On Maximum Weight Clique Algorithms, and How They Are Evaluated

  • Ciaran McCreesh
  • Patrick Prosser
  • Kyle Simpson
  • James Trimble
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10416)

Abstract

Maximum weight clique and maximum weight independent set solvers are often benchmarked using maximum clique problem instances, with weights allocated to vertices by taking the vertex number mod 200 plus 1. For constraint programming approaches, this rule has clear implications, favouring weight-based rather than degree-based heuristics. We show that similar implications hold for dedicated algorithms, and that additionally, weight distributions affect whether certain inference rules are cost-effective. We look at other families of benchmark instances for the maximum weight clique problem, coming from winner determination problems, graph colouring, and error-correcting codes, and introduce two new families of instances, based upon kidney exchange and the Research Excellence Framework. In each case the weights carry much more interesting structure, and do not in any way resemble the 200 rule. We make these instances available in the hopes of improving the quality of future experiments.

References

  1. 1.
    Abraham, D.J., Blum, A., Sandholm, T.: Clearing algorithms for barter exchange markets: enabling nationwide kidney exchanges. In: MacKie-Mason, J.K., Parkes, D.C., Resnick, P. (eds.) Proceedings 8th ACM Conference on Electronic Commerce (EC-2007), 11–15 June 2007, San Diego, California, USA, pp. 295–304. ACM (2007). http://doi.acm.org/10.1145/1250910.1250954
  2. 2.
    Araujo Tavares, W.: Algoritmos exatos para problema da clique maxima ponderada. Ph.D. thesis, Universidade federal do Ceará (2016). http://www.theses.fr/2016AVIG0211
  3. 3.
    Baz, D.E., Hifi, M., Wu, L., Shi, X.: A parallel ant colony optimization for the maximum-weight clique problem. In: 2016 IEEE International Parallel and Distributed Processing Symposium Workshops, IPDPS Workshops 2016, 23–27 May 2016, Chicago, IL, USA, pp. 796–800. IEEE Computer Society (2016). doi:10.1109/IPDPSW.2016.111
  4. 4.
    Benlic, U., Hao, J.: Breakout local search for maximum clique problems. Comput. OR 40(1), 192–206 (2013). doi:10.1016/j.cor.2012.06.002 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Berman, P., Pelc, A.: Distributed probabilistic fault diagnosis for multiprocessor systems. In: Proceedings of the 20th International Symposium on Fault-Tolerant Computing, FTCS 1990, 26–28 June 1990, Newcastle Upon Tyne, UK, pp. 340–346. IEEE Computer Society (1990). doi:10.1109/FTCS.1990.89383
  6. 6.
    Boginski, V., Butenko, S., Shirokikh, O., Trukhanov, S., Gil-Lafuente, J.: A network-based data mining approach to portfolio selection via weighted clique relaxations. Ann. OR 216(1), 23–34 (2014). doi:10.1007/s10479-013-1395-3 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. In: Du, D.Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, pp. 1–74. Springer, Boston (1999). doi:10.1007/978-1-4757-3023-4_1
  8. 8.
    Boussemart, F., Hemery, F., Lecoutre, C., Sais, L.: Boosting systematic search by weighting constraints. In: de Mántaras, R.L., Saitta, L. (eds.) Proceedings of the 16th Eureopean Conference on Artificial Intelligence, ECAI 2004, Including Prestigious Applicants of Intelligent Systems, PAIS 2004, 22–27 August 2004, Valencia, Spain, pp. 146–150. IOS Press (2004)Google Scholar
  9. 9.
    Brockington, M., Culberson, J.C.: Camouflaging independent sets in quasi-random graphs. In: Johnson and Trick [31], pp. 75–88. http://dimacs.rutgers.edu/Volumes/Vol26.html
  10. 10.
    Cai, S., Lin, J.: Fast solving maximum weight clique problem in massive graphs. In: Kambhampati, S. (ed.) Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, IJCAI 2016, 9–15 July 2016, New York, NY, USA, pp. 568–574. IJCAI/AAAI Press (2016). http://www.ijcai.org/Abstract/16/087
  11. 11.
    Carraghan, R., Pardalos, P.M.: An exact algorithm for the maximum clique problem. Oper. Res. Lett. 9, 375–382 (1990)CrossRefMATHGoogle Scholar
  12. 12.
