On the Power of Isolation in Planar Graphs
ACM Transactions on Computation Theory (TOCT)
TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
Space complexity of perfect matching in bounded genus bipartite graphs
Journal of Computer and System Sciences
Green's theorem and isolation in planar graphs
Information and Computation
Planarizing gadgets for perfect matching do not exist
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
Log-Space Algorithms for Paths and Matchings in k-Trees
Theory of Computing Systems
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We present a deterministic Logspace procedure, which, given a bipartite planar graph on n vertices, assigns O(log n) bits long weights to its edges so that the minimum weight perfect matching in the graph becomes unique. The Isolation Lemma as described in Mulmuley et al. (Combinatorica 7(1):105–131, 1987) achieves the same for general graphs using randomness, whereas we can do it deterministically when restricted to bipartite planar graphs. As a consequence, we reduce both decision and construction versions of the perfect matching problem in bipartite planar graphs to testing whether a matrix is singular, under the promise that its determinant is 0 or 1, thus obtaining a highly parallel $\mathsf{SPL}$algorithm for both decision and construction versions of the bipartite perfect matching problem. This improves the earlier known bounds of non-uniform $\mathsf{SPL}$by Allender et al. (J. Comput. Syst. Sci. 59(2):164–181, 1999) and $\mathsf{NC}$2 by Miller and Naor (SIAM J. Comput. 24:1002–1017, 1995), and by Mahajan and Varadarajan (Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing (STOC), pp. 351–357, 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. Further we try to find the lower bound on the number of bits needed for deterministically isolating a perfect matching. We show that our particular method for isolation will require Ω(log n) bits. Our techniques are elementary.