Sitting closer to friends than enemies, revisited

TL;DR

This paper investigates the problem of embedding signed social networks into a line such that friends are closer than enemies, revealing connections to interval graphs and establishing NP-completeness for the general case.

Contribution

It refines the understanding of embedding signed graphs into R^1, relates the problem to proper interval graph recognition, and proves NP-completeness for the general case.

Findings

Embedding into R^1 relates to proper interval graphs

The general embedding problem is NP-complete

Provides bounds on algorithmic complexity for the problem

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 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 R^1 can be expressed as a purely combinatorial problem. In this paper we pursue a deeper study of this particular 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.