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Constant Approximating Disjoint Paths on Acyclic Digraphs Is W[1]-Hard

Author Michał Włodarczyk

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Author Details

Michał Włodarczyk
  • University of Warsaw, Poland

Funding

  • Włodarczyk, Michał: Supported by Polish National Science Centre SONATA-19 grant number 2023/51/D/ST6/00155.

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Michał Włodarczyk. Constant Approximating Disjoint Paths on Acyclic Digraphs Is W[1]-Hard. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 57:1-57:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.57

Abstract

In the Disjoint Paths problem, one is given a graph with a set of k vertex pairs (s_i,t_i) and the task is to connect each s_i to t_i with a path, so that the k paths are pairwise disjoint. In the optimization variant, Max Disjoint Paths, the goal is to maximize the number of vertex pairs to be connected. We study this problem on acyclic directed graphs, where Disjoint Paths is known to be W[1]-hard when parameterized by k. We show that in this setting Max Disjoint Paths is W[1]-hard to c-approximate for any constant c. To the best of our knowledge, this is the first non-trivial result regarding the parameterized approximation for Max Disjoint Paths with respect to the natural parameter k. Our proof is based on an elementary self-reduction that is guided by a certain combinatorial object constructed by the probabilistic method.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • fixed-parameter tractability
  • hardness of approximation
  • disjoint paths

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