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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2019.42
URN: urn:nbn:de:0030-drops-111639
URL: https://drops.dagstuhl.de/opus/volltexte/2019/11163/
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Eiben, Eduard ; Lokshtanov, Daniel ; Mouawad, Amer E.

Bisection of Bounded Treewidth Graphs by Convolutions

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Abstract

In the Bisection problem, we are given as input an edge-weighted graph G. The task is to find a partition of V(G) into two parts A and B such that ||A| - |B|| <= 1 and the sum of the weights of the edges with one endpoint in A and the other in B is minimized. We show that the complexity of the Bisection problem on trees, and more generally on graphs of bounded treewidth, is intimately linked to the (min, +)-Convolution problem. Here the input consists of two sequences (a[i])^{n-1}_{i = 0} and (b[i])^{n-1}_{i = 0}, the task is to compute the sequence (c[i])^{n-1}_{i = 0}, where c[k] = min_{i=0,...,k}(a[i] + b[k - i]). In particular, we prove that if (min, +)-Convolution can be solved in O(tau(n)) time, then Bisection of graphs of treewidth t can be solved in time O(8^t t^{O(1)} log n * tau(n)), assuming a tree decomposition of width t is provided as input. Plugging in the naive O(n^2) time algorithm for (min, +)-Convolution yields a O(8^t t^{O(1)} n^2 log n) time algorithm for Bisection. This improves over the (dependence on n of the) O(2^t n^3) time algorithm of Jansen et al. [SICOMP 2005] at the cost of a worse dependence on t. "Conversely", we show that if Bisection can be solved in time O(beta(n)) on edge weighted trees, then (min, +)-Convolution can be solved in O(beta(n)) time as well. Thus, obtaining a sub-quadratic algorithm for Bisection on trees is extremely challenging, and could even be impossible. On the other hand, for unweighted graphs of treewidth t, by making use of a recent algorithm for Bounded Difference (min, +)-Convolution of Chan and Lewenstein [STOC 2015], we obtain a sub-quadratic algorithm for Bisection with running time O(8^t t^{O(1)} n^{1.864} log n).

BibTeX - Entry

@InProceedings{eiben_et_al:LIPIcs:2019:11163,
  author =	{Eduard Eiben and Daniel Lokshtanov and Amer E. Mouawad},
  title =	{{Bisection of Bounded Treewidth Graphs by Convolutions}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{42:1--42:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Michael A. Bender and Ola Svensson and Grzegorz Herman},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/11163},
  URN =		{urn:nbn:de:0030-drops-111639},
  doi =		{10.4230/LIPIcs.ESA.2019.42},
  annote =	{Keywords: bisection, convolution, treewidth, fine-grained analysis, hardness in P}
}

Keywords: bisection, convolution, treewidth, fine-grained analysis, hardness in P
Seminar: 27th Annual European Symposium on Algorithms (ESA 2019)
Issue Date: 2019
Date of publication: 06.09.2019


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