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Embedding signed graphs in the line

Heuristics to solve MinSA problem

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Abstract

Signed graphs are graphs with an assignment of a positive or a negative sign to each edge. These graphs are helpful to represent different types of networks. For instance, they have been used in social networks, where a positive sign in an edge represents friendship between the two endpoints of that edge, while a negative sign represents enmity. Given a signed graph, an important question is how to embed such a graph in a metric space so that in the embedding every vertex is closer to its positive neighbors than to its negative ones. This problem is known as Sitting Arrangement (SA) problem and it was introduced by Kermarrec et al. (Proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS), pp. 388–399, 2011). Cygan et al. (Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS), 2012) proved that the decision version of SA problem is NP-Complete when the signed graph has to be embedded into the Euclidean line. In this work, we study the minimization version of SA (MinSA) problem in the Euclidean line. We relate MinSA problem to the well known quadratic assignment (QA) problem. We establish such a relation by proving that local minimums in MinSA problem are equivalent to local minimums in a particular case of QA problem. In this document, we design two heuristics based on the combinatorial structure of MinSA problem. We experimentally compare their performances against heuristics designed for QA problem. This comparison favors the proposed heuristics.

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Notes

  1. A signed graph is complete if every pair of vertices is connected by a signed edge.

  2. Note that these percentages are computed including positive and negative edges over the total density parameter.

  3. A table showing all the results of the experiments can be found at http://www.cmm.uchile.cl/~mausoto/Data/signedgraphs/EmbedingSignedGraphDATA.csv.

References

  • Antal T, Krapivsky PL, Redner S (2005) Dynamics of social balance on networks. Phys Rev E 72(3):036–121

    Article  MathSciNet  Google Scholar 

  • Bansal N, Blum A, Chawla S (2004) Correlation clustering. Mach Learn 56(1–3):89–113

    Article  MATH  Google Scholar 

  • Burkard R, Çela E, Karisch S, Rendl F (2012) Qaplib: a qudratic assignment problem library. http://www.opt.math.tu-graz.ac.at/qaplib/

  • Burkard R, Çela E, Pardalos P, Pitsoulis L (1998) The quadratic assignment problem. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Burkard R, Rendl F (1984) A thermodynamically motivated simulation procedure for combinatorial optimization problems. Eur J Oper Res 17(2):169–174

    Article  MATH  Google Scholar 

  • Cartwright D, Harary F (1956) Structural balance: a generalization of Heider’s theory. Psychol Rev 63(5):277–293

    Article  Google Scholar 

  • Çela E (1998) The quadratic assignment problem: theory and algorithms. Kluwer Academic Publishers, Dordrecht

    Book  MATH  Google Scholar 

  • Commander CW (2007) A survey for the quadratic assignment problem. Eur J Oper Res 176(2):657–690

    Article  Google Scholar 

  • Connolly DT (1990) An improved annealing scheme for the qap. Eur J Oper Res 46(1):93–100

    Article  MATH  MathSciNet  Google Scholar 

  • Cygan M, Pilipczuk M, Pilipczuk M, Wojtaszczyk JO (2012) Sitting closer to friends than enemies, revisited. In: Proceedings of the 37th international symposium on mathematical foundations of computer science (MFCS) (2012)

  • Davis JA (1967) Clustering and structural balance in graphs. Hum Relat 20(2):181–187

    Article  Google Scholar 

  • Drezner Z, Hahn PM, Taillard E (2005) Recent advances for the quadratic assignment problem with special emphasis on instances that are difficult for meta-heuristic methods. Ann Oper Res 139(1):65–94

    Article  MATH  MathSciNet  Google Scholar 

  • Feo T, Resende M (1989) A probabilistic heuristic for a computationally difficult set covering problem. Oper Res Lett 8:67–71

    Article  MATH  MathSciNet  Google Scholar 

  • Feo T, Resende M (1995) Greedy randomized adaptive search procedures. J Glob Optim 6:109–133

    Article  MATH  MathSciNet  Google Scholar 

  • Feo T, Resende M, Smith S (1994) A greedy randomized adaptive search procedure for maximum independent set. Oper Res 42:860–878

    Article  MATH  Google Scholar 

  • Harary F (1953) On the notion of balance of a signed graph. Mich Math J 2(2):143–146

    Article  Google Scholar 

  • Harary F, Kabell JA (1980) A simple algorithm to detect balance in signed graphs. Math Soc Sci 1(1): 131–136