    Chvátal, V.: Resolution search. Discrete Appl. Math. 73(1), 81–99 (1997). doi:10.1016/S0166-218X(96)00003-0 MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cohen, D.A., Cooper, M.C., Creed, P., Marx, D., Salamon, A.Z.: The tractability of CSP classes defined by forbidden patterns. J. Artif. Intell. Res. (JAIR) 45, 47–78 (2012). doi:10.1613/jair.3651 MathSciNetMATHGoogle Scholar
  14. 14.
    Cohen, D.A., Jeavons, P., Jefferson, C., Petrie, K.E., Smith, B.M.: Symmetry definitions for constraint satisfaction problems. Constraints 11(2–3), 115–137 (2006). doi:10.1007/s10601-006-8059-8 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Cooper, M.C., Jeavons, P.G., Salamon, A.Z.: Generalizing constraint satisfaction on trees: hybrid tractability and variable elimination. Artif. Intell. 174(9–10), 570–584 (2010). doi:10.1016/j.artint.2010.03.002 MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Cooper, M.C., Zivny, S.: Hybrid tractable classes of constraint problems. In: Krokhin, A.A., Zivny, S. (eds.) The Constraint Satisfaction Problem: Complexity and Approximability, Dagstuhl Follow-Ups, vol. 7, pp. 113–135. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2017). doi:10.4230/DFU.Vol7.15301.4
  17. 17.
    Debroni, J., Eblen, J.D., Langston, M.A., Myrvold, W., Shor, P.W., Weerapurage, D.: A complete resolution of the Keller maximum clique problem. In: Randall, D. (ed.) Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, 23–25 January 2011, San Francisco, California, USA, pp. 129–135. SIAM (2011). doi: 10.1137/1.9781611973082.11
  18. 18.
    Depolli, M., Konc, J., Rozman, K., Trobec, R., Janezic, D.: Exact parallel maximum clique algorithm for general and protein graphs. J. Chem. Inf. Model. 53(9), 2217–2228 (2013). doi:10.1021/ci4002525 CrossRefGoogle Scholar
  19. 19.
    Dickerson, J.P., Procaccia, A.D., Sandholm, T.: Optimizing kidney exchange with transplant chains: theory and reality. In: van der Hoek, W., Padgham, L., Conitzer, V., Winikoff, M. (eds.) International Conference on Autonomous Agents and Multiagent Systems, AAMAS 2012, IFAAMAS, 4–8 June 2012, Valencia, Spain, vol. 3, pp. 711–718 (2012). http://dl.acm.org/citation.cfm?id=2343798
  20. 20.
    Fan, Y., Li, C., Ma, Z., Wen, L., Sattar, A., Su, K.: Local search for maximum vertex weight clique on large sparse graphs with efficient data structures. In: Kang, B.H., Bai, Q. (eds.) AI 2016. LNCS, vol. 9992, pp. 255–267. Springer, Cham (2016). doi:10.1007/978-3-319-50127-7_21 CrossRefGoogle Scholar
  21. 21.
    Fang, Z., Li, C., Qiao, K., Feng, X., Xu, K.: Solving maximum weight clique using maximum satisfiability reasoning. In: Schaub, T., Friedrich, G., ÓSullivan, B. (eds.) ECAI 2014–21st European Conference on Artificial Intelligence, 18–22 August 2014, Prague, Czech Republic - Including Prestigious Applications of Intelligent Systems (PAIS) 2014. Frontiers in Artificial Intelligence and Applications, vol. 263, pp. 303–308. IOS Press (2014). doi:10.3233/978-1-61499-419-0-303
  22. 22.
    Fang, Z., Li, C., Xu, K.: An exact algorithm based on maxsat reasoning for the maximum weight clique problem. J. Artif. Intell. Res. (JAIR) 55, 799–833 (2016). doi:10.1613/jair.4953 MathSciNetMATHGoogle Scholar
  23. 23.
    Gendreau, M., Soriano, P., Salvail, L.: Solving the maximum clique problem using a tabu search approach. Ann. OR 41(4), 385–403 (1993). doi:10.1007/BF02023002 CrossRefMATHGoogle Scholar
  24. 24.
