Planar Disjoint Paths, Treewidth, and Kernels

Michal Wlodarczyk, Meirav Zehavi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations


In the PLANAR DISJOINT PATHS problem, one is given an undirected planar graph with a set of k vertex pairs (s_i, t_i) and the task is to find k pairwise vertex-disjoint paths such that the i-th path connects s_i to t_i. We study the problem through the lens of kernelization, aiming at efficiently reducing the input size in terms of a parameter. We show that PLANAR DISJOINT PATHS does not admit a polynomial kernel when parameterized by k unless coNP ⊆ NP / poly, resolving an open problem by [Bodlaender, Thomassé, Yeo, ESA'09]. Moreover, we rule out the existence of a polynomial Turing kernel unless the WKhierarchy collapses. Our reduction carries over to the setting of edge-disjoint paths, where the kernelization status remained open even in general graphs. On the positive side, we present a polynomial kernel for PLANAR DISJOINT PATHS parameterized by k+tw, where tw denotes the treewidth of the input graph. As a consequence of both our results, we rule out the possibility of a polynomialtime (Turing) treewidth reduction to t w=kO(1) under the same assumptions. To the best of our knowledge, this is the first hardness result of this kind. Finally, combining our kernel with the known techniques [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver, SICOMP'94] yields an alternative (and arguably simpler) proof that PLANAR DISJOINT PATHS can be solved in time 2O(k2) · nO(1), matching the result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].

Original languageEnglish
Title of host publicationProceedings - 2023 IEEE 64th Annual Symposium on Foundations of Computer Science, FOCS 2023
PublisherInstitute of Electrical and Electronics Engineers
Number of pages14
ISBN (Electronic)9798350318944
StatePublished - 1 Jan 2023
Event64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023 - Santa Cruz, United States
Duration: 6 Nov 20239 Nov 2023

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428


Conference64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023
Country/TerritoryUnited States
CitySanta Cruz


  • disjoint paths
  • kernelization
  • parameterized complexity
  • planar graphs
  • treewidth

ASJC Scopus subject areas

  • General Computer Science


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