The maximum balanced subgraph of a signed graph: Applications and solution approaches

The Maximum Balanced Subgraph Problem (MBSP) is the problem of finding a subgraph of a signed graph that is balanced and maximizes the cardinality of its vertex set. This paper is the first one to discuss applications of the MBSP arising in three different research areas: the detection of embedded structures, portfolio analysis in risk management and community structure. The efficient solution of the MBSP is also in the focus of this paper. We discuss pre-processing routines and heuristic solution approaches to the problem. a GRASP metaheuristic is developed and improved versions of a greedy heuristic are discussed. Extensive computational experiments are carried out on a set of instances from the applications previously mentioned as well as on a set of random instances.

[1]  Tobias Achterberg,et al.  The MCF-Separator – Detecting and Exploiting Multi-Commodity Flows in MIPs , 2010 .

[2]  Mauricio G. C. Resende,et al.  A Greedy Randomized Adaptive Search Procedure for Maximum Independent Set , 1994, Oper. Res..

[3]  F. Harary,et al.  STRUCTURAL BALANCE: A GENERALIZATION OF HEIDER'S THEORY1 , 1977 .

[4]  V. Traag,et al.  Community detection in networks with positive and negative links. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Peter Abell,et al.  Structural Balance: A Dynamic Perspective , 2009 .

[6]  Nalan Gülpinar,et al.  Extracting pure network submatrices in linear programs using signed graphs , 2004, Discret. Appl. Math..

[7]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[8]  Shingo Takahashi,et al.  On conditions for a meeting not to reach a deadlock , 1998 .

[9]  Mason A. Porter,et al.  Community Structure in the United Nations General Assembly , 2010, ArXiv.

[10]  Vangelis Th. Paschos.  A (Delta/2)-Approximation Algorithm for the Maximum Independent Set Problem , 1992, Inf. Process. Lett..

[11]  Tobias Achterberg,et al.  The Mcf-separator: detecting and exploiting multi-commodity flow structures in MIPs , 2010, Math. Program. Comput..

[12]  Gregory Gutin,et al.  Fixed-Parameter Algorithms in Analysis of Heuristics for Extracting Networks in Linear Programs , 2009, IWPEC.

[13]  Patrick Doreian,et al.  Partitioning signed social networks , 2009, Soc. Networks.

[14]  Frank Harary,et al.  Signed graphs for portfolio analysis in risk management , 2002 .

[15]  Panos M. Pardalos,et al.  Mining market data: A network approach , 2006, Comput. Oper. Res..

[16]  Egon Balas,et al.  Finding a Maximum Clique in an Arbitrary Graph , 1986, SIAM J. Comput..

[17]  M. Sacramento Quintanilla,et al.  Pre-processing techniques for resource allocation in the heterogeneous case , 1998, Eur. J. Oper. Res..

[18]  Cid C. de Souza,et al.  An exact approach to the problem of extracting an embedded network matrix , 2011, Comput. Oper. Res..

[19]  Ali Ridha Mahjoub,et al.  Facets of the balanced (acyclic) induced subgraph polytope , 1989, Math. Program..

[20]  Manfred W. Padberg,et al.  On the facial structure of set packing polyhedra , 1973, Math. Program..

[21]  Jiming Liu,et al.  Community Mining from Signed Social Networks , 2007, IEEE Transactions on Knowledge and Data Engineering.

[22]  P. Scarf,et al.  Fourth IMA International Conference on Modelling in Industrial Maintenance and Reliability , 2002 .

[23]  Mauricio G. C. Resende,et al.  Grasp: An Annotated Bibliography , 2002 .

[24]  Celso C. Ribeiro,et al.  Multi-start methods for combinatorial optimization , 2013, Eur. J. Oper. Res..

[25]  M. Krawczyk,et al.  Communities in networks - a continuous approach , 2007, 0709.0923.

[26]  Gerhard Reinelt,et al.  A Branch and Cut solver for the maximum stable set problem , 2011, J. Comb. Optim..

[27]  John J. Bartholdi,et al.  A good submatrix is hard to find , 1982, Oper. Res. Lett..

[28]  Hanif D. Sherali,et al.  Linear Programming and Network Flows , 1977 .

[29]  Frank Harary,et al.  A simple algorithm to detect balance in signed graphs , 1980, Math. Soc. Sci..

[30]  P. Doreian,et al.  A partitioning approach to structural balance , 1996 .

[31]  Ronald L. Rivest,et al.  Introduction to Algorithms, 3rd Edition , 2009 .

[32]  David Kempe,et al.  Modularity-maximizing graph communities via mathematical programming , 2007, 0710.2533.

[33]  Avrim Blum,et al.  Correlation Clustering , 2004, Machine Learning.

[34]  Rolf Niedermeier,et al.  Separator-based data reduction for signed graph balancing , 2010, J. Comb. Optim..

[35]  Rosa Figueiredo,et al.  An improved Branch-and-cut code for the maximum balanced subgraph of a signed graph , 2013, ArXiv.

[36]  F. Heider.  Attitudes and cognitive organization. , 1946, The Journal of psychology.

[37]  Fabrizio Rossi,et al.  A branch-and-cut algorithm for the maximum cardinality stable set problem , 2001, Oper. Res. Lett..

[38]  Eduardo D. Sontag,et al.  Algorithmic and complexity results for decompositions of biological networks into monotone subsystems , 2007, Biosyst..