Journal of Combinatorial Optimization

, Volume 29, Issue 2, pp 451–471 | Cite as

Embedding signed graphs in the line

Heuristics to solve MinSA problem
  • Eduardo G. Pardo
  • Mauricio Soto
  • Christopher ThravesEmail author


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.


Signed graphs Graph embedding Graph drawing Heuristics Quadratic assignment problem 



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.


  1. Antal T, Krapivsky PL, Redner S (2005) Dynamics of social balance on networks. Phys Rev E 72(3):036–121CrossRefMathSciNetGoogle Scholar
  2. Bansal N, Blum A, Chawla S (2004) Correlation clustering. Mach Learn 56(1–3):89–113CrossRefzbMATHGoogle Scholar
  3. Burkard R, Çela E, Karisch S, Rendl F (2012) Qaplib: a qudratic assignment problem library.
  4. Burkard R, Çela E, Pardalos P, Pitsoulis L (1998) The quadratic assignment problem. Kluwer Academic Publishers, DordrechtGoogle Scholar
  5. Burkard R, Rendl F (1984) A thermodynamically motivated simulation procedure for combinatorial optimization problems. Eur J Oper Res 17(2):169–174CrossRefzbMATHGoogle Scholar
  6. Cartwright D, Harary F (1956) Structural balance: a generalization of Heider’s theory. Psychol Rev 63(5):277–293CrossRefGoogle Scholar
  7. Çela E (1998) The quadratic assignment problem: theory and algorithms. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar
  8. Commander CW (2007) A survey for the quadratic assignment problem. Eur J Oper Res 176(2):657–690CrossRefGoogle Scholar
  9. Connolly DT (1990) An improved annealing scheme for the qap. Eur J Oper Res 46(1):93–100CrossRefzbMATHMathSciNetGoogle Scholar
  10. 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)Google Scholar
  11. Davis JA (1967) Clustering and structural balance in graphs. Hum Relat 20(2):181–187CrossRefGoogle Scholar
  12. 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):65–94CrossRefzbMATHMathSciNetGoogle Scholar
  13. Feo T, Resende M (1989) A probabilistic heuristic for a computationally difficult set covering problem. Oper Res Lett 8:67–71CrossRefzbMATHMathSciNetGoogle Scholar
  14. Feo T, Resende M (1995) Greedy randomized adaptive search procedures. J Glob Optim 6:109–133CrossRefzbMATHMathSciNetGoogle Scholar
  15. Feo T, Resende M, Smith S (1994) A greedy randomized adaptive search procedure for maximum independent set. Oper Res 42:860–878CrossRefzbMATHGoogle Scholar
  16. Harary F (1953) On the notion of balance of a signed graph. Mich Math J 2(2):143–146CrossRefGoogle Scholar
  17. Harary F, Kabell JA (1980) A simple algorithm to detect balance in signed graphs. Math Soc Sci 1(1): 131–136CrossRefzbMATHMathSciNetGoogle Scholar
  18. 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 388–399Google Scholar
  19. Koopmans TC, Beckmann M (1957) Assignment problems and the location of economic activities. Econometrica 25(1):53–76CrossRefzbMATHMathSciNetGoogle Scholar
  20. 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 559–571Google Scholar
  21. 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 641–650Google Scholar
  22. 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 1361–1370Google Scholar
  23. 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):657–690CrossRefzbMATHGoogle Scholar
  24. Nehi HM, Gelareh S (2007) A survey of meta-heuristic solution methods for the quadratic assignment problem. Appl Math Sci 1(45–48):2293–2312zbMATHMathSciNetGoogle Scholar
  25. 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 1–42Google Scholar
  26. 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 219–249Google Scholar
  27. Sahni S, Gonzalez T (1976) P-complete approximation problems. J ACM 23(3):555–565CrossRefzbMATHMathSciNetGoogle Scholar
  28. Skorin-Kapov J (1990) Tabu search applied to the quadratic assignment problem. ORSA J Comput 2(1): 33–45CrossRefzbMATHGoogle Scholar
  29. 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
  30. Taillard E (1991) Robust taboo search for the quadratic assignment problem. Parallel Comput 17(45): 443–455CrossRefMathSciNetGoogle Scholar
  31. Taillard E (1998) Fant: fast ant system. Technical report, Dalle Molle Institute for Artificial Intelligence, LuganoGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Eduardo G. Pardo
    • 1
  • Mauricio Soto
    • 2
  • Christopher Thraves
    • 3
    Email author
  1. 1.Universidad Rey Juan CarlosMóstoles, MadridSpain
  2. 2.LIFOUniversité d’OrléansOrleans Cedex 2France
  3. 3.Universidad Rey Juan CarlosFuenlabrada, MadridSpain

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