TY - GEN

T1 - Parity problems in planar graphs

AU - Braverman, Mark

AU - Kulkarni, Raghav

AU - Roy, Sambuddha

PY - 2007/10/2

Y1 - 2007/10/2

N2 - 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 2k, 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 2k, we also exhibit a highly parallel ⊕L algorithm for finding the value of a Permanent modulo 2k. Previously, the best known result in this direction was Valiant's result that this problem lies in P.

AB - 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 2k, 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 2k, we also exhibit a highly parallel ⊕L algorithm for finding the value of a Permanent modulo 2k. Previously, the best known result in this direction was Valiant's result that this problem lies in P.

UR - http://www.scopus.com/inward/record.url?scp=34748867980&partnerID=8YFLogxK

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U2 - 10.1109/CCC.2007.23

DO - 10.1109/CCC.2007.23

M3 - Conference contribution

AN - SCOPUS:34748867980

SN - 0769527809

SN - 9780769527802

T3 - Proceedings of the Annual IEEE Conference on Computational Complexity

SP - 222

EP - 235

BT - Proceedings - Twenty-Second Annual IEEE Conference on Computational Complexity, CCC 2007

T2 - 22nd Annual IEEE Conference on Computational Complexity, CCC 2007

Y2 - 13 June 2007 through 16 June 2007

ER -