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Experimental Evaluation of Approximation and Heuristic Algorithms for Maximum Distance-Bounded Subgraph Problems

  • Yuichi Asahiro
  • Tomohiro Kubo
  • Eiji MiyanoEmail author
Article
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Abstract

In this paper, we consider two distance-based relaxed variants of the maximum clique problem (Max Clique), named Maxd-Clique and Maxd-Club for positive integers d. Max 1-Clique and Max 1-Club cannot be efficiently approximated within a factor of \(n^{1-\varepsilon }\) for any real \(\varepsilon > 0\) unless \({{{\mathcal {P}}}} = {{\mathcal {NP}}}\), since they are identical to Max Clique (Håstad in Acta Math 182(1):105–142, 1999; Zuckerman in Theory Comput 3:103–128, 2007). In addition, it is \({{\mathcal {NP}}}\)-hard to approximate Maxd-Clique and Maxd-Club to within a factor of \(n^{1/2 - \varepsilon }\) for any fixed integer \(d\ge 2\) and any real \(\varepsilon > 0\) (Asahiro et al. in Approximating maximum diameter-bounded subgraphs. In: Proc of LATIN 2010, Springer, pp 615–626, 2010; Asahiro et al. in Optimal approximation algorithms for maximum distance-bounded subgraph problems. In: Proc of COCOA, Springer, pp 586–600, 2015). As for approximability of Maxd-Clique and Maxd-Club, a polynomial-time algorithm, called ReFindStar\(_d\), that achieves an optimal approximation ratio of \(O(n^{1/2})\) for Maxd-Clique and Maxd-Club was designed for any integer \(d\ge 2\) in Asahiro et al. (2015, Algorithmica 80(6):1834–1856, 2018). Moreover, a simpler algorithm, called ByFindStar\(_d\), was proposed and it was shown in Asahiro et al. (2010, 2018) that although the approximation ratio of ByFindStar\(_d\) is much worse for any odd\(d\ge 3\), its time complexity is better than ReFindStar\(_d\). In this paper, we implement those approximation algorithms and evaluate their quality empirically for random graphs. The experimental results show that (1) ReFindStar\(_d\) can find larger d-clubs (d-cliques) than ByFindStar\(_d\) for odd d, (2) the size of d-clubs (d-cliques) output by ByFindStar\(_d\) is the same as ones by ReFindStar\(_d\) for even d, and (3) ByFindStar\(_d\) can find the same size of d-clubs (d-cliques) much faster than ReFindStar\(_d\). Furthermore, we propose and implement two new heuristics, Hclub\(_d\) for Maxd-Club and Hclique\(_d\) for Maxd-Clique. Then, we present the experimental evaluation of the solution size of ReFindStar\(_d\), Hclub\(_d\), Hclique\(_d\) and previously known heuristic algorithms for random graphs and Erdős collaboration graphs.

Keywords

Maximum distance-bounded subgraph problems d-Clique d-Club Approximation algorithms Heuristic algorithms 

Notes

Acknowledgements

This work was partially supported by JST CREST JPMJR1402 and the Grants-in-Aid for Scientific Research of Japan (KAKENHI) Grant numbers JP17K00016 and JP17K00024.

