Document Open Access Logo

Shortest Two Disjoint Paths in Conservative Graphs

Author Ildikó Schlotter

PDF
Thumbnail PDF

File

LIPIcs.STACS.2024.57.pdf
  • Filesize: 0.86 MB
  • 17 pages

Document Identifiers

Author Details

Ildikó Schlotter
  • Centre for Economic and Regional Studies, Budapest, Hungary
  • Budapest University of Technology and Economics, Hungary

Funding

  • Schlotter, Ildikó: supported by the Hungarian Academy of Sciences under its Momentum Programme (LP2021-2) and its János Bolyai Research Scholarship.

Cite As Get BibTex

Ildikó Schlotter. Shortest Two Disjoint Paths in Conservative Graphs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 57:1-57:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.STACS.2024.57

Abstract

We consider the following problem that we call the Shortest Two Disjoint Paths problem: given an undirected graph G = (V,E) with edge weights w:E → ℝ, two terminals s and t in G, find two internally vertex-disjoint paths between s and t with minimum total weight. As shown recently by Schlotter and Sebő (2022), this problem becomes NP-hard if edges can have negative weights, even if the weight function is conservative, i.e., there are no cycles in G with negative total weight. We propose a polynomial-time algorithm that solves the Shortest Two Disjoint Paths problem for conservative weights in the case when the negative-weight edges form a constant number of trees in G.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Shortest paths
  • Theory of computation → Dynamic programming
Keywords
  • Shortest paths
  • disjoint paths
  • conservative weights
  • graph algorithm

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Isolde Adler, Stavros G. Kolliopoulos, Philipp Klaus Krause, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Irrelevant vertices for the Planar Disjoint Paths problem. Journal of Combinatorial Theory, Series B, 122:815-843, 2017. URL: https://doi.org/10.1016/j.jctb.2016.10.001.
  2. Aaron Bernstein, Danupon Nanongkai, and Christian Wulff-Nilsen. Negative-weight single-source shortest paths in near-linear time. In FOCS 2022: Proceedings of the 63rd Annual IEEE Symposium on Foundations of Computer Science, pages 600-611. IEEE Computer Society, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00063.
  3. Andreas Björklund and Thore Husfeldt. Shortest two disjoint paths in polynomial time. SIAM Journal on Computing, 48(6):1698-1710, 2019. URL: https://doi.org/10.1137/18M1223034.
  4. Marek Cygan, Dániel Marx, Marcin Pilipczuk, and Michal Pilipczuk. The planar directed k-vertex-disjoint paths problem is fixed-parameter tractable. In FOCS 2013: Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, pages 197-206. IEEE Computer Society, 2013. URL: https://doi.org/10.1109/FOCS.2013.29.
  5. Éric Colin de Verdière and Alexander Schrijver. Shortest vertex-disjoint two-face paths in planar graphs. ACM Trans. Algorithms, 7(2):19:1-19:12, 2011. URL: https://doi.org/10.1145/1921659.1921665.
  6. Guoli Ding, A. Schrijver, and P. D. Seymour. Disjoint paths in a planar graph - a general theorem. SIAM Journal on Discrete Mathematics, 5(1):112-116, 1992. URL: https://doi.org/10.1137/0405009.
  7. Richard M. Karp. On the computational complexity of combinatorial problems. Networks, 5(1):45-68, 1975. Google Scholar
  8. Yusuke Kobayashi and Christian Sommer. On shortest disjoint paths in planar graphs. Discrete Optimization, 7(4):234-245, 2010. URL: https://doi.org/10.1016/j.disopt.2010.05.002.
  9. Daniel Lokshtanov, Pranabendu Misra, Michał Pilipczuk, Saket Saurabh, and Meirav Zehavi. An exponential time parameterized algorithm for planar disjoint paths. In STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 1307-1316. Association for Computing Machinery, 2020. URL: https://doi.org/10.1145/3357713.3384250.
  10. James F. Lynch. The equivalence of theorem proving and the interconnection problem. ACM SIGDA Newsletter, 5:31-36, 1975. URL: https://doi.org/10.1145/1061425.1061430.
  11. Neil Robertson and P.D. Seymour. Graph minors. XIII. The disjoint paths problem. Journal of Combinatorial Theory, Series B, 63(1):65-110, 1995. URL: https://doi.org/10.1006/jctb.1995.1006.
  12. Ildikó Schlotter. Shortest two disjoint paths in conservative graphs, 2023. URL: https://arxiv.org/abs/2307.12602.
  13. Ildikó Schlotter and András Sebő. Odd paths, cycles and T-joins: Connections and algorithms, 2022. URL: https://arxiv.org/abs/2211.12862.
  14. Alexander Schrijver. Finding k disjoint paths in a directed planar graph. SIAM Journal on Computing, 23(4):780-788, 1994. URL: https://doi.org/10.1137/S0097539792224061.
  15. Alexander Schrijver. Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin, 2003. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact