二部平面图中完美匹配的确定性隔离
Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs
摘要：
我们提出了一个确定性的方式分配小（日志位）的权重的二分平面图的边缘，使最小重量完美匹配成为唯一的。隔离引理（Mulmuley et al. 1987）中所描述的实现了相同的一般图使用随机加权方案，而我们可以做到这一点时，确定性限制到二分平面图。因此，我们减少了决策和建设版本的匹配问题，以测试是否矩阵是奇异的，根据承诺，其行列式是0或1，从而获得一个高度并行的SPL算法的二分平面图。这改进了先前已知的非均匀SPL的界限（Allender et al. 1999）和$NC^2$的界限（Miller and Naor 1995，Mahajan and Varadarajan 2000）。这也重新点燃了希望，获得一个确定性的并行算法，构造一个完美的匹配在非二部平面图，这已经开放了很长一段时间。我们的技术是基本和简单的。
英文摘要：
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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