Skip to main content
SpringerLink
Log in

Exact algorithms for the minimum s-club partitioning problem

  • Computational Biomedicine
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

Graph clustering (partitioning) is a helpful tool in understanding complex systems and analyzing their structure and internal properties. One approach for graph clustering is based on partitioning the graph into cliques. However, clique models are too restrictive and prone to errors given imperfect data. Thus, using clique relaxations instead may provide a more reasonable and applicable partitioning of the graph. An s-club is a distance-based relaxation of a clique and is formally defined as a subset of vertices inducing a subgraph with a diameter of at most s. In this work, we study the minimum s-club partitioning problem, which is to partition the graph into a minimum number of non-overlapping s-club clusters. Integer programming techniques and combinatorial branch-and-bound framework are employed to develop exact algorithms to solve this problem. We also study and compare the computational performance of the proposed algorithms for the special cases of \(s=2\) and \(s=3\) on a test-bed of randomly generated instances and real-life graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Abbasi, A. A., & Younis, M. (2007). A survey on clustering algorithms for wireless sensor networks. Computer Communications, 30(14), 2826–2841.

    Article  Google Scholar 

  • Abbas, N., & Stewart, L. (1999). Clustering bipartite and chordal graphs: Complexity, sequential and parallel algorithms. Discrete Applied Mathematics, 91(1), 1–23.

    Article  Google Scholar 

  • Balasundaram, B., Butenko, S., & Trukhanov, S. (2005). Novel approaches for analyzing biological networks. Journal of Combinatorial Optimization, 10(1), 23–39.

    Article  Google Scholar 

  • Berry, N., Ko, T., Moy, T., Smrcka, J., Turnley, J., & Wu, B. (2004) Emergent clique formation in terrorist recruitment. The AAAI-04 workshop on agent organizations: Theory and practice, July 25, 2004, San Jose, California. http://www.cs.uu.nl/~virginia/aotp/papers.htm.

  • Bourjolly, J. M., Laporte, G., & Pesant, G. (2000). Heuristics for finding \(k\)-clubs in an undirected graph. Computers and Operations Research, 27(6), 559–569.

    Article  Google Scholar 

  • Bourjolly, J. M., Laporte, G., & Pesant, G. (2002). An exact algorithm for the maximum \(k\)-club problem in an undirected graph. European Journal of Operational Research, 138(1), 21–28.

    Article  Google Scholar 

  • Buchanan, A., Sung, J., Butenko, S., & Pasiliao, E. L. (2015). An integer programming approach for fault-tolerant connected dominating sets. INFORMS Journal on Computing, 27, 178–188.

    Article  Google Scholar 

  • Chang, J. M., Yang, J. S., & Peng, S. L. (2014). On the complexity of graph clustering with bounded diameter. In 2014 international computer science and engineering conference (ICSEC) (pp. 18–22). IEEE.

  • Csardi, G., & Nepusz, T. (2006). The igraph software package for complex network research. InterJournal Complex Systems, 1695(5), 1–9.

    Google Scholar 

  • De Amorim, S. G., Barthélemy, J. P., & Ribeiro, C. C. (1992). Clustering and clique partitioning: Simulated annealing and tabu search approaches. Journal of Classification, 9(1), 17–41.

    Article  Google Scholar 

  • Deogun, J. S., Kratsch, D., & Steiner, G. (1997). An approximation algorithm for clustering graphs with dominating diametral path. Information Processing Letters, 61(3), 121–127.

    Article  Google Scholar 

  • De, T., Pal, A., & Sengupta, I. (2010). Traffic grooming, routing, and wavelength assignment in an optical wdm mesh networks based on clique partitioning. Photonic Network Communications, 20(2), 101–112.

    Article  Google Scholar 

  • DIMACS (2012) Algorithm implementation challenge: Graph partitioning and graph clustering. The tenth DIMACS implementation challenge. http://dimacs.rutgers.edu/Challenges/.

  • Dorndorf, U., Jaehn, F., & Pesch, E. (2008). Modelling robust flight-gate scheduling as a clique partitioning problem. Transportation Science, 42(3), 292–301.

    Article  Google Scholar 

  • Dorndorf, U., & Pesch, E. (1994). Fast clustering algorithms. ORSA Journal on Computing, 6(2), 141–153.

    Article  Google Scholar 

  • Du, D. Z., & Wan, P. J. (2013). Connected dominating set: Theory and applications. New York: Springer.

    Book  Google Scholar 

  • European Bioinformatics Institute. (2017). IntAct molecular interaction database. http://www.ebi.ac.uk/intact/. Accessed May, 2017.

  • Fernandess, Y., & Malkhi, D. (2002). K-clustering in wireless ad hoc networks. In Proceedings of the second ACM international workshop on principles of mobile computing (pp. 31–37). ACM.

  • Fortunato, S. (2010). Community detection in graphs. Physics Reports, 486(3), 75–174.

    Article  Google Scholar 

  • Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. New York: WH Freeman.

    Google Scholar 

  • Gendreau, M., Soriano, P., & Salvail, L. (1993). Solving the maximum clique problem using a tabu search approach. Annals of Operations Research, 41(4), 385–403.

    Article  Google Scholar 

  • González, R. C., & Tou, J. T. (1974). Pattern recognition principles. Applied mathematics and computation. Reading, MA: Addison-Wesley.

    Google Scholar 

  • Grötschel, M., & Wakabayashi, Y. (1989). A cutting plane algorithm for a clustering problem. Mathematical Programming, 45(1–3), 59–96.

    Article  Google Scholar 

  • Grötschel, M., & Wakabayashi, Y. (1990). Facets of the clique partitioning polytope. Mathematical Programming, 47(1–3), 367–387.

