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Algorithmica

, Volume 80, Issue 6, pp 1834–1856 | Cite as

Optimal Approximation Algorithms for Maximum Distance-Bounded Subgraph Problems

  • Yuichi Asahiro
  • Yuya Doi
  • Eiji Miyano
  • Kazuaki Samizo
  • Hirotaka Shimizu
Article

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.

Keywords

d-clique d-club Maximum distance-bounded subgraph problems (In)approximability 

Notes

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.

References

  1. 1.
    Abello, J., Resende, M.G., Sudarsky, S.: Massive quasi-clique detection. In: Proceedings of LATIN 2002, LNCS 2286, pp. 598–612 (2002)Google Scholar
  2. 2.
    Alba, R.: A graph-theoretic definition of a sociometric clique. J. Math. Sociol. 3, 113–126 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Asahiro, Y., Doi, Y., Miyano, E., Shimizu, H.: Optimal approximation algorithms for maximum distance-bounded subgraph problems. In: Proceedings of COCOA 2015, LNCS 9486, pp. 586–600 (2015)Google Scholar
  4. 4.
    Asahiro, Y., Miyano, E., Samizo, K.: Approximating maximum diameter-bounded subgraphs. In: Proceedings of LATIN 2010, LNCS 6034, pp. 615–626 (2010)Google Scholar
  5. 5.
    Balasundaram, B., Butenko, S., Trukhanov, S.: Novel approaches for analyzing biological networks. J. Comb. Optim. 10(1), 23–39 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bellare, M., Goldreich, O., Sudan, M.: Free bits, PCPs and non-approximability—towards tight results. SIAM J. Comput. 27(3), 804–915 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bourjolly, J.-M., Laporte, G., Pesant, G.: An exact algorithm for the maximum \(k\)-club problem in an undirected graph. Eur. J. Oper. Res. 138, 21–28 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chang, M.-S., Hung, L.-J., Lin, C.-R., Su, P.-C.: Finding large \(k\)-clubs in undirected graphs. Computing 95, 739–758 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Erdös, P., Pach, J., Pollack, R., Tuza, Z.: Radius, diameter, and minimum degree. J. Comb. Theory 47, 73–79 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Feige, U.: Approximating maximum clique by removing subgraphs. SIAM J. Discrete Math. 18, 219–225 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Galil, Z., Margalit, O.: All pairs shortest distances for graphs with small integer length edges. Inf. Comput. 134, 103–139 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Galil, Z., Margalit, O.: All pairs shortest paths for graphs with small integer length edges. J. Comput. Syst. Sci. 54, 243–254 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gavril, F.: Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM J. Comput. 1(2), 180–187 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Golovach, P.A., Heggernes, P., Kratsch, D., Rafiey, A.: Finding clubs in graph classes. Discrete Appl. Math. 174, 57–65 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hartung, S., Komusiewicz, C., Nichterlein, A.: Parameterized algorithmics and computational experiments for finding \(2\)-clubs. J. Graph Algorithm Appl. 19(1), 155–190 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hartung, S., Komusiewicz, C., Nichterlein, A., Suchý, O.: On structural parameterizations for the \(2\)-club problem. Discrete Appl. Math. 185, 79–92 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Håstad, J.: Clique is hard to approximate within \(n^{1-\varepsilon }\). Acta Math. 182(1), 105–142 (1999)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kahruman-Anderoglu, S., Buchanan, A., Butenko, S.: On provably best construction heuristics for hard combinatorial optimization problems. Networks 67(3), 238–245 (2016)CrossRefGoogle Scholar
  19. 19.
    Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103 (1972)Google Scholar
  20. 20.
    Luce, R.D., Perry, A.D.: A method of matrix analysis of group structure. Psychometrika 14, 95–116 (1949)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Luce, R.D.: Connectivity and generalized cliques in sociometric group structure. Psychometrika 15(2), 169–190 (1950)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Marinc̆ek, J., Mohar, B.: On approximating the maximum diameter ratio of graphs. Discrete Math. 244, 323–330 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mokken, R.J.: Cliques, clubs and clans. Qual. Quant. 13, 161–173 (1979)CrossRefGoogle Scholar
  24. 24.
    Mardavi Pajouh, F., Balasundaram, B.: On inclusionwise maximal and maximum cardinality \(k\)-clubs in graphs. Discrete Optim. 9, 84–97 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pasupuleti, S.: Detection of protein complexes in protein interaction networks using \(n\)-clubs. In: Proceedings of EvoBIO 2008, pp. 153–164 (2008)Google Scholar
  26. 26.
    Schäfer, A., Komusiewicz, C., Moser, H., Niedermeier, R.: Parameterized computational complexity of finding small-diameter subgraphs. Optim. Lett. 6(5), 883–891 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Seidel, R.: On the all-pairs-shortest-path problem in unweighted undirected graphs. J. Comput. Syst. Sci. 51, 400–403 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Seidman, S.B., Foster, B.L.: A graph-theoretic generalization of the clique concept. J. Math. Sociol. 6(1), 139–154 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Seidman, S.B.: Network structure and minimum degree. Soc. Netw. 5(3), 269–287 (1983)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Vazirani, V.V.: Approximation Algorithms. Springer, New York (2003)CrossRefGoogle Scholar
  31. 31.
    Veremyev, A., Boginski, V.: Identifying large robust network clusters via new compact formulations of maximum \(k\)-club problems. Eur. J. Oper. Res. 218, 316–326 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Williams, V.V.: Multiplying matrices faster than Coppersmith–Winograd. In: Proceedings of STOC 2012, pp. 887–898 (2012)Google Scholar
  33. 33.
    Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3, 103–128 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Yuichi Asahiro
    • 1
  • Yuya Doi
    • 2
  • Eiji Miyano
    • 2
  • Kazuaki Samizo
    • 2
  • Hirotaka Shimizu
    • 2
  1. 1.Department of Information ScienceKyushu Sangyo UniversityFukuokaJapan
  2. 2.Department of Systems Design and InformaticsKyushu Institute of TechnologyFukuokaJapan

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