. These parameters measure in some sense how chordal-like a graph is—the last parameter was not previously considered in literature. The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many NP-hard problems, when restricted to graphs for which at least one of the three complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.

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Notes

  1. The theorem holds also for k-perfectly groupable graphs since they are k-simplicial (Observation 2).

  2. By Observation 2 and Lemma 9 the result concerning the minimum vertex coloring holds also for k-perfectly groupable and k-simplicial graphs. This is also true for the remaining problems but not necessary since we have already shown better approximation results on k-perfectly groupable and k-simplicial graphs.

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Acknowledgements

The authors thank the anonymous reviewers for their valuable comments and suggestions that helped to improve the readability of the paper. In particular, we are grateful for the hint that the approximation ratio of the minimum clique partition problem can be improved from O(klog2 n) shown in the conference version of this paper to O(klogn) as shown in this paper.

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  1. Institut für Informatik, Augsburg University, Universitätsstr. 6a, 86159, Augsburg, Germany

    Frank Kammer & Torsten Tholey

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  1. Frank Kammer
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Correspondence to Torsten Tholey.

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A preliminary version of this paper appeared in [31].

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Kammer, F., Tholey, T. Approximation Algorithms for Intersection Graphs. Algorithmica 68, 312–336 (2014). https://doi.org/10.1007/s00453-012-9671-1

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