    Article  Google Scholar 

  • GUROBI. (2016). Gurobi optimizer. http://www.gurobi.com.

  • Kershenbaum, A. (1993). Telecommunications network design algorithms. New York: McGraw-Hill.

    Google Scholar 

  • Kochenberger, G., Glover, F., Alidaee, B., & Wang, H. (2005). Clustering of microarray data via clique partitioning. Journal of Combinatorial Optimization, 10(1), 77–92.

    Article  Google Scholar 

  • Krishna, P., Vaidya, N. H., Chatterjee, M., & Pradhan, D. K. (1997). A cluster-based approach for routing in dynamic networks. ACM SIGCOMM Computer Communication Review, 27(2), 49–64.

    Article  Google Scholar 

  • Levy, E. D., Pereira-Leal, J. B., Chothia, C., & Teichmann, S. A. (2006). 3D complex: A structural classification of protein complexes. PLoS Computational Biology, 2(11), e155.

    Article  Google Scholar 

  • Li, Y., Lao, L., & Cui, J. H. (2006). Sdc: A distributed clustering protocol for peer-to-peer networks. In International conference on research in networking (pp. 1234–1239). Springer.

  • Mahdavi Pajouh, F., & Balasundaram, B. (2012). On inclusionwise maximal and maximum cardinality \(k\)-clubs in graphs. Discrete Optimization, 9(2), 84–97.

    Article  Google Scholar 

  • Mehrotra, A., & Trick, M. A. (1998). Cliques and clustering: A combinatorial approach. Operations Research Letters, 22(1), 1–12.

    Article  Google Scholar 

  • Mokken, R. J. (1979). Cliques, clubs and clans. Quality and Quantity, 13(2), 161–173.

    Article  Google Scholar 

  • Newman, M. E. J., & Girvan, M. (2004). Finding and evaluating community structure in networks. Physical Review E, 69(2), 026,113.

    Article  Google Scholar 

  • Oosten, M., Rutten, J. H. G. C., & Spieksma, F. C. R. (2001). The clique partitioning problem: Facets and patching facets. Networks, 38(4), 209–226.

    Article  Google Scholar 

  • Parley, A., Hedetniemi, S., & Proskurowski, A. (1981). Partitioning trees: Matching, domination, and maximum diameter. International Journal of Computer and Information Sciences, 10(1), 55–61.

    Article  Google Scholar 

  • Pasupuleti, S. (2008). Detection of protein complexes in protein interaction networks using \(n\)-clubs. In Proceedings of the 6th European conference on evolutionary computation, machine learning and data mining in bioinformatics. Lecture notes in computer science (Vol. 4973, pp. 153–164). Springer.

  • Pattillo, J., Youssef, N., & Butenko, S. (2013). On clique relaxation models in network analysis. European Journal of Operational Research, 226(1), 9–18.

    Article  Google Scholar 

  • Rothenberg, R. B., Potterat, J. J., & Woodhouse, D. E. (1996). Personal risk taking and the spread of disease: Beyond core groups. Journal of Infectious Diseases, 174(Supp. 2), S144–S149.

    Article  Google Scholar 

  • Sageman, M. (2004). Understanding terrorist networks. Philadelphia, PA: University of Pennsylvania Press.

    Google Scholar 

  • Sampson, R. J., & Groves, B. W. (1989). Community structure and crime: Testing social-disorganization theory. American Journal of Sociology, 94, 774–802.

    Article  Google Scholar 

  • Shahinpour, S., & Butenko, S. (2013a). Algorithms for the maximum \(k\)-club problem in graphs. Journal of Combinatorial Optimization, 26(3), 520–554.

    Article  Google Scholar 

  • Shahinpour, S., & Butenko, S. (2013b). Dinstance-based clique relaxations in networks: \(s\)-cliques and \(s\)-clubs. In B. I. Goldengorin, V. A. Kalyagin, & P. M. Pardalos (Eds.), Models, algorithms, and technologies for network analysis. Springer proceedings in mathematics and statistics (Vol. 59, pp. 149–174). New York: Springer.

    Google Scholar 

  • Sharan, R., Maron-Katz, A., & Shamir, R. (2003). Click and expander: A system for clustering and visualizing gene expression data. Bioinformatics, 19(14), 1787–1799.

    Article  Google Scholar 

  • Wu, Z., & Leahy, R. (1993). An optimal graph theoretic approach to data clustering: Theory and its application to image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(11), 1101–1113.

    Article  Google Scholar 

Download references

Acknowledgements

This material is based upon work supported by the AFRL Mathematical Modeling and Optimization Institute. Partial support by AFOSR under Grant FA8651-14-2-0005, and NSF Grant CMMI-1538493 is also gratefully acknowledged.

Author information

Authors and Affiliations

  1. Industrial and Systems Engineering, Texas A&M University, College Station, TX, 77843-3131, USA

    Oleksandra Yezerska & Sergiy Butenko

  2. Department of Management Science and Information Systems, University of Massachusetts Boston, Boston, MA, 02125-3393, USA

    Foad Mahdavi Pajouh

  3. Department of Industrial Engineering and Management Systems, University of Central Florida, 12800 Pegasus Dr., Orlando, FL, 32816-2993, USA

    Alexander Veremyev

Authors
  1. Oleksandra Yezerska
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Foad Mahdavi Pajouh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexander Veremyev
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sergiy Butenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergiy Butenko.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yezerska, O., Mahdavi Pajouh, F., Veremyev, A. et al. Exact algorithms for the minimum s-club partitioning problem. Ann Oper Res 276, 267–291 (2019). https://doi.org/10.1007/s10479-017-2665-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-017-2665-2

Keywords

Search

Navigation