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Graph Theoretic Clique Relaxations and Applications

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Handbook of Combinatorial Optimization

Abstract

Cliques and graph theoretic clique relaxations are used to model clusters in graph-based data mining, where data is modeled by a graph in which an edge implies some relationship between the entities represented by its endpoints. The need for relaxations of the clique model arises in practice when dealing with massive data sets which are error prone, resulting in false or missing edges. The clique definition which requires complete pairwise adjacency in the cluster becomes overly restrictive in such situations. Graph theoretic clique relaxations address this need by relaxing structural properties of a clique in a controlled manner via user-specified parameters. This chapter surveys such clique relaxations available in the literature primarily focusing on polyhedral results, complexity studies, approximability, and exact algorithmic approaches.

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Notes

  1. 1.

    It should be noted that a subset \(S \subseteq V\) such that \(\delta (G[S]) \geq \gamma (\vert S\vert - 1)\) has also been called a \(\gamma\)-quasi-clique in some literature (see for instance [39, 87, 154]).

Recommended Reading

  1. J. Abello, P.M. Pardalos, M.G.C. Resende, On maximum clique problems in very large graphs, in External Memory Algorithms and Visualization, ed. by J. Abello, J. Vitter, DIMACS Series on Discrete Mathematics and Theoretical Computer Science, vol. 50 (American Mathematical Society, Providence, 1999), pp. 119–130

    Google Scholar 

  2. J. Abello, M.G.C. Resende, S. Sudarsky, Massive quasi-clique detection, in LATIN 2002: Proceedings of the 5th Latin American Symposium on Theoretical Informatics, ed. by S. Rajsbaum (Springer, London, 2002), pp. 598–612

    Google Scholar 

  3. R.D. Alba, A graph-theoretic definition of a sociometric clique. J. Math. Sociol. 3(1), 113–126 (1973)

    MathSciNet  MATH  Google Scholar 

  4. M.T. Almeida, F.D. Carvalho, The k-club problem: new results for k = 3. Technical report CIO working paper 3/2008, CIO-Centro de Investigação Operacional, 2008

    Google Scholar 

  5. M.T. Almeida, F.D. Carvalho, Integer models and upper bounds for the 3-club problem. Networks 60(3), 155–166 (2012)

    MathSciNet  MATH  Google Scholar 

  6. A.T. Amin, S.L. Hakimi, Upper bounds on the order of a clique of a graph. SIAM J. Appl. Math. 22, 569–573 (1972)

    MathSciNet  MATH  Google Scholar 

  7. D. Applegate, D.S. Johnson, dfmax.c [c program, second dimacs implementation challenge], ftp://dimacs.rutgers.edu/pub/challenge/graph/solvers/

  8. S. Arora, D. Karger, M. Karpinski, Polynomial time approximation schemes for dense instances of -hard problems. J. Comput. Syst. Sci. 58(1), 193–210 (1999)

    MathSciNet  MATH  Google Scholar 

  9. J. Arquilla, D. Ronfeldt, What next for networks and netwars? in Networks and Netwars: The Future of Terror, Crime, and Militancy, ed. by J. Arquilla, D. Ronfeldt (RAND Corporation, Santa Monica, 2001), pp. 311–361

    Google Scholar 

  10. Y. Asahiro, K. Iwama, H. Tamaki, T. Tokuyama, Greedily finding a dense subgraph. J. Algorithm 34, 203–221 (2000)

    MathSciNet  MATH  Google Scholar 

  11. Y. Asahiro, R. Hassin, K. Iwama, Complexity of finding dense subgraphs. Discret. Appl. Math. 121(1–3), 15–26 (2002)

    MathSciNet  MATH  Google Scholar 

  12. Y. Asahiro, E. Miyano, K. Samizo, Approximating maximum diameter-bounded subgraphs, in LATIN 2010: Theoretical Informatics, ed. by A. López-Ortiz. Lecture Notes in Computer Science, vol. 6034 (Springer, Berlin/Heidelberg, 2010), pp. 615–626

    Google Scholar 

  13. L. Babel, Finding maximum cliques in arbitrary and in special graphs. Computing 46(4), 321–341 (1991)

    MathSciNet  MATH  Google Scholar 

  14. G.D. Bader, C.W.V. Hogue, An automated method for finding molecular complexes in large protein interaction networks. BMC Bioinf. 4(2), (2003)

    Google Scholar 

  15. E. Balas, J. Xue, Weighted and unweighted maximum clique algorithms with upper bounds from fractional coloring. Algorithmica 15, 397–412 (1996)

