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

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.

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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.

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Correspondence to Eiji Miyano.

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A preliminary version of this paper appeared in Proceedings of the Joint 8th International Conference on Soft Computing and Intelligent Systems and the 17th International Symposium on Advanced Intelligent Systems, 892–897, 2016 [5].

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Asahiro, Y., Kubo, T. & Miyano, E. Experimental Evaluation of Approximation and Heuristic Algorithms for Maximum Distance-Bounded Subgraph Problems. Rev Socionetwork Strat 13, 143–161 (2019). https://doi.org/10.1007/s12626-019-00036-2

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Keywords

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