Parity Separation: A Scientifically Proven Method for Permanent Weight Loss
Given an edge weighted graph G, let PerfMatch(G) denote the weighted sum over all perfect matchings M in G, weighting each matching M by the product of weights of edges in M. If G is unweighted, this plainly counts the perfect matchings of G. In this paper, we introduce parity separation, a new method for reducing PerfMatch to unweighted instances: For graphs G with edge weights 1 and 1, we construct two unweighted graphs G1 and G2 such that PerfMatch(G) PerfMatch(G1) PerfMatch(G2) This yields a novel weight removal technique for counting perfect matchings, in addition to those known from classical #P hardness proofs. We derive the following applications: 1. An alternative #P completeness proof for counting unweighted perfect matchings. 2. C=P completeness for deciding whether two given unweighted graphs have the same number of perfect matchings. To the best of our knowledge, this is the first C=P completeness result for the "equality testing version" of any natural counting problem that is not already #P hard under parsimonious reductions. 3. An alternative tight lower bound for counting unweighted perfect matchings under the counting exponential time hypothesis #ETH. Our technique is based upon matchgates and the Holant framework. To make our #P hardness proof self contained, we also apply matchgates for an alternative #P hardness proof of PerfMatch on graphs with edge weights 1 and 1.
ICALP