    MathSciNet  MATH  Google Scholar 

  16. E. Balas, C. Yu, Finding a maximum clique in an arbitrary graph. SIAM J. Comput. 15, 1054–1068 (1986)

    MathSciNet  MATH  Google Scholar 

  17. B. Balasundaram, Graph theoretic generalizations of clique: optimization and extensions. Ph.D. thesis, Texas A&M University, College Station, TX, USA, 2007

    Google Scholar 

  18. B. Balasundaram, S. Butenko, Network clustering, in Analysis of Biological Networks, ed. by B.H. Junker, F. Schreiber (Wiley, New York, 2008), pp. 113–138

    Google Scholar 

  19. B. Balasundaram, S. Butenko, S. Trukhanov, Novel approaches for analyzing biological networks. J. Comb. Optim. 10(1), 23–39 (2005)

    MathSciNet  MATH  Google Scholar 

  20. B. Balasundaram, S. Butenko, I.V. Hicks, Clique relaxations in social network analysis: the maximum k-plex problem. Oper. Res. 59(1), 133–142 (2011)

    MathSciNet  MATH  Google Scholar 

  21. B. Balasundaram, S.S. Chandramouli, S. Trukhanov, Approximation algorithms for finding and partioning unit-disk graphs into co-k-plexes. Optim. Lett. 4(3), 311–320 (2010)

    MathSciNet  MATH  Google Scholar 

  22. A.-L. Barabási, R. Albert, Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)

    MathSciNet  Google Scholar 

  23. F. Barahona, A.R. Mahjoub, Compositions of graphs and polyhedra II: stable sets. SIAM J. Discret. Math. 7, 359–371 (1994)

    MathSciNet  MATH  Google Scholar 

  24. F. Barahona, A.R. Mahjoub, Compositions of graphs and polyhedra III: graphs with no W 4 minor. SIAM J. Discret. Math. 7, 372–389 (1994)

    MathSciNet  MATH  Google Scholar 

  25. A. Bhaskara, M. Charikar, E. Chlamtac, U. Feige, A. Vijayaraghavan, Detecting high log-densities – an o(n 1 ∕ 4) approximation for densest k-subgraph. CoRR, abs/1001.2891, 2010

    Google Scholar 

  26. A. Billionnet, Different formulations for solving the heaviest k-subgraph problem. Inf. Syst. Oper. Res. 43(3), 171–186 (2005)

    MathSciNet  Google Scholar 

  27. V. Boginski, Network-based data mining: operations research techniques and applications, in Encyclopedia of Operations Research and Management Science (Wiley, New York, 2011, to appear)

    Google Scholar 

  28. V. Boginski, S. Butenko, P.M. Pardalos, On structural properties of the market graph, in Innovation in Financial and Economic Networks, ed. by A. Nagurney (Edward Elgar, London, 2003)

    Google Scholar 

  29. V. Boginski, S. Butenko, P. Pardalos, Statistical analysis of financial networks. Comput. Stat. Data Anal. 48, 431–443 (2005)

    MathSciNet  MATH  Google Scholar 

  30. V. Boginski, S. Butenko, P. Pardalos, Mining market data: a network approach. Comput. Oper. Res. 33, 3171–3184 (2006)

    MATH  Google Scholar 

  31. I.M. Bomze, M. Budinich, P.M. Pardalos, M. Pelillo, The maximum clique problem, in Handbook of Combinatorial Optimization, ed. by D.-Z. Du, P.M. Pardalos (Kluwer Academic, Dordrecht 1999), pp. 1–74

    Google Scholar 

  32. F. Bonomo, J. Maranco, D. Saban, N. Stier-Moses, A polyhedral study of the maximum edge subgraph problem. Discret. Appl. Math. (2011). doi:10.1016/j.dam.2011.10.011

    Google Scholar 

  33. R. Borndörfer, Aspects of Set Packing, Partitioning, and Covering (Shaker Verlag, Aachen, 1998). Ph.D. thesis, Technische Universität Berlin

    Google Scholar 

  34. J.-M. Bourjolly, G. Laporte, G. Pesant, Heuristics for finding k-clubs in an undirected graph. Comput. Oper. Res. 27, 559–569 (2000)

    MathSciNet  MATH  Google Scholar 

  35. J.-M. Bourjolly, G. Laporte, G. Pesant, An exact algorithm for the maximum k-club problem in an undirected graph. Eur. J. Oper. Res. 138, 21–28 (2002)

