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Length-Bounded Cuts: Proper Interval Graphs and Structural Parameters

Authors Matthias Bentert, Klaus Heeger , Dušan Knop



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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

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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

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