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- W2145664670 abstract "We consider the problem of counting the number of spanning trees in planar graphs. We prove tight bounds on the complexity of the problem, both in general and especially in the modular setting. We exhibit the problem to be complete for Logspace when the modulus is 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sup> , for constant k. On the other hand, we show that for any other modulus and in the non-modular case, our problem is as hard in the planar case as for the case of arbitrary graphs. This completely settles the question regarding the complexity of modular computation of the number of spanning trees in planar graphs. The techniques used rely heavily on algebraic-topology. In the spirit of counting problems modulo 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sup> , we also exhibit a highly parallel oplusL algorithm for finding the value of a Permanent modulo 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sup> . Previously, the best known result in this direction was Valiant's result that this problem lies in P." @default.
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- W2145664670 date "2007-06-01" @default.
- W2145664670 modified "2023-09-25" @default.
- W2145664670 title "Parity Problems in Planar Graphs" @default.
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- W2145664670 doi "https://doi.org/10.1109/ccc.2007.23" @default.
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