License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.2
URN: urn:nbn:de:0030-drops-75511
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Bérczi, Kristóf ; Chandrasekaran, Karthekeyan ; Király, Tamás ; Lee, Euiwoong ; Xu, Chao

Global and Fixed-Terminal Cuts in Digraphs

LIPIcs-APPROX-RANDOM-2017-2.pdf (0.9 MB)


The computational complexity of multicut-like problems may vary significantly depending on whether the terminals are fixed or not. In this work we present a comprehensive study of this phenomenon in two types of cut problems in directed graphs: double cut and bicut.

1. Fixed-terminal edge-weighted double cut is known to be solvable efficiently. We show that fixed-terminal node-weighted double cut cannot be approximated to a factor smaller than 2 under the Unique Games Conjecture (UGC), and we also give a 2-approximation algorithm. For the global version of the problem, we prove an inapproximability bound of 3/2 under UGC.

2. Fixed-terminal edge-weighted bicut is known to have an approximability factor of 2 that is tight under UGC. We show that the global edge-weighted bicut is approximable to
a factor strictly better than 2, and that the global node-weighted bicut cannot be approximated to a factor smaller than 3/2 under UGC.

3. In relation to these investigations, we also prove two results on undirected graphs which are of independent interest. First, we show NP-completeness and a tight inapproximability bound of 4/3 for the node-weighted 3-cut problem under UGC. Second, we show that for constant k, there exists an efficient algorithm to solve the minimum {s,t}-separating k-cut problem.

Our techniques for the algorithms are combinatorial, based on LPs and based on the enumeration of approximate min-cuts. Our hardness results are based on combinatorial reductions and integrality gap instances.

BibTeX - Entry

  author =	{Krist{\'o}f B{\'e}rczi and Karthekeyan Chandrasekaran and Tam{\'a}s Kir{\'a}ly and Euiwoong Lee and Chao Xu},
  title =	{{Global and Fixed-Terminal Cuts in Digraphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{2:1--2:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Klaus Jansen and Jos{\'e} D. P. Rolim and David Williamson and Santosh S. Vempala},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-75511},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.2},
  annote =	{Keywords: Directed Graphs, Arborescence, Graph Cuts, Hardness of Approximation}

Keywords: Directed Graphs, Arborescence, Graph Cuts, Hardness of Approximation
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)
Issue Date: 2017
Date of publication: 11.08.2017

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