License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.STACS.2022.12
URN: urn:nbn:de:0030-drops-158221
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15822/
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Bilò, Davide ; D'Angelo, Gianlorenzo ; Gualà, Luciano ; Leucci, Stefano ; Proietti, Guido ; Rossi, Mirko

Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers

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Abstract

Let G be a directed graph with n vertices, m edges, and non-negative edge costs. Given G, a fixed source vertex s, and a positive integer p, we consider the problem of computing, for each vertex t≠ s, p edge-disjoint paths of minimum total cost from s to t in G. Suurballe and Tarjan [Networks, 1984] solved the above problem for p = 2 by designing a O(m+nlog n) time algorithm which also computes a sparse single-source 2-multipath preserver, i.e., a subgraph containing 2 edge-disjoint paths of minimum total cost from s to every other vertex of G. The case p ≥ 3 was left as an open problem.
We study the general problem (p ≥ 2) and prove that any graph admits a sparse single-source p-multipath preserver with p(n-1) edges. This size is optimal since the in-degree of each non-root vertex v must be at least p. Moreover, we design an algorithm that requires O(pn² (p + log n)) time to compute both p edge-disjoint paths of minimum total cost from the source to all other vertices and an optimal-size single-source p-multipath preserver. The running time of our algorithm outperforms that of a natural approach that solves n-1 single-pair instances using the well-known successive shortest paths algorithm by a factor of Θ(m/(np)) and is asymptotically near optimal if p = O(1) and m = Θ(n²). Our results extend naturally to the case of p vertex-disjoint paths.

BibTeX - Entry

@InProceedings{bilo_et_al:LIPIcs.STACS.2022.12,
  author =	{Bil\`{o}, Davide and D'Angelo, Gianlorenzo and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Rossi, Mirko},
  title =	{{Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{12:1--12:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15822},
  URN =		{urn:nbn:de:0030-drops-158221},
  doi =		{10.4230/LIPIcs.STACS.2022.12},
  annote =	{Keywords: multipath spanners, graph sparsification, edge-disjoint paths, min-cost flow}
}

Keywords: multipath spanners, graph sparsification, edge-disjoint paths, min-cost flow
Collection: 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)
Issue Date: 2022
Date of publication: 09.03.2022

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