    Glorie, K., Haase-Kromwijk, B., van de Klundert, J., Wagelmans, A., Weimar, W.: Allocation and matching in kidney exchange programs. Transpl. Int. 27(4), 333–343 (2014)CrossRefGoogle Scholar
  25. 25.
    Gouveia, L., Martins, P.: Solving the maximum edge-weight clique problem in sparse graphs with compact formulations. EURO J. Comput. Optim. 3(1), 1–30 (2015). doi:10.1007/s13675-014-0028-1 MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Held, S., Cook, W.J., Sewell, E.C.: Maximum-weight stable sets and safe lower bounds for graph coloring. Math. Program. Comput. 4(4), 363–381 (2012). doi:10.1007/s12532-012-0042-3 MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Hosseinian, S., Fontes, D., Butenko, S.: A quadratic approach to the maximum edge weight clique problem. In: XIII Global Optimization Workshop GOW 2016, pp. 125–128 (2016)Google Scholar
  28. 28.
    Jégou, P.: Decomposition of domains based on the micro-structure of finite constraint-satisfaction problems. In: Fikes, R., Lehnert, W.G. (eds.) Proceedings of the 11th National Conference on Artificial Intelligence, 11–15 July 1993, Washington, DC, USA, pp. 731–736. AAAI Press/The MIT Press (1993). http://www.aaai.org/Library/AAAI/1993/aaai93-109.php
  29. 29.
    Jiang, H., Li, C., Manyà, F.: An exact algorithm for the maximum weight clique problem in large graphs. In: Singh, S.P., Markovitch, S. (eds.) Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, 4–9 February 2017, San Francisco, California, USA, pp. 830–838. AAAI Press (2017). http://aaai.org/ocs/index.php/AAAI/AAAI17/paper/view/14370
  30. 30.
    Johnson, D.S., Trick, M.A.: Introduction to the second DIMACS challenge: cliques, coloring, and satisfiability. In: Cliques, Coloring, and Satisfiability, Proceedings of a DIMACS Workshop, 11–13 October 1993, New Brunswick, New Jersey, USA, [31], pp. 1–10. http://dimacs.rutgers.edu/Volumes/Vol26.html
  31. 31.
    Johnson, D.S., Trick, M.A. (eds.): Cliques, coloring, and satisfiability. In: Proceedings of a DIMACS Workshop, DIMACS/AMS, 11–13 October 1993, New Brunswick, New Jersey, USA. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 26 (1996). http://dimacs.rutgers.edu/Volumes/Vol26.html
  32. 32.
    Kumlander, D.: On importance of a special sorting in the maximum-weight clique algorithm based on colour classes. In: An, L.T.H., Bouvry, P., Tao, P.D. (eds.) Modelling, Computation and Optimization in Information Systems and Management Sciences, MCO 2008. Communications in Computer and Information Science, vol. 14, pp. 165–174. Springer, Heidelberg (2008). doi:10.1007/978-3-540-87477-5-18
  33. 33.
    Lau, H.C., Goh, Y.G.: An intelligent brokering system to support multi-agent web-based 4th-party logistics. In: 14th IEEE International Conference on Tools with Artificial Intelligence (ICTAI), 4–6 November 2002, Washington, DC, USA, p. 154. IEEE Computer Society (2002). doi:10.1109/TAI.2002.1180800
  34. 34.
    Malladi, K.T., Mitrovic-Minic, S., Punnen, A.P.: Clustered maximum weight clique problem: algorithms and empirical analysis. Comput. Oper. Res. 85, 113–128 (2017). http://www.sciencedirect.com/science/article/pii/S0305054817300837
  35. 35.
    Manlove, D.F., O’Malley, G.: Paired and altruistic kidney donation in the UK: algorithms and experimentation. ACM J. Exper. Algorithmics 19(1) (2014). http://doi.acm.org/10.1145/2670129
  36. 36.
    Mannino, C., Sassano, A.: Solving hard set covering problems. Oper. Res. Lett. 18(1), 1–5 (1995). doi:10.1016/0167-6377(95)00034-H MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Mannino, C., Stefanutti, E.: An augmentation algorithm for the maximum weighted stable set problem. Comput. Opt. Appl. 14(3), 367–381 (1999). doi:10.1023/A: 1026456624746 MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Mattei, N., Walsh, T.: Preflib: a library of preference data. In: Perny, P., Pirlot, M., Tsoukiàs, A. (eds.) ADT2013, vol. 8176, pp. 259–270. Springer, Heidelberg (2013). doi:10.1007/978-3-642-41575-3_20. http://www.preflib.org
  39. 39.
