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PTAS for Densest \(k\)-Subgraph in Interval Graphs

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Abstract

Given an interval graph and integer \(k\), we consider the problem of finding a subgraph of size \(k\) with a maximum number of induced edges, called densest k -subgraph problem in interval graphs. This problem is NP-hard even for chordal graphs (Perl and Corneil in Discret Appl Math 9(1):27–39, 1984), and there is probably no PTAS for general graphs (Khot and Subhash in SIAM J Comput 36(4):1025–1071, 2006). However, the exact complexity status for interval graphs is a long-standing open problem (Perl and Corneil in Discret Appl Math 9(1):27–39, 1984), and the best known approximation result is a \( 3 \)-approximation algorithm (Liazi et al. in Inf Process Lett 108(1):29–32, 2008). We shed light on the approximation complexity of finding a densest \( k \)-subgraph in interval graphs by presenting a polynomial-time approximation scheme (PTAS), that is, we show that there is an \( (1+\epsilon ) \)-approximation algorithm for any \( \epsilon > 0 \), which is the first such approximation scheme for the densest \( k \)-subgraph problem in an important graph class without any further restrictions.

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  1. IBM Research, Zurich, Switzerland

    Tim Nonner

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  1. Tim Nonner
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Correspondence to Tim Nonner.

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A preliminary version of this paper has appeared at WADS’11.

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Nonner, T. PTAS for Densest \(k\)-Subgraph in Interval Graphs. Algorithmica 74, 528–539 (2016). https://doi.org/10.1007/s00453-014-9956-7

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  • DOI: https://doi.org/10.1007/s00453-014-9956-7

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