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.2015.212
URN: urn:nbn:de:0030-drops-53040
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Cygan, Marek ; Kociumaka, Tomasz

Approximating Upper Degree-Constrained Partial Orientations

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In the Upper Degree-Constrained Partial Orientation (UDPO) problem we are given an undirected graph G=(V,E), together with two degree constraint functions d^-,d^+:V -> N. The goal is to orient as many edges as possible, in such a way that for each vertex v in V the number of arcs entering v is at most d^-(v), whereas the number of arcs leaving v is at most d^+(v). This problem was introduced by Gabow [SODA'06], who proved it to be MAXSNP-hard (and thus APX-hard). In the same paper Gabow presented an LP-based iterative rounding 4/3-approximation algorithm.

As already observed by Gabow, the problem in question is a special case of the classic 3-Dimensional Matching, which in turn is a special case of the k-Set Packing problem. Back in 2006 the best known polynomial time approximation algorithm for 3-Dimensional Matching was a simple local search by Hurkens and Schrijver [SIDMA'89], the approximation ratio of which is (3+epsilon)/2; hence the algorithm of Gabow was an improvement over the approach brought from the more general problems.

In this paper we show that the UDPO problem when cast as 3-Dimensional Matching admits a special structure, which is obliviously exploited by the known approximation algorithms for k-Set Packing. In fact, we show that already the local-search routine of Hurkens and Schrijver gives (4+epsilon)/3-approximation when used for the instances coming from UDPO. Moreover, the recent approximation algorithm for 3-Set Packing [Cygan, FOCS'13] turns out to be a (5+epsilon)/4-approximation for UDPO. This improves over 4/3 as the best ratio known up to date for UDPO.

BibTeX - Entry

  author =	{Marek Cygan and Tomasz Kociumaka},
  title =	{{Approximating Upper Degree-Constrained Partial Orientations}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{212--224},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Naveen Garg and Klaus Jansen and Anup Rao and Jos{\'e} D. P. Rolim},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-53040},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.212},
  annote =	{Keywords: graph orientations, degree-constrained orientations, approximation algorithm, local search}

Keywords: graph orientations, degree-constrained orientations, approximation algorithm, local search
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)
Issue Date: 2015
Date of publication: 13.08.2015

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