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The Maximum Clique Problem in Multiple Interval Graphs

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Multiple interval graphs are variants of interval graphs where instead of a single interval, each vertex is assigned a set of intervals on the real line. We study the complexity of the MAXIMUM CLIQUE problem in several classes of multiple interval graphs. The MAXIMUM CLIQUE problem, or the problem of finding the size of the maximum clique, is known to be NP-complete for t-interval graphs when t≥3 and polynomial-time solvable when t=1. The problem is also known to be NP-complete in t-track graphs when t≥4 and polynomial-time solvable when t≤2. We show that MAXIMUM CLIQUE is already NP-complete for unit 2-interval graphs and unit 3-track graphs. Further, we show that the problem is APX-complete for 2-interval graphs, 3-track graphs, unit 3-interval graphs and unit 4-track graphs. We also introduce two new classes of graphs called t-circular interval graphs and t-circular track graphs and study the complexity of the MAXIMUM CLIQUE problem in them. On the positive side, we present a polynomial time t-approximation algorithm for MAXIMUM WEIGHTED CLIQUE on t-interval graphs, improving earlier work with approximation ratio 4t.

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Acknowledgements

This work was partially supported by the grant ANR-09-JCJC-0041.

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Authors and Affiliations

  1. Indian Statistical Institute, Chennai Centre, SETS, MGR Knowledge City, CIT Campus, Taramani, Chennai, 600113, India

    Mathew C. Francis

  2. LIRMM, CNRS et Université Montpellier 2, 161 rue Ada, 34392, Montpellier Cedex 05, France

    Daniel Gonçalves & Pascal Ochem

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  1. Mathew C. Francis
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  2. Daniel Gonçalves
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Correspondence to Mathew C. Francis.

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An extended abstract of this paper has been accepted for publication in the proceedings of WG 2012.

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Francis, M.C., Gonçalves, D. & Ochem, P. The Maximum Clique Problem in Multiple Interval Graphs. Algorithmica 71, 812–836 (2015). https://doi.org/10.1007/s00453-013-9828-6

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