References

  1. 1.
    Abello, J., Resende, M.G., Sudarsky, S. (2002). Massive quasi-clique detection. In: Proc of LATIN 2002 (pp. 598–612). Springer.Google Scholar
  2. 2.
    Alba, R. D. (1973). A graph-theoretic definition of a sociometric clique. Journal of Mathematical Sociology, 3(1), 113–126.CrossRefGoogle Scholar
  3. 3.
    Asahiro, Y., Doi, Y., Miyano, E., Shimizu, H. (2015). Optimal approximation algorithms for maximum distance-bounded subgraph problems. In: Proc of COCOA (pp. 586–600). Springer.Google Scholar
  4. 4.
    Asahiro, Y., Doi, Y., Miyano, E., & Shimizu, H. (2018). Optimal approximation algorithms for maximum distance-bounded subgraph problems. Algorithmica, 80(6), 1834–1856.CrossRefGoogle Scholar
  5. 5.
    Asahiro, Y., Kubo, T., Miyano, E. (2016). Experimental evaluation of approximation algorithms for maximum distance-bounded subgraph problems. In: Proc of SCIS & ISIS (pp. 892–897).Google Scholar
  6. 6.
    Asahiro, Y., Miyano, E., Samizo, K. (2010). Approximating maximum diameter-bounded subgraphs. In: Proc of LATIN 2010 (pp. 615–626). Springer.Google Scholar
  7. 7.
    Batagelj, V., Mrvar, A.: Graph files in bajek datasets. http://vlado.fmf.uni-lj.si/pub/networks/pajek/data/gphs.htm. Accessed Dec 2018.
  8. 8.
    Bollobás, B. (2001). Random graphs. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  9. 9.
    boost C++ Libraries – johnson\_all\_pairs\_shortest\_paths: http://www.boost.org/doc/libs/1_60_0/libs/graph/doc/johnson_all_pairs_shortest.html. Accessed Nov 2017.
  10. 10.
    Bourjolly, J. M., Laporte, G., & Pesant, G. (2000). Heuristics for finding \(k\)-clubs in an undirected graph. Computers & Operations Research, 27, 559–569.CrossRefGoogle Scholar
  11. 11.
    Carraghan, R., & Pardalos, P. (1990). An exact algorithm for the maximum clique problem. Operations Research Letters, 9(6), 375–382.CrossRefGoogle Scholar
  12. 12.
    Erdős, P., & Rényi, A. (1959). On random graphs I. Publicationes Mathematicae, 6, 290–297.Google Scholar
  13. 13.
    Galil, Z., & Margalit, O. (1977). All pairs shortest distances for graphs with small integer length edges. Information & Computation, 134, 103–139.CrossRefGoogle Scholar
  14. 14.
    Galil, Z., & Margalit, O. (1977). All pairs shortest paths for graphs with small integer length edges. Journal of Computer and System Sciences, 54, 243–254.CrossRefGoogle Scholar
  15. 15.
    Grossman, J., Ion, P., Castro, R.: Erdős number project. https://oakland.edu/enp/. Accessed Dec 2018.
  16. 16.
    Grosso, A., Locatelli, M., & Croce, F. (2004). Combining swaps and node weights in an adaptive greedy approach for the maximum clique problem. Journal of Heuristics, 10(2), 135–152.CrossRefGoogle Scholar
  17. 17.
    Håstad, J. (1999). Clique is hard to approximate within \(n^{1-\varepsilon }\). Acta Mathematics, 182(1), 105–142.CrossRefGoogle Scholar
  18. 18.
    Karp, R. (1972). Reducibility among combinatorial problems. Complexity of computer computations (pp. 85–103). Boston: Springer.CrossRefGoogle Scholar
  19. 19.
    Katayama, K., Hamamoto, A., & Narihisa, H. (2005). An effective local search for the maximum clique problem. Information Processing Letters, 95(5), 503–511.CrossRefGoogle Scholar
  20. 20.
    Le Gall, F. (2014) Powers of tensors and fast matrix multiplication. In: Proc of ISAAC, pp. 296–303.Google Scholar
  21. 21.
    Luce, R., & Perry, A. (1949). A method of matrix analysis of group structure. Psychometrika, 14, 95–116.CrossRefGoogle Scholar
  22. 22.
    Luce, R. D. (1950). Connectivity and generalized cliques in sociometric group structure. Psychometrika, 15(2), 169–190.CrossRefGoogle Scholar
  23. 23.
    Marinček, J., & Mohar, B. (2002). On approximating the maximum diameter ratio of graphs. Discrete Mathematics, 244, 323–330.CrossRefGoogle Scholar
  24. 24.
    Maslov, E., Batsyn, M., & Pardalos, P. (2014). Speeding up branch and bound algorithms for solving the maximum clique problem. Journal of Global Optimization, 59(1), 1–21.CrossRefGoogle Scholar
  25. 25.
    Mokken, R. J. (1979). Cliques, clubs and clans. Quality and Quantity, 13, 161–173.CrossRefGoogle Scholar
  26. 26.
    Moore, C., & Mertens, S. (2011). The nature of computation. Oxford: Oxford University Press.CrossRefGoogle Scholar
  27. 27.
    Östergȧrd, P. (2002). A fast algorithm for the maximum clique problem. Discrete Applied Mathematics, 120(1), 197–207.CrossRefGoogle Scholar
  28. 28.
    Pajouh, F. M., & Balasundaram, B. (2012). On inclusionwise maximal and maximum cardinality \(k\)-clubs in graphs. Descrete Optimization, 9, 84–97.CrossRefGoogle Scholar
  29. 29.
    Pattabiraman, B., Patwary, M. M. A., Gebremedhin, A. H., Liao, Wk, & Choudhary, A. (2015). Fast algorithms for the maximum clique problem on massive graphs with applications to overlapping community detection. Internet Mathematics, 11(4–5), 421–448.CrossRefGoogle Scholar
  30. 30.
    Seidel, R. (1995). On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences, 51, 400–403.CrossRefGoogle Scholar
  31. 31.
    Seidman, S. B. (1983). Network structure and minimum degree. Social Networks, 5(3), 269–287.CrossRefGoogle Scholar
  32. 32.
    Seidman, S. B., & Foster, B. L. (1978). A graph-theoretic generalization of the clique concept. Journal of Mathematical Sociology, 6(1), 139–154.CrossRefGoogle Scholar
  33. 33.
    Shahinpour, S., & Butenko, S. (2013). Algorithms for the maximum \(k\)-club problem in graphs. Journal of Combinatorial Optimization, 26, 520–554.CrossRefGoogle Scholar
  34. 34.
    Tomita, E., Seki, T. (2003). An efficient branch-and-bound algorithm for finding a maximum clique. In: Proc of DMTCS (pp. 278–289).Google Scholar
  35. 35.
    Tomita, E., Sutani, Y., Higashi, T., Takahashi, S., Wakatsuki, M. (2010) A simple and faster branch-and-bound algorithm for finding a maximum clique. In: Proc of WALCOM (pp. 191–203).Google Scholar
  36. 36.
    Zuckerman, D. (2007). Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing, 3, 103–128.CrossRefGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Information ScienceKyushu Sangyo UniversityFukuokaJapan
  2. 2.Department of Artificial IntelligenceKyushu Institute of TechnologyFukuokaJapan

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