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Abstract:

The study of cohesive subgroups is an important aspect of social network analysis. Cohesive subgroups are studied using different relaxations of the notion of clique in a graph. For instance, given a graph and an integer k, the maximum edge subgraph problem consists of finding a k-vertex subset such that the number of edges within the subset is maximum. This work proposes a polyhedral approach for this NP-hard problem. We study the polytope associated to an integer programming formulation of the problem, present several families of facet-inducing valid inequalities, and discuss the separation problem associated to these families. Finally, we implement a branch and cut algorithm for this problem. This computational study illustrates the effectiveness of the classes of inequalities presented in this work. © 2011 Elsevier B.V. All rights reserved.

Registro:

Documento: Artículo
Título:A polyhedral study of the maximum edge subgraph problem
Autor:Bonomo, F.; Marenco, J.; Saban, D.; Stier-Moses, N.E.
Filiación:Depto. de Computación, FCEyN, IMAS-CONICET, Argentina
Instituto de Ciencias, Universidad Nacional de General Sarmiento, Argentina
Depto. de Computación, FCEyN, Universidad de Buenos Aires, Argentina
Graduate School of Business, Columbia University, United States
Palabras clave:Maximum edge subgraph problem; Polyhedral combinatorics; Quasi-cliques; Branch-and-cut algorithms; Computational studies; Integer programming formulations; Polyhedral approach; Polyhedral combinatorics; Polyhedral studies; Polytopes; Quasi-cliques; Separation problems; Social Network Analysis; Subgraph problems; Valid inequality; Integer programming; Linear programming; Social networking (online); Computational complexity
Año:2012
Volumen:160
Número:18
Página de inicio:2573
Página de fin:2590
DOI: http://dx.doi.org/10.1016/j.dam.2011.10.011
Título revista:Discrete Applied Mathematics
Título revista abreviado:Discrete Appl Math
ISSN:0166218X
CODEN:DAMAD
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_0166218X_v160_n18_p2573_Bonomo.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0166218X_v160_n18_p2573_Bonomo

Referencias:

