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DOI: 10.4230/LIPIcs.STACS.2008.1346
URN: urn:nbn:de:0030-drops-13465
URL: http://drops.dagstuhl.de/opus/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 =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2008.1346},
  annote =	{Keywords: }
}

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


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