License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.ESA.2021.11
URN: urn:nbn:de:0030-drops-145925
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Bang-Jensen, Jørgen ; Klinkby, Kristine Vitting ; Saurabh, Saket

k-Distinct Branchings Admits a Polynomial Kernel

LIPIcs-ESA-2021-11.pdf (0.9 MB)


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 2^{𝒪(k² log² k)}n^{𝒪(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 2^{𝒪(klog k)}+ n^{𝒪(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.

BibTeX - Entry

  author =	{Bang-Jensen, J{\o}rgen and Klinkby, Kristine Vitting and Saurabh, Saket},
  title =	{{k-Distinct Branchings Admits a Polynomial Kernel}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-145925},
  doi =		{10.4230/LIPIcs.ESA.2021.11},
  annote =	{Keywords: Digraphs, Polynomial Kernel, In-branching, Out-Branching}

Keywords: Digraphs, Polynomial Kernel, In-branching, Out-Branching
Collection: 29th Annual European Symposium on Algorithms (ESA 2021)
Issue Date: 2021
Date of publication: 31.08.2021

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