Sitting Closer to Friends Than Enemies, Revisited

  • Marek Cygan
  • Marcin Pilipczuk
  • Michał Pilipczuk
  • Jakub Onufry Wojtaszczyk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)

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 ℝ l 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.

Keywords

Interval Graph Chordal Graph Signed Graph Positive Edge Negative Edge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Marek Cygan
    • 1
  • Marcin Pilipczuk
    • 2
  • Michał Pilipczuk
    • 3
  • Jakub Onufry Wojtaszczyk
    • 4
  1. 1.IDSIAUniversity of LuganoSwitzerland
  2. 2.Institute of InformaticsUniversity of WarsawPoland
  3. 3.Department of InformaticsUniversity of BergenNorway
  4. 4.Google Inc.WarsawPoland

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