Deterministically isolating a perfect matching in bipartite planar graphs

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 [MVV87] 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 [ARZ99] and NC2 by [MN95, MV00]. 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.