# Optimal Approximation Algorithms for Maximum Distance-Bounded Subgraph Problems

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9486))

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## Abstract

A d-clique in a graph $$G = (V, E)$$ is a subset $$S\subseteq V$$ of vertices such that for pairs 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. Max 1-Clique and Max 1-Club cannot be efficiently approximated within a factor of $$n^{1-\varepsilon }$$ for any $$\varepsilon > 0$$ unless $$\mathcal{P} = \mathcal{NP}$$ since they are identical to Max Clique [14, 21]. Also, it is known [3] that it is $$\mathcal{NP}$$-hard to approximate Max d-Club to within a factor of $$n^{1/2 - \varepsilon }$$ for any fixed $$d\ge 2$$ and for any $$\varepsilon > 0$$. As for approximability of Max d-Club, there exists a polynomial-time algorithm which achieves an optimal approximation ratio of $$O(n^{1/2})$$ for any even $$d\ge 2$$ [3]. For any odd $$d\ge 3$$, however, there still remains a gap between the $$O(n^{2/3})$$-approximability and the $$\varOmega (n^{1/2-\varepsilon })$$-inapproximability for Max d-Club [3]. In this paper, we first strengthen the approximability result for Max d-Club; we design a polynomial-time algorithm which achieves an optimal approximation ratio of $$O(n^{1/2})$$ for Max d-Club for any odd $$d\ge 3$$. Then, by using the similar ideas, we show the $$O(n^{1/2})$$-approximation algorithm for Max d-Clique on general graphs 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. Furthermore, we study the tractability of Max d-Clique and Max d-Club on subclasses of graphs.

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## Acknowledgments

This work is partially supported by KAKENHI grant numbers 25330018 and 26330017.

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Correspondence to Eiji Miyano .

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### Cite this paper

Asahiro, Y., Doi, Y., Miyano, E., Shimizu, H. (2015). Optimal Approximation Algorithms for Maximum Distance-Bounded Subgraph Problems. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, DZ. (eds) Combinatorial Optimization and Applications. Lecture Notes in Computer Science(), vol 9486. Springer, Cham. https://doi.org/10.1007/978-3-319-26626-8_43

• DOI: https://doi.org/10.1007/978-3-319-26626-8_43

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• Publisher Name: Springer, Cham

• Print ISBN: 978-3-319-26625-1

• Online ISBN: 978-3-319-26626-8

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