    MathSciNet  MATH  Google Scholar 

  36. J. Brimberg, N. Mladenović, D. Urosević, E. Ngai, Variable neighborhood search for the heaviest k-subgraph. Comput. Oper. Res. 36(11), 2885–2891 (2009)

    MathSciNet  MATH  Google Scholar 

  37. A. Broido, K.C. Claffy, Internet topology: connectivity of ip graphs, in Scalability and Traffic Control in IP Networks, ed. by S. Fahmy, K. Park (SPIE, Bellingham, 2001), pp. 172–187

    Google Scholar 

  38. C. Bron, J. Kerbosch, Algorithm 457: finding all cliques on an undirected graph. Commun. ACM 16, 575–577 (1973)

    MATH  Google Scholar 

  39. M. Brunato, H. Hoos, R. Battiti, On effectively finding maximal quasi-cliques in graphs, in Learning and Intelligent Optimization, ed. by V. Maniezzo, R. Battiti, J.-P. Watson. Lecture Notes in Computer Science, vol. 5313 (Springer, Berlin/Heidelberg, 2008), pp. 41–55

    Google Scholar 

  40. S. Butenko, O. Prokopyev, On k-club and k-clique numbers in graphs. Technical report, Texas A&M University, 2007

    Google Scholar 

  41. S. Butenko, W. Wilhelm, Clique-detection models in computational biochemistry and genomics. Eur. J. Oper. Res. 173, 1–17 (2006)

    MathSciNet  MATH  Google Scholar 

  42. L.M. Camarinha-matos, H. Afsarmanesh, Collaborative networks: a new scientific discipline. J. Intell. Manuf. 16, 439–452 (2005)

    Google Scholar 

  43. R. Carraghan, P. Pardalos, An exact algorithm for the maximum clique problem. Oper. Res. Lett. 9, 375–382 (1990)

    MATH  Google Scholar 

  44. F.D. Carvalho, M.T. Almeida, Upper bounds and heuristics for the 2-club problem. Eur. J. Oper. Res. 210(3), 489–494 (2011)

    MathSciNet  MATH  Google Scholar 

  45. G.J. Chang, G.L. Nemhauser, The k-domination and k-stability problems on sun-free chordal graphs. SIAM J. Algebr. Discret. Method 5, 332–345 (1984)

    MathSciNet  MATH  Google Scholar 

  46. M. Charikar, Greedy approximation algorithms for finding dense components in a graph, in Approximation Algorithms for Combinatorial Optimization, ed. by K. Jansens, S. Khuller. Lecture Notes in Computer Science, vol. 1913 (Springer, Berlin/Heidelberg, 2000), pp. 139–152

    Google Scholar 

  47. H. Chen, D. Zeng, H. Atabakhsh, W. Wyzga, J. Schroeder, COPLINK: managing law enforcement data and knowledge. Commun. ACM 46(1), 28–34 (2003)

    Google Scholar 

  48. H. Chen, W. Chung, J.J. Xu, G. Wang, Y. Qin, M. Chau, Crime data mining: a general framework and some examples. Computer 37(4), 50–56 (2004)

    Google Scholar 

  49. E. Cheng, W.H. Cunningham, Wheel inequalities for stable set polytopes. Math. Program. 77, 389–421 (1997)

    MathSciNet  MATH  Google Scholar 

  50. E. Cheng, S. de Vries, On the facet-inducing antiweb-wheel inequalities for stable set polytopes. SIAM J. Discret Math. 15(4), 470–487 (2002)

    MATH  Google Scholar 

  51. E.J. Chesler, M.A. Langston, Combinatorial genetic regulatory network analysis tools for high throughput transcriptomic data. Technical report ut-cs-06-575, CS Technical reports, University of Tennessee, Knoxville, 2006

    Google Scholar 

  52. F. Chung, L. Lu, Complex Graphs and Networks. CBMS Lecture Series (American Mathematical Society, Providence, 2006)

    MATH  Google Scholar 

  53. V. Chvátal, On certain polytopes associated with graphs. J. Comb. Theory (B) 18, 138–154 (1975)

    MATH  Google Scholar 

  54. B.N. Clark, C.J. Colbourn, D.S. Johnson, Unit disk graphs. Discret. Math. 86, 165–177 (1990)

    MathSciNet  MATH  Google Scholar 

  55. D.J. Cook, L.B. Holder, Graph-based data mining. IEEE Intell. Syst. 15(2), 32–41 (2000)

    Google Scholar 

  56. W. Cook, W. Cunningham, W. Pulleyblank, A. Schrijver, Combinatorial Optimization (Wiley, New York, 1998)

    MATH  Google Scholar 

  57. D. Corneil, Y. Perl, Clustering and domination in perfect graphs. Discret. Appl. Math. 9, 27–39 (1984)

    MathSciNet  MATH  Google Scholar 

  58. G. Cornuéjols, Combinatorial Optimization: Packing and Covering. CBMS-NSF Regional Conference Series in Applied Mathematics (SIAM, Philadelphia, 2001)