    McCreesh, C., Ndiaye, S.N., Prosser, P., Solnon, C.: Clique and constraint models for maximum common (connected) subgraph problems. In: Rueher, M. (ed.) CP 2016. LNCS, vol. 9892, pp. 350–368. Springer, Cham (2016). doi:10.1007/978-3-319-44953-1_23 CrossRefGoogle Scholar
  40. 40.
    Mehrotra, A., Trick, M.A.: A column generation approach for graph coloring. INFORMS J. Comput. 8(4), 344–354 (1996). doi:10.1287/ijoc.8.4.344 CrossRefMATHGoogle Scholar
  41. 41.
    Nogueira, B., Pinheiro, R.G.S., Subramanian, A.: A hybrid iterated local search heuristic for the maximum weight independent set problem. Optim. Lett. 1–17 (2017). doi:10.1007/s11590-017-1128-7
  42. 42.
    Östergård, P.R.J.: A new algorithm for the maximum-weight clique problem. Nord. J. Comput. 8(4), 424–436 (2001). http://www.cs.helsinki.fi/njc/References/ostergard2001: 424.html
  43. 43.
    Östergård, P.R.J.: A fast algorithm for the maximum clique problem. Discrete Appl. Math. 120(1–3), 197–207 (2002). doi:10.1016/S0166-218X(01)00290-6 MathSciNetCrossRefGoogle Scholar
  44. 44.
    Prosser, P.: CSPLib problem 063: Winner determination problem (combinatorial auction)Google Scholar
  45. 45.
    Pullan, W.J.: Approximating the maximum vertex/edge weighted clique using local search. J. Heuristics 14(2), 117–134 (2008). doi:10.1007/s10732-007-9026-2 CrossRefMATHGoogle Scholar
  46. 46.
    Refalo, P.: Impact-based search strategies for constraint programming. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 557–571. Springer, Heidelberg (2004). doi:10.1007/978-3-540-30201-8_41 CrossRefGoogle Scholar
  47. 47.
    Roth, A.E., Sönmez, T., Ünver, M.U.: Kidney exchange. Q. J. Econ. 119(2), 457 (2004). doi:10.1162/0033553041382157 CrossRefMATHGoogle Scholar
  48. 48.
    Roth, A.E., Sönmez, T., Ünver, M.U.: Efficient kidney exchange: coincidence of wants in markets with compatibility-based preferences. Am. Econ. Rev. 97(3), 828–851 (2007). http://www.aeaweb.org/articles?id=10.1257/aer.97.3.828
  49. 49.
    Saidman, S.L., Roth, A.E., Sonmez, T., Unver, M.U., Delmonico, F.L.: Increasing the opportunity of live kidney donation by matching for two- and three-way exchanges. Transplantation 81(5), 773–782 (2006)CrossRefGoogle Scholar
  50. 50.
    Sanchis, L.A.: Test case construction for the vertex cover problem. In: Dean, N., Shannon, G.E. (eds.) Computational Support for Discrete Mathematics, Proceedings of a DIMACS Workshop, 12–14 March 1992, Piscataway, New Jersey, USA. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, DIMACS/AMS, vol. 15, pp. 315–326 (1992). http://dimacs.rutgers.edu/Volumes/Vol15.html
  51. 51.
    Sanchis, L.A.: Generating hard and diverse test sets for NP-hard graph problems. Discrete Appl. Math. 58(1), 35–66 (1995). doi:10.1016/0166-218X(93)E0140-T MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Sandholm, T., Suri, S.: BOB: improved winner determination in combinatorial auctions and generalizations. Artif. Intell. 145(1–2), 33–58 (2003). doi:10.1016/S0004-3702(03)00015-8 MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Sandholm, T., Suri, S., Gilpin, A., Levine, D.: CABOB: a fast optimal algorithm for winner determination in combinatorial auctions. Manag. Sci. 51(3), 374–390 (2005). doi:10.1287/mnsc.1040.0336 CrossRefMATHGoogle Scholar
  54. 54.
