Embedding signed graphs in the line
- 311 Downloads
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.
KeywordsSigned 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.
- 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, DordrechtGoogle 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)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 388–399Google 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 559–571Google Scholar
- 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
- 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
- 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
- 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
- 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 (1998) Fant: fast ant system. Technical report, Dalle Molle Institute for Artificial Intelligence, LuganoGoogle Scholar