K-distinct branchings admits a polynomial kernel

Jørgen Bang-Jensen*, Kristine Vitting Klinkby, Saket Saurabh

*Corresponding author for this work

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Unlike the problem of deciding whether a digraph D = (V, A) has ℓ in-branchings (or ℓ out-branchings) is polynomial time solvable, the problem of deciding whether a digraph D = (V, A) has an in-branching B and an out-branching B+ which are arc-disjoint is NP-complete. Motivated by this, a natural optimization question that has been studied in the realm of Parameterized Complexity is called Rooted k-Distinct Branchings. In this problem, a digraph D = (V, A) with two prescribed vertices s, t are given as input and the question is whether D has an in-branching rooted at t and an out-branching rooted at s such that they differ on at least k arcs. Bang-Jensen et al. [Algorithmica, 2016 ] showed that the problem is fixed parameter tractable (FPT) on strongly connected digraphs. Gutin et al. [ICALP, 2017; JCSS, 2018 ] completely resolved this problem by designing an algorithm with running time 2O(k2 log2 k)nO(1). Here, n denotes the number of vertices of the input digraph. In this paper, answering an open question of Gutin et al., we design a polynomial kernel for Rooted k-Distinct Branchings. In particular, we obtain the following: Given an instance (D, k, s, t) of Rooted k-Distinct Branchings, in polynomial time we obtain an equivalent instance (D, k, s, t) of Rooted k-Distinct Branchings such that |V (D)| ≤ O(k2) and the treewidth of the underlying undirected graph is at most O(k). This result immediately yields an FPT algorithm with running time 2O(k log k) + nO(1); improving upon the previous running time of Gutin et al. For our algorithms, we prove a structural result about paths avoiding many arcs in a given in-branching or out-branching. This result might turn out to be useful for getting other results for problems concerning in-and out-branchings.

Original languageEnglish
Title of host publication29th Annual European Symposium on Algorithms, ESA 2021
EditorsPetra Mutzel, Rasmus Pagh, Grzegorz Herman
Number of pages15
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Publication dateSept 2021
Article number11
ISBN (Electronic)9783959772044
Publication statusPublished - Sept 2021
Event29th Annual European Symposium on Algorithms, ESA 2021 - Vitual, Lisbon, Portugal
Duration: 6. Sept 20218. Sept 2021


Conference29th Annual European Symposium on Algorithms, ESA 2021
CityVitual, Lisbon
SeriesLeibniz International Proceedings in Informatics, LIPIcs


  • Digraphs
  • In-branching
  • Out-branching
  • Polynomial Kernel


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