Hidden Circuits and Exact Counting in Ordered Graphs

Abstract: We prove that counting perfect matchings is $#P$-complete under polynomial-time Turing reductions on each of three classes of simple, unweighted graphs: monotone graphs, unit interval graphs, and chordal permutation graphs. The monotone result settles the exact-counting complexity left open by Dyer, Jerrum, and Müller (JACM 2017), complementing their rapid-mixing theorem. Inspired by quantum circuits, our reductions implement a circuit simulation using globally coupled matching-transfer operators. The key construction is an exact projection, implemented by a polynomial-length sequence of normalized transfers, that restores tensor-product locality and makes encoded gates composable. Interpolation-based cancellation then reduces circuit evaluation to unweighted perfect-matching counts in all three classes. We also place Dyer and Müller's class QChains within the distance-hereditary graphs and give an $O(n<sup>2)$-arithmetic-operation counting algorithm for the latter, improving the $O(n<sup>4)$ bound obtainable from Curticapean and Marx (SODA 2016). Together with prior results, these advances complete the exact-counting classification of the graph classes in Dyer and Müller's diagram (SIDMA 2019).