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Sitting closer to friends than enemies, revisited
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Cygan, Marek, Pilipczuk, Marcin, Pilipczuk, Michał and Wojtaszczyk, Jakub Onufry (2013) Sitting closer to friends than enemies, revisited. Theory of Computing Systems . doi:10.1007/s00224-014-9558-4 (In Press)
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Official URL: http://dx.doi.org/10.1007/s00224-014-9558-4
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 (2011) initiated the study of the problem of appropriately visualising the network: They asked whether any signed graph can be embedded into the metric space TeX 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 TeX 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.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software | ||||||||
| Divisions: | Faculty of Science > Computer Science | ||||||||
| Library of Congress Subject Headings (LCSH): | Graph theory, Online social networks | ||||||||
| Journal or Publication Title: | Theory of Computing Systems | ||||||||
| Publisher: | Springer | ||||||||
| ISSN: | 1432-4350 | ||||||||
| Official Date: | 1 July 2013 | ||||||||
| Dates: |
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| Date of first compliant deposit: | 28 December 2015 | ||||||||
| Number of Pages: | 12 | ||||||||
| DOI: | 10.1007/s00224-014-9558-4 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | In Press | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Funder: | Seventh Framework Programme (European Commission) (FP7), European Research Council (ERC), Fundacja na rzecz Nauki Polskiej [Foundation for Polish Science] (FNP), Narodowe Centrum Nauki (NCN) | ||||||||
| Grant number: | 267959 (ERC), 279352 (ERC), N206567140 (NCN) |
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