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Reports (Research Report) Year : 2021

On the complexity of finding $k$ shortest dissimilar paths in a graph

Abstract

The similarity between two paths can be measured according to the proportion of arcs they share. We study the complexity of several variants of the problem of computing "dissimilar" paths (whose measure of similarity does not exceed a certain threshold) between two given vertices of a weighted directed graph. For four of the most studied measures in the literature, we give a unified and simple proof of the fact that finding $k$ shortest dissimilar paths is NP-complete. We then consider the problem of finding an alternative to one or more given paths. We show that finding a path that is dissimilar to another given path can be done in polynomial time for one of the four considered measures while it is NP-complete for the three remaining measures. In addition, we show that if $k$ = 2 paths are given, finding a new path that is dissimilar to the given ones is NP-complete even on DAGs for the four considered measures. Moreover, for the four considered measures, we show that if a path $P$ is given, finding a shortest path among those that are dissimilar to P is NP-complete in DAGs.
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Dates and versions

hal-03187276 , version 1 (31-03-2021)
hal-03187276 , version 2 (29-09-2021)

Identifiers

  • HAL Id : hal-03187276 , version 2

Cite

Ali Al Zoobi, David Coudert, Nicolas Nisse. On the complexity of finding $k$ shortest dissimilar paths in a graph. [Research Report] Inria; CNRS; I3S; Université Côte d’Azur. 2021, pp.9. ⟨hal-03187276v2⟩
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