    Google Scholar 

  59. L. Cowen, R. Cowen, D. Woodall, Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valence. J. Graph Theory 10, 187–195 (1986)

    MathSciNet  MATH  Google Scholar 

  60. R.H. Davis, Social network analysis: an aid in conspiracy investigations. FBI Law Enforc. Bull. 50(12), 11–19 (1981)

    Google Scholar 

  61. A. Dessmark, K. Jansen, A. Lingas, The maximum k-dependent and f-dependent set problem, in Proceedings of the 4th International Symposium on Algorithms and Computation:ISAAC ’93, ed. by K.W. Ng, P. Raghavan, N.V. Balasubramanian, F.Y.L. Chin. Lecture Notes in Computer Science, vol. 762 (Springer, Berlin, 1993), pp. 88–97

    Google Scholar 

  62. R. Diestel, Graph Theory (Springer, Berlin, 1997)

    MATH  Google Scholar 

  63. Dimacs, Cliques, coloring, and satisfiability: second Dimacs implementation challenge (1995), Online: http://dimacs.rutgers.edu/Challenges/. Accessed Mar 2007

  64. H. Djidev, O. Garrido, C. Levcopoulos, A. Lingas, On the maximum k-dependent set problem. Technical report LU-CS-TR:92-91, Department of Computer Science, Lund University, Sweden, 1992

    Google Scholar 

  65. R.G. Downey, M.R. Fellows, Fixed-parameter tractability and completeness II: on completeness for W[1]. Theor. Comput. Sci. 141(1–2), 109–131 (1995)

    MathSciNet  MATH  Google Scholar 

  66. J. Edachery, A. Sen, F.J. Brandenburg, in Graph Clustering Using Distance-k Cliques. Lecture Notes in Computer Science, vol. 1731 (Springer, Berlin/New York, 1999), pp. 98–106

    Google Scholar 

  67. J. Edmonds, Maximum matching and a polyhedron with 0,1-vertices. J. Res. Natl. Bur. Stand. B 69B, 125–130 (1965)

    MathSciNet  Google Scholar 

  68. U. Feige, G. Kortsarz, D. Peleg, The dense k-subgraph problem. Algorithmica 29, 410–421 (2001)

    MathSciNet  MATH  Google Scholar 

  69. U. Feige, M. Seltser, On the densest k-subgraph problem. Technical report CS97-16, Weizmann Institute, 1997

    Google Scholar 

  70. M.R. Fellows, J. Guo, H. Moser, R. Niedermeier, A generalization of nemhauser and Trotter’s local optimization theorem. J. Comput. Syst. Sci. 77(6), 1141–1158 (2011)

    MathSciNet  MATH  Google Scholar 

  71. D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra. Math. Program. 1, 168–194 (1971)

    MathSciNet  MATH  Google Scholar 

  72. J. Gagneur, R. Krause, T. Bouwmeester, G. Casari, Modular decomposition of protein-protein interaction networks. Genome Biol. 5(8), R57.1–R57.12 (2004)

    Google Scholar 

  73. G. Gallo, M.D. Grigoriadis, R.E. Tarjan, A fast parametric maximum flow algorithm and applications. SIAM J. Comput. 18(1), 30–55 (1989)

    MathSciNet  MATH  Google Scholar 

  74. M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (W.H. Freeman, New York, 1979)

    MATH  Google Scholar 

  75. R. Giles, L.E. Trotter, On stable set polyhedra for k 1, 3-free graphs. J. Combin. Theory B 31(3), 313–326 (1981)

    MathSciNet  MATH  Google Scholar 

  76. L.M. Glass, R.J. Glass, Social contact networks for the spread of pandemic influenza in children and teenagers. BMC Public Health 8, 61 (2008)

    MathSciNet  Google Scholar 

  77. J. Grossman, P. Ion, R. De Castro, The Erdös number project (1995), Online: http://www.oakland.edu/enp/. Accessed Mar 2007

  78. R. Hassin, S. Rubinstein, A. Tamir, Approximation algorithms for maximum dispersion. Oper. Res. Lett. 21(3), 133–137 (1997)

