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Latin American Symposium on Theoretical Informatics

LATIN 2018: LATIN 2018: Theoretical Informatics pp 544–557Cite as

Efficient Algorithms for Listing k Disjoint st-Paths in Graphs

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10807))

Abstract

Given a connected graph G of m edges and n vertices, we consider the basic problem of listing all the choices of k vertex-disjoint st-paths, for any two input vertices s, t of G and a positive integer k. Our algorithm takes O(m) time per solution, using O(m) space and requiring \(O(F_k(G))\) setup time, where \(F_k(G) = O(m \min \{k, n^{2/3} \log n, \sqrt{m} \log n\} )\) is the cost of running a max-flow algorithm on G to compute a flow of size k. The proposed techniques are simple and apply to other related listing problems discussed in the paper.

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Notes

  1. 1.

    Connect a dummy source s to each \(s_i\), and each \(t_i\) to a dummy target t; then, run a max flow algorithm from s to t.

  2. 2.

    Note that the certificate can be seen as the residual network in max-flow on arcs with 0−1 capacities.

  3. 3.

    This flow can be achieved by creating a dummy vertex \(s'\) connected to all the vertices in I with capacity 1 and to s with an arc of capacity \(k-|I|\).

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Authors and Affiliations

  1. Università di Pisa, Pisa, Italy

    Roberto Grossi, Andrea Marino & Luca Versari

Authors
  1. Roberto Grossi
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  2. Andrea Marino
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  3. Luca Versari
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Corresponding authors

Correspondence to Roberto Grossi , Andrea Marino or Luca Versari .

Editor information

Editors and Affiliations

  1. Stony Brook University, Stony Brook, New York, USA

    Michael A. Bender

  2. Rutgers University, New Brunswick, New Jersey, USA

    Martín Farach-Colton

  3. Pace University, New York, New York, USA

    Miguel A. Mosteiro

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Grossi, R., Marino, A., Versari, L. (2018). Efficient Algorithms for Listing k Disjoint st-Paths in Graphs. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_40

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  • DOI: https://doi.org/10.1007/978-3-319-77404-6_40

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-77403-9

  • Online ISBN: 978-3-319-77404-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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