    Article  MATH  MathSciNet  Google Scholar 

  • Kermarrec AM, Thraves C (2011) Can everybody sit closer to their friends than their enemies? In: Proceedings of the 36th international symposium on mathematical foundations of computer science (MFCS), pp 388–399

  • Koopmans TC, Beckmann M (1957) Assignment problems and the location of economic activities. Econometrica 25(1):53–76

    Article  MATH  MathSciNet  Google Scholar 

  • Kunegis J, Schmidt S, Lommatzsch A, Lerner J, Luca EWD, Albayrak S (2010) Spectral analysis of signed graphs for clustering, prediction and visualization. In: Proceedings of the SIAM international conference on data mining (SDM), pp 559–571

  • Leskovec J, Huttenlocher DP, Kleinberg J (2010) Predicting positive and negative links in online social networks. In: Proceedings of the 19th international conference on world wide web (WWW), pp 641–650

  • Leskovec J, Huttenlocher DP, Kleinberg J (2010) Signed networks in social media. In: Proceedings of the 28th international conference on human factors in computing systems (CHI), pp 1361–1370

  • Loiola EM, de Abreu NMM, Boaventura-Netto PO, Hahn P, Querido T (2007) A survey for the quadratic assignment problem. Eur J Oper Res 176(2):657–690

    Article  MATH  Google Scholar 

  • Nehi HM, Gelareh S (2007) A survey of meta-heuristic solution methods for the quadratic assignment problem. Appl Math Sci 1(45–48):2293–2312

    MATH  MathSciNet  Google Scholar 

  • Pardalos PM, Rendl F. Wolkowicz H (1994) The quadratic assignment problem: a survey and recent developments. In: Proceedings of the DIMACS workshop on quadratic assignment problems, volume 16 of DIMACS series in discrete mathematics and theoretical computer science. American Mathematical Society, Providence, pp 1–42

  • Resende M, Ribeiro C (2003) Greedy randomized adaptive search procedures. In: Glover F, Kochenberger FS, Hillier CC, Price (eds) Handbook of metaheuristics, international series in operations research and management science, vol 57. Springer, New York, pp 219–249

  • Sahni S, Gonzalez T (1976) P-complete approximation problems. J ACM 23(3):555–565

    Article  MATH  MathSciNet  Google Scholar 

  • Skorin-Kapov J (1990) Tabu search applied to the quadratic assignment problem. ORSA J Comput 2(1): 33–45

    Article  MATH  Google Scholar 

  • Szell M, Lambiotte R, Thurner S (2010) Multirelational organization of large-scale social networks in an online world. Proc Natl Acad Sci USA (PNAS) 107(31), 13, 636–13, 641 (2010)

    Google Scholar 

  • Taillard E (1991) Robust taboo search for the quadratic assignment problem. Parallel Comput 17(45): 443–455

    Article  MathSciNet  Google Scholar 

  • Taillard E (1998) Fant: fast ant system. Technical report, Dalle Molle Institute for Artificial Intelligence, Lugano

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Acknowledgments

Christopher Thraves is supported by Spanish MICINN Grant Juan de la Cierva, Comunidad de Madrid Grant S2009TIC-1692 and Spanish MICINN Grant TIN2008–06735-C02-01. Eduardo G. Pardo is supported by Spanish MICINN Grant TIN2008-06890-C02-02 and TIN2012-35632-C02-02.

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Authors and Affiliations

  1. Universidad Rey Juan Carlos, Office 131, Dept-II, C/Tulipán s/n, 28933 , Móstoles, Madrid, Spain

    Eduardo G. Pardo

  2. LIFO, Université d’Orléans, Bat. 3IA, Rue Léonard de Vinci, B.P. 6759, 45067 , Orleans Cedex 2, France

    Mauricio Soto

  3. Universidad Rey Juan Carlos, Office 101, Dept-III, Camino del Molino s/n, 28943 , Fuenlabrada, Madrid, Spain

    Christopher Thraves

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  1. Eduardo G. Pardo
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  2. Mauricio Soto
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  3. Christopher Thraves
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Correspondence to Christopher Thraves.

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Pardo, E.G., Soto, M. & Thraves, C. Embedding signed graphs in the line. J Comb Optim 29, 451–471 (2015). https://doi.org/10.1007/s10878-013-9604-1

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