    MathSciNet  MATH  Google Scholar 

  79. J. Håstad, Clique is hard to approximate within \({n}^{1-\epsilon }\). Acta Math. 182, 105–142 (1999)

    MathSciNet  MATH  Google Scholar 

  80. F. Havet, R.J. Kang, J.-S. Sereni, Improper colouring of unit disk graphs. Electron. Notes Discret. Math. 22, 123–128 (2005)

    Google Scholar 

  81. F. Havet, R.J. Kang, J.-S. Sereni, Improper colouring of unit disk graphs. Technical report RR-6206, Institute National de Recherche en Informatique et en Automatique (INRIA), France, May 2007. To appear in Networks. Available at: http://hal.inria.fr/inria-00150464_v2/

  82. S. Hill, F. Provost, C. Volinsky, Network-based marketing: identifying likely adopters via consumer networks. Stat. Sci. 22, 256–275 (2006)

    MathSciNet  Google Scholar 

  83. D.S. Hochbaum, Efficient bounds for the stable set, vertex cover and set packing problems. Discret. Appl. Math. 6(3), 243–254 (1983)

    MathSciNet  MATH  Google Scholar 

  84. H.B. Hunt, M.V. Marathe, V. Radhakrishnan, S.S. Ravi, D.J. Rosenkrantz, R.E. Stearns, NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs. J. Algor. 26(2), 238–274 (1998)

    MathSciNet  MATH  Google Scholar 

  85. D. Iacobucci, N. Hopkins, Modelling dyadic interactions and networks in marketing. J. Mark. Res. 24, 5–17 (1992)

    Google Scholar 

  86. T. Ito, T. Chiba, R. Ozawa, M. Yoshida, M. Hattori, Y. Sakaki, A comprehensive two-hybrid analysis to explore the yeast protein interactome. Proc. Natl. Acad. Sci. USA 98(8), 4569–4574 (2001)

    Google Scholar 

  87. D. Jiang, J. Pei, Mining frequent cross-graph quasi-cliques. ACM Trans. Knowl. Discov. Data 2(4), 16:1–42 (2009)

    MathSciNet  Google Scholar 

  88. R. Kang, Improper coloring of graphs. Ph.D. thesis, University of Oxford, 2007

    Google Scholar 

  89. J.M. Keil, T.B. Brecht, The complexity of clustering in planar graphs. J. Combin. Math. Combin. Comput. 9, 155–159 (1991)

    MathSciNet  MATH  Google Scholar 

  90. S. Khot, Ruling out ptas for graph min-bisection, densest subgraph and bipartite clique, in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, Vancouver, pp. 136–145 (2004)

    Google Scholar 

  91. C. Komusiewicz, F. Hüffner, H. Moser, R. Niedermeier, Isolation concepts for efficiently enumerating dense subgraphs. Theor. Comput. Sci. 410(38–40), 3640–3654 (2009)

    MATH  Google Scholar 

  92. G. Kortsarz, D. Peleg, On choosing a dense subgraph, in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science (IEEE, Piscataway, 1993), pp. 692–701

    Google Scholar 

  93. M. Liazi, I. Milis, V. Zissimopoulos, A constant approximation algorithm for the densest k-subgraph problem on chordal graphs. Inf. Process. Lett. 108(1), 29–32 (2008)

    MathSciNet  MATH  Google Scholar 

  94. R.D. Luce, Connectivity and generalized cliques in sociometric group structure. Psychometrika 15(2), 169–190 (1950)

    MathSciNet  Google Scholar 

  95. R.D. Luce, A.D. Perry, A method of matrix analysis of group structure. Psychometrika 14(2), 95–116 (1949)

    MathSciNet  Google Scholar 

  96. C. Lund, M. Yannakakis, The approximation of maximum subgraph problems, in Proceedings of the 20th International Colloquium on Automata, Languages and Programming, ICALP ’93 (Springer, London, 1993), pp. 40–51

    Google Scholar 

  97. F. Mahdavi Pajouh, B. Balasundaram, On inclusionwise maximal and maximum cardinality k-clubs in graphs. Discret. Optim. 9(2), 84–97 (2012). doi:10.1016/j.disopt.2012.02.002

    MathSciNet  MATH  Google Scholar 

  98. F. Mahdavi Pajouh, Polyhedral combinatorics, complexity & algorithms for k-clubs in graphs. Oklahoma State University, Stillwater (2012)

    Google Scholar 

  99. F. Mahdavi Pajouh, Z. Miao, B. Balasundaram, A branch-and-bound approach for maximum quasi-cliques. Ann. Oper. Res. (2012). doi:10.1007/s10479-012-1242-y

