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arXiv · 0802.2850

Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs

Samir Datta·Raghav Kulkarni·Sambuddha Roy
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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.

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Samir Datta, Raghav Kulkarni, Sambuddha Roy. 2008-02-20. Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs. https://arxiv.org/abs/0802.2850

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