Is matching in NC, i.e., is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in TCS for over three decades, ever since the discovery of Random NC matching algorithms. Within this question, the case of planar graphs has remained an enigma: On the one hand, counting the number of perfect matchings is far harder than finding one (the former is #P-complete and the latter is in P), and on the other, for planar graphs, counting has long been known to be in NC whereas finding one has resisted a. solution! The case of bipartite planar graphs was solved b. IEEE Xplore, delivering full text access to the world's highest quality technical literature in engineering and technology. | IEEE Xplore. Is matching in NC, i.e., is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in TCS for over three decades. Sep 06,  · Is matching in NC, i.e., is there a deterministic fast parallel algorithm for it? This has What Are Planar Graphs Used For been an outstanding Planar Graph Problems And Solutions open question in TCS for over three decades, e. In this paper, we give an NC algorithm for finding a perfect matching in a planar graph. Our algorithm uses the abovestated fact about counting matchings in a crucial way. Our main new idea is an NC algorithm for finding a face of the perfect matching polytope at which many new conditions, involving constraints of the polytope, are. In this paper, we give an NC algorithm for finding a perfect matching in a planar graph. Our algorithm uses the above-stated fact about counting matchings in a crucial way. Several new ideas are needed, such as finding a point in the interior of the minimum weight face of the perfect matching polytope and finding a balanced odd tight set in NC.
In order to solve this problem we need to combine ideas from Cut Perfect Circles In Wood 40 the previous section with the ideas used for computing maximum size bipartite matching, i. In the case when we have many blossoms we observe that they divide the graph into regions and each lowest level recursive call goes to one of these regions, i. Types of Graphs- Before you go through this article, make sure that you have gone through the previous article on various Types of Graphs in Graph Theory. The weight of this path is equal to the number of edges from M s on it. Now, we want to construct a graph G L that will represent all shortest alternating paths from s with respect to M. In comparison, our paper gives a simpler algorithm for constructing a perfect matchings in Planar Graph NC than the one in [ 2 ]. Hence, the terms in det A G , s , w of minimum degree needs to correspond to pairs of minimum weight perfect matchings.

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