    Google Scholar 

  100. M.V. Marathe, H. Breu, H.B. Hunt III, S.S. Ravi, D.J. Rosenkrantz, Simple heuristics for unit disk graphs. Networks 25, 59–68 (1995)

    MathSciNet  MATH  Google Scholar 

  101. J. Marincek, B. Mohar, On approximating the maximum diameter ratio of graphs. Discret. Math. 244(1–3), 323–330 (2002)

    MathSciNet  MATH  Google Scholar 

  102. B. McClosky, Independence systems and stable set relaxations. Ph.D. thesis, Rice University, 2008

    Google Scholar 

  103. B. McClosky, Clique relaxations, in Wiley Encyclopedia of Operations Research and Management Science Cochran, ed. by J.J. Cochran, L.A. Cox, P. Keskinocak, J.P. Kharoufeh, J.C. Smith (Wiley, New York, 2010)

    Google Scholar 

  104. B. McClosky, J.D. Arellano, I.V. Hicks, Co-k-plex vertex partitions. Rice University, Houston (2012)

    Google Scholar 

  105. B. McClosky, I.V. Hicks, The co-2-plex polytope and integral systems. SIAM J. Discret. Math 23(3), 1135–1148 (2009)

    MathSciNet  MATH  Google Scholar 

  106. B. McClosky, I.V. Hicks, Combinatorial algorithms for the maximum k-plex problem. J. Combin. Opt. 23, 29–49 (2012)

    MathSciNet  MATH  Google Scholar 

  107. S.T. McCormick, Optimal approximation of sparse hessians and its equivalence to a graph coloring problem. Math. Program. 26, 153–171 (1983)

    MathSciNet  MATH  Google Scholar 

  108. N. Memon, H.L. Larsen, Structural analysis and mathematical methods for destabilizing terrorist networks using investigative data mining, in Advanced Data Mining and Applications, ed. by X. Li, O. Zaïane, Z. Li. Lecture Notes in Computer Science, vol. 4093 (Springer, Berlin/Heidelberg, 2006), pp. 1037–1048

    Google Scholar 

  109. N. Memon, K.C. Kristoffersen, D.L. Hicks, H.L. Larsen, Detecting critical regions in covert networks: a case study of 9/11 terrorists network, in The Second International Conference on Availability, Reliability and Security, ARES 2007, Vienna, 2007, pp. 861–870

    Google Scholar 

  110. J. Miao, D. Berleant, From paragraph networks to document networks, in Proceedings of the International Conference on Information Technology: Coding and Computing, (ITCC 2004), Los vegas, vol. 1, 2004, pp. 295–302

    Google Scholar 

  111. N. Mladenović, P. Hansen, Variable neighborhood search. Comput. Oper. Res. 24, 1097–1100 (1997)

    MathSciNet  MATH  Google Scholar 

  112. R.J. Mokken, Cliques, clubs and clans. Qual. Quant. 13(2), 161–173 (1979)

    Google Scholar 

  113. H. Moser, Finding optimal solutions for covering and matching problems. Ph.D. thesis, Institut für Informatik, Friedrich-Schiller-Universität Jena, 2009

    Google Scholar 

  114. H. Moser, R. Niedermeier, M. Sorge, Exact combinatorial algorithms and experiments for finding maximum k-plexes. J. Combin. Opt. 1–27 (2011). doi:10.1007/s10878-011-9391-5

    Google Scholar 

  115. T.S. Motzkin, E.G. Straus, Maxima for graphs and a new proof of a theorem of Turán. Canad. J. Math. 17, 533–540 (1965)

    MathSciNet  MATH  Google Scholar 

  116. G.L. Nemhauser, L.E. Trotter, Properties of vertex packings and independence system. Math. Program. 6, 48–61 (1974)

    MathSciNet  MATH  Google Scholar 

  117. G.L. Nemhauser, L.E. Trotter, Vertex packing: structural properties and algorithms. Math. Program. 8, 232–248 (1975)

    MathSciNet  MATH  Google Scholar 

  118. N. Nishimura, P. Ragde, D.M. Thilikos, Fast fixed-parameter tractable algorithms for nontrivial generalizations of vertex cover. Discret. Appl. Math. 152(1–3), 229–245 (2005)

    MathSciNet  MATH  Google Scholar 

  119. Office of Technology Assessment U.S. Congress, Technologies for detecting money laundering, in Information Technologies for the Control of Money Laundering (U.S. Government Printing Office, Washington, DC, 1995), pp. 51–74. Online: http://www.wws.princeton.edu/ota/disk1/1995/9529/9529.PDF. Accessed May 2006

