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Optimal Approximation Algorithms for Maximum Distance-Bounded Subgraph Problems

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Abstract

In this paper we study the (in)approximability of two distance-based relaxed variants of the maximum clique problem (Max Clique), named Max d-Clique and Max d-Club: A d-clique in a graph \(G = (V, E)\) is a subset \(S\subseteq V\) of vertices such that for every pair of vertices \(u, v\in S\), the distance between u and v is at most d in G. A d-club in a graph \(G = (V, E)\) is a subset \(S'\subseteq V\) of vertices that induces a subgraph of G of diameter at most d. Given a graph G with n vertices, the goal of Max d-Clique (Max d-Club, resp.) is to find a d-clique (d-club, resp.) of maximum cardinality in G. Since Max 1-Clique and Max 1-Club are identical to Max Clique, the inapproximabilty for Max Clique shown by Zuckerman in 2007 is transferred to them. Namely, Max 1-Clique and Max 1-Club cannot be efficiently approximated within a factor of \(n^{1-\varepsilon }\) for any \(\varepsilon > 0\) unless \(\mathcal{P} = \mathcal{NP}\). Also, in 2002, Marin\(\breve{\mathrm{c}}\)ek and Mohar showed that it is \(\mathcal{NP}\)-hard to approximate Max d-Club to within a factor of \(n^{1/3-\varepsilon }\) for any \(\varepsilon >0\) and any fixed \(d\ge 2\). In this paper, we strengthen the hardness result; we prove that, for any \(\varepsilon > 0\) and any fixed \(d\ge 2\), it is \(\mathcal{NP}\)-hard to approximate Max d-Club to within a factor of \(n^{1/2-\varepsilon }\). Then, we design a polynomial-time algorithm which achieves an optimal approximation ratio of \(O(n^{1/2})\) for any integer \(d\ge 2\). By using the similar ideas, we show the \(O(n^{1/2})\)-approximation algorithm for Max d-Clique for any \(d\ge 2\). This is the best possible in polynomial time unless \(\mathcal{P} = \mathcal{NP}\), as we can prove the \(\varOmega (n^{1/2-\varepsilon })\)-inapproximability.

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Acknowledgements

This work is partially supported by JSPS KAKENHI Grant Numbers JP25330018, JP26330017, JP17K00016, and JP17K00024. The authors would like to thank the anonymous reviewers for their suggestions and detailed comments that helped to improve the presentation of the paper.

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Authors and Affiliations

  1. Department of Information Science, Kyushu Sangyo University, Fukuoka, 813-8503, Japan

    Yuichi Asahiro

  2. Department of Systems Design and Informatics, Kyushu Institute of Technology, Fukuoka, 820-8502, Japan

    Yuya Doi, Eiji Miyano, Kazuaki Samizo & Hirotaka Shimizu

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  1. Yuichi Asahiro
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  2. Yuya Doi
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  3. Eiji Miyano
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  4. Kazuaki Samizo
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  5. Hirotaka Shimizu
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Correspondence to Yuichi Asahiro.

Additional information

This paper combines results from two extended abstracts that appeared in Proceedings of LATIN 2010, LNCS 6034, pp. 615–625, 2010 [4] and COCOA 2015, LNCS 9486, pp. 586–600, 2015 [3].

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Asahiro, Y., Doi, Y., Miyano, E. et al. Optimal Approximation Algorithms for Maximum Distance-Bounded Subgraph Problems. Algorithmica 80, 1834–1856 (2018). https://doi.org/10.1007/s00453-017-0344-y

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