# K-distinct branchings admits a polynomial kernel

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

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingArticle in proceedingsResearchpeer-review

## Abstract

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 language English 29th Annual European Symposium on Algorithms, ESA 2021 Petra Mutzel, Rasmus Pagh, Grzegorz Herman 15 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing Sept 2021 11 9783959772044 https://doi.org/10.4230/LIPIcs.ESA.2021.11 Published - Sept 2021 29th Annual European Symposium on Algorithms, ESA 2021 - Vitual, Lisbon, PortugalDuration: 6. Sept 2021 → 8. Sept 2021

### Conference

Conference 29th Annual European Symposium on Algorithms, ESA 2021 Portugal Vitual, Lisbon 06/09/2021 → 08/09/2021
Series Leibniz International Proceedings in Informatics, LIPIcs 204 1868-8969

## Keywords

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

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