  120. P.R.J. Östergård, A fast algorithm for the maximum clique problem. Discret. Appl. Math. 120, 197–207 (2002)

    MATH  Google Scholar 

  121. M.W. Padberg, On the facial structure of set packing polyhedra. Math. Programm. 5(1), 199–215 (1973)

    MathSciNet  MATH  Google Scholar 

  122. P.M. Pardalos, J. Xue, The maximum clique problem. J. Global Opt. 4, 301–328 (1994)

    MathSciNet  MATH  Google Scholar 

  123. S. Pasupuleti, Detection of protein complexes in protein interaction networks using n-clubs, in EvoBIO 2008: Proceedings of the 6th European Conference on Evolutionary Computation, Machine Learning and Data Mining in Bioinformatics, Naples. Lecture Notes in Computer Science, vol. 4973 (Springer, Berlin, 2008), pp. 153–164

    Google Scholar 

  124. J. Patillo, A. Veremyev, S. Butenko, V. Boginski, On the maximum quasi-clique problem. Discret. Appl. Math. 161(1–2), 244–257 (2013). doi:10.1016/j.dam.2012.07.019

    Google Scholar 

  125. J. Pattillo, N. Youssef, S. Butenko, Clique relaxation models in social network analysis, in Handbook of Optimization in Complex Networks, ed. by M.T. Thai, P.M. Pardalos. Springer Optimization and its Applications, vol. 58 (Springer, New York, 2012), pp. 143–162

    Google Scholar 

  126. J. Pei, D. Jiang, A. Zhang, Mining cross-graph quasi-cliques in gene expression and protein interaction data, in Proceedingsof the 21st International Conference on Data Engineering, ICDE 2005, Tokyo, 2005, pp. 353–356

    Google Scholar 

  127. X. Peng, M.A. Langston, A.M. Saxton, N.E. Baldwin, J.R. Snoddy, Detecting network motifs in gene co-expression networks through integration of protein domain information, in Methods of Microarray Data Analysis V, ed. by P. McConnell, S.M. Lin, P. Hurban (Springer, New York, 2007), pp. 89–102

    Google Scholar 

  128. O.A. Prokopyev, V. Boginski, W. Chaovalitwongse, P.M. Pardalos, J.C. Sackellares, P.R. Carney, Network-based techniques in eeg data analysis and epileptic brain modeling, in Data Mining in Biomedicine, ed. by P.M. Pardalos, V. Boginski, A. Vazacopoulos (Springer, Berlin, 2007), pp. 559–573

    Google Scholar 

  129. S.S. Ravi, D.J. Rosenkrantz, G.K. Tayi, Heuristics and special case algorithms for dispersion problems. Oper. Res. 42(2), 299–310 (1994)

    MATH  Google Scholar 

  130. F. Roupin, A. Billionnet, A deterministic approximation algorithm for the densest k-subgraph problem. Int. J. Oper. Res. 3(3), 301–314 (2008)

    MathSciNet  MATH  Google Scholar 

  131. A. Schäfer, Exact algorithms for s-club finding and related problems. Master’s thesis, Diplomarbeit, Institut für Informatik, Friedrich-Schiller-Universität Jena, 2009

    Google Scholar 

  132. A. Schäfer, C. Komusiewicz, H. Moser, R. Niedermeier, Parameterized computational complexity of finding small-diameter subgraphs. Opt. Lett. 1–9 (2011). doi:10.1007/s11590-011-0311-5

    Google Scholar 

  133. S.B. Seidman, B.L. Foster, A graph theoretic generalization of the clique concept. J. Math. Sociol. 6, 139–154 (1978)

    MathSciNet  MATH  Google Scholar 

  134. E.C. Sewell, A branch and bound algorithm for the stability number of a sparse graph. INFORMS J. Comput. 10(4), 438–447 (1998)

    MathSciNet  Google Scholar 

  135. T. Smieszek, L. Fiebig, R.W. Scholz, Models of epidemics: when contact repetition and clustering should be included. Theor. Biol. Med. Model. 6(1), 11 (2009)

    Google Scholar 

  136. M.K. Sparrow, The application of network analysis to criminal intelligence: an assessment of the prospects. Soc. Netw. 13, 251–274 (1991)

    Google Scholar 

  137. V. Spirin, L.A. Mirny, Protein complexes and functional modules in molecular networks. Proc. Natl. Acad. Sci. 100(21), 12123–12128 (2003)

