Abstract
Signed graphs, i.e., undirected graphs with edges labelled with a plus or minus sign, are commonly used to model relationships in social networks. Recently, Kermarrec and Thraves [13] initiated the study of the problem of appropriately visualising the network: They asked whether any signed graph can be embedded into the metric space ℝl in such a manner that every vertex is closer to all its friends (neighbours via positive edges) than to all its enemies (neighbours via negative edges). Interestingly, embeddability into ℝ1 can be expressed as a purely combinatorial problem. In this paper we pursue a deeper study of this case, answering several questions posed by Kermarrec and Thraves.
First, we refine the approach of Kermarrec and Thraves for the case of complete signed graphs by showing that the problem is closely related to the recognition of proper interval graphs. Second, we prove that the general case, whose polynomial-time tractability remained open, is in fact NP-complete. Finally, we provide lower and upper bounds for the time complexity of the general case: we prove that the existence of a subexponential time (in the number of vertices and edges of the input signed graph) algorithm would violate the Exponential Time Hypothesis, whereas a simple dynamic programming approach gives a running time single-exponential in the number of vertices.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Antal, T., Krapivsky, P.L., Redner, S.: Dynamics of social balance on networks. Phys. Rev. E 72(3), 036121 (2005)
Belmonte, R., Vatshelle, M.: Graph Classes with Structured Neighborhoods and Algorithmic Applications. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 47–58. Springer, Heidelberg (2011)
Cartwright, D., Harary, F.: Structural balance: a generalization of heider’s theory. Psychological Review 63(5), 277–293 (1956)
Corneil, D.G., Kim, H., Natarajan, S., Olariu, S., Sprague, A.P.: Simple linear time recognition of unit interval graphs. Inf. Process. Lett. 55(2), 99–104 (1995)
Davis, J.A.: Clustering and structural balance in graphs. Human Relations 20(2), 181 (1967)
Eppstein, D., Mumford, E.: Self-overlapping curves revisited. In: Mathieu, C. (ed.) SODA, pp. 160–169. SIAM (2009)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)
Guillemot, S., Jansson, J., Sung, W.K.: Computing a smallest multilabeled phylogenetic tree from rooted triplets. IEEE/ACM Trans. Comput. Biology Bioinform. 8(4), 1141–1147 (2011)
Ibarra, L.: A simple algorithm to find hamiltonian cycles in proper interval graphs. Inf. Process. Lett. 109(18), 1105–1108 (2009)
Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)
Ioannidou, K., Mertzios, G.B., Nikolopoulos, S.D.: The Longest Path Problem Is Polynomial on Interval Graphs. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 403–414. Springer, Heidelberg (2009)
Kermarrec, A.M., Thraves, C.: Can Everybody Sit Closer to Their Friends Than Their Enemies? In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 388–399. Springer, Heidelberg (2011)
Kunegis, J., Schmidt, S., Lommatzsch, A., Lerner, J., Luca, E.W.D., Albayrak, S.: Spectral analysis of signed graphs for clustering, prediction and visualization. In: SDM, p. 559–570. SIAM (2010)
Leskovec, J., Huttenlocher, D.P., Kleinberg, J.M.: Governance in social media: A case study of the wikipedia promotion process. In: Cohen, W.W., Gosling, S. (eds.) ICWSM. The AAAI Press (2010)
Leskovec, J., Huttenlocher, D.P., Kleinberg, J.M.: Predicting positive and negative links in online social networks. In: Rappa, M., Jones, P., Freire, J., Chakrabarti, S. (eds.) WWW, pp. 641–650. ACM (2010)
Leskovec, J., Huttenlocher, D.P., Kleinberg, J.M.: Signed networks in social media. In: Mynatt, E.D., Schoner, D., Fitzpatrick, G., Hudson, S.E., Edwards, W.K., Rodden, T. (eds.) CHI, pp. 1361–1370. ACM (2010)
Mertzios, G.B.: A polynomial algorithm for the k-cluster problem on the interval graphs. Electronic Notes in Discrete Mathematics 26, 111–118 (2006)
Szell, M., Lambiotte, R., Thurner, S.: Multirelational organization of large-scale social networks in an online world. PNAS 107(31), 13636–13641 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O. (2012). Sitting Closer to Friends Than Enemies, Revisited. In: Rovan, B., Sassone, V., Widmayer, P. (eds) Mathematical Foundations of Computer Science 2012. MFCS 2012. Lecture Notes in Computer Science, vol 7464. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32589-2_28
Download citation
DOI: https://doi.org/10.1007/978-3-642-32589-2_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32588-5
Online ISBN: 978-3-642-32589-2
eBook Packages: Computer ScienceComputer Science (R0)