    Sethuraman, S., Butenko, S.: The maximum ratio clique problem. Comput. Manag. Sci. 12(1), 197–218 (2015). doi:10.1007/s10287-013-0197-z MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Shimizu, S., Yamaguchi, K., Saitoh, T., Masuda, S.: Fast maximum weight clique extraction algorithm: optimal tables for branch-and-bound. Discrete Appl. Math. 223, 120–134 (2017). http://www.sciencedirect.com/science/article/pii/S0166218X1730063X
  56. 56.
    Soriano, P., Gendreau, M.: Tabu search algorithms for the maximum clique problem. In: Johnson and Trick [31], pp. 221–244. http://dimacs.rutgers.edu/Volumes/Vol26.html
  57. 57.
    Strash, D.: On the power of simple reductions for the maximum independent set problem. In: Dinh, T.N., Thai, M.T. (eds.) COCOON 2016. LNCS, vol. 9797, pp. 345–356. Springer, Cham (2016). doi:10.1007/978-3-319-42634-1_28 CrossRefGoogle Scholar
  58. 58.
    Suters, W.H., Abu-Khzam, F.N., Zhang, Y., Symons, C.T., Samatova, N.F., Langston, M.A.: A new approach and faster exact methods for the maximum common subgraph problem. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 717–727. Springer, Heidelberg (2005). doi:10.1007/11533719_73 CrossRefGoogle Scholar
  59. 59.
    Tavares, W.A., Neto, M.B.C., Rodrigues, C.D., Michelon, P.: Um algoritmo de branch and bound para o problema da clique máxima ponderada. In: Proceedings of XLVII SBPO, vol. 1 (2015)Google Scholar
  60. 60.
    Verfaillie, G., Lemaître, M., Schiex, T.: Russian doll search for solving constraint optimization problems. In: Clancey, W.J., Weld, D.S. (eds.) Proceedings of the Thirteenth National Conference on Artificial Intelligence and Eighth Innovative Applications of Artificial Intelligence Conference, AAAI 1996, IAAI 1996, 4–8 August 1996, Portland, Oregon, vol. 1, pp. 181–187. AAAI Press/The MIT Press (1996). http://www.aaai.org/Library/AAAI/1996/aaai96-027.php
  61. 61.
    Wang, Y., Hao, J., Glover, F., Lü, Z., Wu, Q.: Solving the maximum vertex weight clique problem via binary quadratic programming. J. Comb. Optim. 32(2), 531–549 (2016)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Wang, Y., Cai, S., Yin, M.: Two efficient local search algorithms for maximum weight clique problem. In: Schuurmans, D., Wellman, M.P. (eds.) Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, 12–17 February 2016, Phoenix, Arizona, USA, pp. 805–811. AAAI Press (2016). http://www.aaai.org/ocs/index.php/AAAI/AAAI16/paper/view/11915
  63. 63.
    Wang, Y., Cai, S., Yin, M.: Local search for minimum weight dominating set with two-level configuration checking and frequency based scoring function. J. Artif. Intell. Res. (JAIR) 58, 267–295 (2017). doi:10.1613/jair.5205 MathSciNetMATHGoogle Scholar
  64. 64.
    Wu, Q., Hao, J.: Solving the winner determination problem via a weighted maximum clique heuristic. Expert Syst. Appl. 42(1), 355–365 (2015). doi:10.1016/j.eswa.2014.07.027 MathSciNetCrossRefGoogle Scholar
  65. 65.
    Wu, Q., Hao, J., Glover, F.: Multi-neighborhood tabu search for the maximum weight clique problem. Ann. OR 196(1), 611–634 (2012). doi:10.1007/s10479-012-1124-3 MathSciNetCrossRefMATHGoogle Scholar
  66. 66.
    Zhou, Y., Hao, J., Goëffon, A.: PUSH: a generalized operator for the maximum vertex weight clique problem. Eur. J. Oper. Res. 257(1), 41–54 (2017). doi:10.1016/j.ejor.2016.07.056 MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Ciaran McCreesh
    • 1
  • Patrick Prosser
    • 1
  • Kyle Simpson
    • 1
  • James Trimble
    • 1
  1. 1.University of GlasgowGlasgowScotland

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