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Computing Paths of Large Rank in Planar Frameworks Deterministically

Authors Fedor V. Fomin , Petr A. Golovach , Tuukka Korhonen , Giannos Stamoulis



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Fedor V. Fomin
  • Department of Informatics, University of Bergen, Norway
Petr A. Golovach
  • Department of Informatics, University of Bergen, Norway
Tuukka Korhonen
  • Department of Informatics, University of Bergen, Norway
Giannos Stamoulis
  • LIRMM, Université de Montpellier, CNRS, Montpellier, France

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Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, and Giannos Stamoulis. Computing Paths of Large Rank in Planar Frameworks Deterministically. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 32:1-32:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.32

Abstract

A framework consists of an undirected graph G and a matroid M whose elements correspond to the vertices of G. Recently, Fomin et al. [SODA 2023] and Eiben et al. [ArXiV 2023] developed parameterized algorithms for computing paths of rank k in frameworks. More precisely, for vertices s and t of G, and an integer k, they gave FPT algorithms parameterized by k deciding whether there is an (s,t)-path in G whose vertex set contains a subset of elements of M of rank k. These algorithms are based on Schwartz-Zippel lemma for polynomial identity testing and thus are randomized, and therefore the existence of a deterministic FPT algorithm for this problem remains open. We present the first deterministic FPT algorithm that solves the problem in frameworks whose underlying graph G is planar. While the running time of our algorithm is worse than the running times of the recent randomized algorithms, our algorithm works on more general classes of matroids. In particular, this is the first FPT algorithm for the case when matroid M is represented over rationals. Our main technical contribution is the nontrivial adaptation of the classic irrelevant vertex technique to frameworks to reduce the given instance to one of bounded treewidth. This allows us to employ the toolbox of representative sets to design a dynamic programming procedure solving the problem efficiently on instances of bounded treewidth.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Planar graph
  • longest path
  • linear matroid
  • irrelevant vertex

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