License: Creative Commons Attribution-NoDerivs 3.0 Unported license (CC BY-ND 3.0)
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DOI: 10.4230/LIPIcs.STACS.2008.1346
URN: urn:nbn:de:0030-drops-13465
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2008/1346/
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Datta, Samir ; Kulkarni, Raghav ; Roy, Sambuddha

Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs

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Abstract

We present a deterministic way of assigning small (log bit) weights
to the edges of a bipartite planar graph so that the minimum weight
perfect matching becomes unique. The isolation lemma as described
in (Mulmuley et al. 1987) achieves the same for general graphs
using a randomized weighting scheme, whereas we can do it
deterministically when restricted to bipartite planar graphs. As a
consequence, we reduce both decision and construction versions of
the matching problem to testing whether a matrix is singular, under
the promise that its determinant is $0$ or $1$, thus obtaining a
highly parallel SPL algorithm for bipartite planar graphs. This
improves the earlier known bounds of non-uniform SPL by (Allender
et al. 1999) and $NC^2$ by (Miller and Naor 1995, Mahajan and
Varadarajan 2000). It also rekindles the hope of obtaining a
deterministic parallel algorithm for constructing a perfect
matching in non-bipartite planar graphs, which has been open for a
long time. Our techniques are elementary and simple.

BibTeX - Entry

@InProceedings{datta_et_al:LIPIcs:2008:1346,
  author =	{Samir Datta and Raghav Kulkarni and Sambuddha Roy},
  title =	{{Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{229--240},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Susanne Albers and Pascal Weil},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2008/1346},
  URN =		{urn:nbn:de:0030-drops-13465},
  doi =		{10.4230/LIPIcs.STACS.2008.1346},
  annote =	{Keywords: }
}

Collection: 25th International Symposium on Theoretical Aspects of Computer Science
Issue Date: 2008
Date of publication: 06.02.2008

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