  • Abello, J.M., Pardalos, P., Resende, M.G.C., On maximum clique problems in very large graphs (1999) External Memory Algorithms, 50, pp. 119-130. , DIMACS Series in Discrete Mathematics and Theoretical Computer Science American Mathematical Society Boston, MA, USA
  • Albert, R., Barabási, A.-L., Statistical mechanics of complex networks (2002) Reviews of Modern Physics, 74, pp. 47-97
  • Asahiro, Y., Hassin, R., Iwama, K., Complexity of finding dense subgraphs (2002) Discrete Applied Mathematics, 121, pp. 15-26
  • Asahiro, Y., Iwama, K., Tamaki, H., Tokuyama, T., Greedily finding dense subgraphs (2000) Journal of Algorithms, 34, pp. 203-221
  • Barabási, A.-L., Albert, R., Emergence of scaling in random networks (1999) Science, 286, pp. 509-512
  • Billionnet, A., Different formulations for solving the heaviest k-subgraph problem (2005) Information Systems and Operational Research, 43 (3), pp. 171-186
  • Bonomo, F., Marenco, J., Saban, D., Stier-Moses, N.E., A polyhedral study of the maximum edge subgraph problem (2009) Proceedings of the v Latin-American Algorithms, Graphs and Optimization Symposium, LAGOS'09, 35, pp. 197-202. , Gramado, Brazil, in: Electronic Notes in Discrete Mathematics
  • Corneil, D., Perl, Y., Clustering and domination in perfect graphs (1984) Discrete Applied Mathematics, 9, pp. 27-39
  • Edmonds, J., Maximum matching and a polyhedron with (0, 1)-vertices (1965) Journal of Research of the National Bureau of Standards, 69, pp. 125-130
  • Edmonds, J., Paths, trees, and flowers (1965) Canadian Journal of Mathematics, 17, pp. 449-467
  • Erds, P., Rényi, A., On random graphs (1959) Publicationes Mathematicae, 6, pp. 290-297
  • Feige, U., Kortsarz, G., Peleg, D., The dense k-subgraph problem (2001) Algorithmica, 29 (3), pp. 410-421
  • Fischetti, M., Hamacher, H., Jornsten, K., Maffioli, F., Weighted k-cardinality trees: Complexity and polyhedral structure (1994) Networks, 24, pp. 11-21
  • Hagberg, A.A., Schult, D.A., Swart, P.J., Exploring network structure, dynamics, and function using NetworkX (2008) Proceedings of the 7th Python in Science Conference, SciPy'08, pp. 11-15. , Pasadena, CA, USA, August
  • Han, Q., Ye, Y., Zhang, J., Approximation of dense-k-subgraph (2000) Technical Report, , Department of Management Sciences, Henry B. Tippie College of Business, the University of Iowa
  • Jünger, M., Thienel, S., The ABACUS system for branch-and-cut-and-price algorithms in integer programming and combinatorial optimization (2000) Software: Practice and Experience, 30, pp. 1325-1352
  • Johnson, E., Mehrotra, A., Nemhauser, G., Min-cut clustering (1993) Mathematical Programming, 62, pp. 133-151
  • Johnson, D.S., Trick, M.A., (1996) Cliques, Coloring, and Satisfiability: Second Dimacs Implementation Challenge, 26. , October 11-13, 1993 DIMACS Series in Discrete Mathematics and Theoretical Computer Science American Mathematical Society Providence, RI
  • Liazi, M., Milis, I., Pascual, F., Zissimopoulos, V., The densest k-subgraph problem on clique graphs (2007) Journal of Combinatorial Optimization, 14 (4), pp. 465-474
  • Liazi, M., Milis, I., Zissimopoulos, V., A constant approximation algorithm for the densest k-subgraph problem on chordal graphs (2008) Information Processing Letters, 108 (1), pp. 29-32
  • Mehrotra, A., Cardinality constrained Boolean quadratic polytope (1997) Discrete Applied Mathematics, 79, pp. 137-154
  • Padberg, M., The boolean quadric polytope: Some characteristics, facets and relatives (1989) Mathematical Programming, 45 (1), pp. 139-172
  • Roupin, F., Billionnet, A., A deterministic approximation algorithm for the densest k-subgraph problem (2008) International Journal of Operational Research, 3 (3), pp. 301-314
  • Saban, D., Bonomo, F., Stier-Moses, N.E., Analysis and models of bilateral investment treaties using a social networks approach (2010) Physica A: Statistical Mechanics and Its Applications, 389, pp. 3661-3673
  • Sherali, H., Cole Smith, J., A polyhedral study of the generalized vertex packing problem (2006) Mathematical Programming, 107 (3), pp. 367-390
  • Wasserman, S., Faust, K., (1994) Social Network Analysis: Methods and Applications, 8. , Structural Analysis in the Social Sciences Cambridge University Press

Citas:

---------- APA ----------
Bonomo, F., Marenco, J., Saban, D. & Stier-Moses, N.E. (2012) . A polyhedral study of the maximum edge subgraph problem. Discrete Applied Mathematics, 160(18), 2573-2590.
http://dx.doi.org/10.1016/j.dam.2011.10.011
---------- CHICAGO ----------
Bonomo, F., Marenco, J., Saban, D., Stier-Moses, N.E. "A polyhedral study of the maximum edge subgraph problem" . Discrete Applied Mathematics 160, no. 18 (2012) : 2573-2590.
http://dx.doi.org/10.1016/j.dam.2011.10.011
---------- MLA ----------
Bonomo, F., Marenco, J., Saban, D., Stier-Moses, N.E. "A polyhedral study of the maximum edge subgraph problem" . Discrete Applied Mathematics, vol. 160, no. 18, 2012, pp. 2573-2590.
http://dx.doi.org/10.1016/j.dam.2011.10.011
---------- VANCOUVER ----------
Bonomo, F., Marenco, J., Saban, D., Stier-Moses, N.E. A polyhedral study of the maximum edge subgraph problem. Discrete Appl Math. 2012;160(18):2573-2590.
http://dx.doi.org/10.1016/j.dam.2011.10.011