Length-Bounded Cuts: Proper Interval Graphs and Structural Parameters

Authors Matthias Bentert, Klaus Heeger , Dušan Knop



PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2020.36.pdf
  • Filesize: 0.57 MB
  • 14 pages

Document Identifiers

Author Details

Matthias Bentert
  • Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Germany
Klaus Heeger
  • Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Germany
Dušan Knop
  • Faculty of Information Technology, Czech Technical University in Prague, Czech Republic

Cite AsGet BibTex

Matthias Bentert, Klaus Heeger, and Dušan Knop. Length-Bounded Cuts: Proper Interval Graphs and Structural Parameters. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ISAAC.2020.36

Abstract

In the presented paper, we study the Length-Bounded Cut problem for special graph classes as well as from a parameterized-complexity viewpoint. Here, we are given a graph G, two vertices s and t, and positive integers β and λ. The task is to find a set F of edges of size at most β such that every s-t-path of length at most λ in G contains some edge in F. Bazgan et al. [Networks, 2019] conjectured that Length-Bounded Cut admits a polynomial-time algorithm if the input graph G is a proper interval graph. We confirm this conjecture by providing a dynamic-programming based polynomial-time algorithm. Moreover, we strengthen the W[1]-hardness result of Dvořák and Knop [Algorithmica, 2018] for Length-Bounded Cut parameterized by pathwidth. Our reduction is shorter, and the target of the reduction has stronger structural properties. Consequently, we give W[1]-hardness for the combined parameter pathwidth and maximum degree of the input graph. Finally, we prove that Length-Bounded Cut is W[1]-hard for the feedback vertex number. Both our hardness results complement known XP algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Dynamic programming
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Combinatorial optimization
Keywords
  • Edge-disjoint paths
  • pathwidth
  • feedback vertex number

Metrics

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

References

  1. Jiří Adámek and Václav Koubek. Remarks on flows in network with short paths. Comment. Math. Univ. Carolinae, 12:661-667, 1971. Google Scholar
  2. Georg Baier, Thomas Erlebach, Alexander Hall, Ekkehard Köhler, Petr Kolman, Ondřej Pangrác, Heiko Schilling, and Martin Skutella. Length-bounded cuts and flows. ACM Trans. Algorithms, 7(1):4:1-4:27, 2010. Google Scholar
  3. Cristina Bazgan, Till Fluschnik, André Nichterlein, Rolf Niedermeier, and Maximilian Stahlberg. A more fine-grained complexity analysis of finding the most vital edges for undirected shortest paths. Networks, 73(1):23-37, 2019. Google Scholar
  4. Matthias Bentert, Klaus Heeger, and Dusan Knop. Length-bounded cuts: Proper interval graphs and structural parameters. CoRR, abs/1910.03409, 2019. URL: http://arxiv.org/abs/1910.03409.
  5. Andreas Brandstädt, Van Bang Le, and Jeremy P. Spinrad. Graph Classes: A Survey. Philadelphia, PA: SIAM, 1999. Google Scholar
  6. Jianer Chen, Benny Chor, Mike Fellows, Xiuzhen Huang, David W. Juedes, Iyad A. Kanj, and Ge Xia. Tight lower bounds for certain parameterized np-hard problems. Inf. Comput., 201(2):216-231, 2005. Google Scholar
  7. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  8. Yefim Dinitz. Dinitz' algorithm: The original version and Even’s version. In Theoretical Computer Science, Essays in Memory of Shimon Even, pages 218-240, 2006. Google Scholar
  9. Pavel Dvořák and Dušan Knop. Parameterized complexity of length-bounded cuts and multicuts. Algorithmica, 80(12):3597-3617, 2018. Google Scholar
  10. Till Fluschnik, Danny Hermelin, André Nichterlein, and Rolf Niedermeier. Fractals for kernelization lower bounds. SIAM J. Discrete Math., 32(1):656-681, 2018. Google Scholar
  11. Lester Randolph Ford and Delbert Ray Fulkerson. Maximal flow through a network. Canadian Journal of Mathematics, 8:399–404, 1956. Google Scholar
  12. Petr A. Golovach and Dimitrios M. Thilikos. Paths of bounded length and their cuts: Parameterized complexity and algorithms. Discrete Optimization, 8(1):72-86, 2011. Google Scholar
  13. Luís Gouveia, Pedro Patrício, and Amaro de Sousa. Hop-constrained node survivable network design: An application to mpls over wdm. Networks and Spatial Economics, 8(1):3-21, March 2008. URL: https://doi.org/10.1007/s11067-007-9038-3.
  14. David Huygens, Martine Labbé, A. Ridha Mahjoub, and Pierre Pesneau. The two-edge connected hop-constrained network design problem: Valid inequalities and branch-and-cut. Networks, 49(1):116-133, 2007. URL: https://doi.org/10.1002/net.20146.
  15. David Huygens and A. Ridha Mahjoub. Integer programming formulations for the two 4-hop-constrained paths problem. Networks, 49(2):135-144, 2007. URL: https://doi.org/10.1002/net.20147.
  16. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. J. Comput. Syst. Sci., 62(2):367-375, 2001. Google Scholar
  17. Petr Kolman. On algorithms employing treewidth for l-bounded cut problems. J. Graph Algorithms Appl., 22(2):177-191, 2018. Google Scholar
  18. Petr Kolman and Christian Scheideler. Improved bounds for the unsplittable flow problem. J. Algorithms, 61(1):20-44, 2006. Google Scholar
  19. Ali Ridha Mahjoub and S. Thomas McCormick. Max flow and min cut with bounded-length paths: complexity, algorithms, and approximation. Math. Program., 124(1-2):271-284, 2010. Google Scholar
  20. Vishv M. Malhotra, M. Pramodh Kumar, and S. N. Maheshwari. An O(|V|³) algorithm for finding maximum flows in networks. Inf. Process. Lett., 7(6):277-278, 1978. Google Scholar
  21. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer Science & Business Media, 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