    Google Scholar 

  138. A. Srivastav, K. Wolf, Finding dense subgraphs with semidefinite programming, in Approximation Algorithms for Combinatiorial Optimization, ed. by K. Jansen, J. Rolim. Lecture Notes in Computer Science, vol. 1444 (Springer, Berlin/Heidelberg, 1998), pp. 181–191. doi: 10.1007/BFb0053974

    Google Scholar 

  139. P.-N. Tan, M. Steingach, V. Kumar, Introduction to Data Mining (Addison-Wesley, Boston, 2006)

    Google Scholar 

  140. L. Terveen, W. Hill, B. Amento, Constructing, organizing, visualizing collections of topically related, web resources. ACM Trans. Comput. Human Int. 6, 67–94 (1999)

    Google Scholar 

  141. E. Tomita, T. Kameda, An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments. J. Global Opt. 37(1), 95–111 (2007)

    MathSciNet  MATH  Google Scholar 

  142. L.E. Trotter, A class of facet producing graphs for vertex packing polyhedra. Discret. Math. 12, 373–388 (1975)

    MathSciNet  MATH  Google Scholar 

  143. S. Trukhanov, Novel approaches for solving large-scale optimization problems on graphs. Ph.D. thesis, Texas A&M University, 2008

    Google Scholar 

  144. S. Trukhanov, B. Balasundaram, S. Butenko, Exact algorithms for hard node deletion problems arising in network-based data mining (2011, submitted)

    Google Scholar 

  145. A. Veremyev, V. Boginski, Identifying large robust network clusters via new compact formulations of maximum k-club problems. Eur. J. Oper. Res. 218(2), 316–326 (2012)

    MathSciNet  MATH  Google Scholar 

  146. A. Veremyev, V. Boginski, P. Krokhmal, D.E. Jeffcoat, Asymptotic behavior and phase transitions for clique relaxations in random graphs, working paper, 2010

    Google Scholar 

  147. B.H. Voy, J.A. Scharff, A.D. Perkins, A.M. Saxton, B. Borate, E.J. Chesler, L.K. Branstetter, M.A. Langston, Extracting gene networks for low-dose radiation using graph theoretical algorithms. PLoS Comput Biol. 2(7), e89 (2006)

    Google Scholar 

  148. T. Washio, H. Motoda, State of the art of graph-based data mining. SIGKDD Explor. Newsl. 5(1), 59–68 (2003)

    Google Scholar 

  149. S. Wasserman, K. Faust, Social Network Analysis (Cambridge University Press, New York, 1994)

    Google Scholar 

  150. D. West, Introduction to Graph Theory (Prentice-Hall, Upper Saddle River, 2001)

    Google Scholar 

  151. D.R. Wood, An algorithm for finding a maximum clique in a graph. Oper. Res. Lett. 21(5), 211–217 (1997)

    MathSciNet  MATH  Google Scholar 

  152. A.G. Woodside, M.W. DeLozier, Effects of word of mouth advertising on consumer risk taking. J. Advert. 5(4), 12–19 (1976)

    Google Scholar 

  153. M. Yannakakis, Node-and edge-deletion NP-complete problems, in STOC ’78: Proceedings of the 10th Annual ACM Symposium on Theory of Computing (ACM, New York, 1978), pp. 253–264

    Google Scholar 

  154. Z. Zeng, J. Wang, L. Zhou, G. Karypis, Coherent closed quasi-clique discovery from large dense graph databases, in Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’06 (ACM, New York, 2006), pp. 797–802

    Google Scholar 

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Acknowledgements

This work was partially supported by the US Department of Energy Grant DE-SC0002051.

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Authors and Affiliations

  1. School of Industrial Engineering and Management, Oklahoma State University, 322 Engineering North, 74078, Stillwater, OK, USA

    Balabhaskar Balasundaram

  2. School of Industrial Engineering and Management, Oklahoma State University, 322 Engineering North, 74078, Stillwater, OK, USA

    Foad Mahdavi Pajouh

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  1. Balabhaskar Balasundaram
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  2. Foad Mahdavi Pajouh
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Correspondence to Balabhaskar Balasundaram or Foad Mahdavi Pajouh .

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  1. Department of Industrial and Systems Eng, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Computer Science, University of Texas, Dallas, Richardson, Texas, USA

    Ding-Zhu Du

  3. Dept. Comp. Sci. & Engineering, University of California, San Diego, La Jolla, California, USA

    Ronald L. Graham

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Balasundaram, B., Pajouh, F.M. (2013). Graph Theoretic Clique Relaxations and Applications. In: Pardalos, P., Du, DZ., Graham, R. (eds) Handbook of Combinatorial Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7997-1_9

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