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International Workshop on Graph-Theoretic Concepts in Computer Science

WG 2016: Graph-Theoretic Concepts in Computer Science pp 13–24Cite as

Approximate Association via Dissociation

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9941))

Abstract

A vertex set X of a graph G is an association set if each component of \(G - X\) is a clique, or a dissociation set if each component of \(G - X\) is a single vertex or a single edge. Interestingly, \(G - X\) is then precisely a graph containing no induced \(P_3\)’s or containing no \(P_3\)’s, respectively. We observe some special structures and show that if none of them exists, then the minimum association set problem can be reduced to the minimum (weighted) dissociation set problem. This yields the first nontrivial approximation algorithm for the association set problem, with approximation ratio is 2.5. The reduction is based on a combinatorial study of modular decomposition of graphs free of these special structures. Further, a novel algorithmic use of modular decomposition enables us to implement this approach in \(O(m n + n^2)\) time.

Supported in part by NSFC under grants 61572414 and 61420106009, and RGC under grant 252026/15E.

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Notes

  1. 1.

    If G is a clique or an independent set, then \(\widetilde{Q}(G)\) is isomorphic to G and is the largest quotient graph of G; if \(\widetilde{Q}(G)\) is prime, then it is the smallest nontrivial quotient graph of G, both cardinality-wise and inclusion-wise (see Lemma 5). Otherwise, there can be other quotient graph larger or smaller than \(\widetilde{Q}(G)\).

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Authors and Affiliations

  1. School of Information Science and Engineering, Central South University, Changsha, China

    Jie You & Jianxin Wang

  2. Department of Computing, Hong Kong Polytechnic University, Hong Kong, China

    Jie You & Yixin Cao

Authors
  1. Jie You
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  2. Jianxin Wang
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  3. Yixin Cao
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Correspondence to Yixin Cao .

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Editors and Affiliations

  1. University of Bergen, Bergen, Norway

    Pinar Heggernes

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You, J., Wang, J., Cao, Y. (2016). Approximate Association via Dissociation. In: Heggernes, P. (eds) Graph-Theoretic Concepts in Computer Science. WG 2016. Lecture Notes in Computer Science(), vol 9941. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53536-3_2

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  • DOI: https://doi.org/10.1007/978-3-662-53536-3_2

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