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LIPIcs - Leibniz International Proceedings in Informatics. 25th International Symposium on Theoretical Aspects of Computer Science
Doi
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 NC2 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
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