A fully polynomial randomized approximation scheme for counting perfect matchings in general graphs

Theorem 1.1: a uniform classical randomized algorithm that, for any finite simple undirected graph $G$ and rationals $0<\varepsilon<1$, $0<\delta<1/2$, returns $\widehat Z\ge0$ with $\Pr[(1-\varepsilon)Z(G)\le\widehat Z\le(1+\varepsilon)Z(G)]\ge1-\delta$, returns 0 with certainty when $Z(G)=0$, and runs in worst-case bit time polynomial in the input length, $\varepsilon^{-1}$ and $\log\delta^{-1}$. Corollaries give counting schemes for edge sets with prescribed degrees and for matchings of a given size. It is for unweighted simple graphs: it does not claim an FPRAS for hafnians with arbitrary nonnegative weights, and the polynomial's degree is not optimised.

Theorem 1.1: a uniform classical randomized algorithm that, for any finite simple undirected graph $G$ and rationals $0<\varepsilon<1$, $0<\delta<1/2$, returns $\widehat Z\ge0$ with $\Pr[(1-\varepsilon)Z(G)\le\widehat Z\le(1+\varepsilon)Z(G)]\ge1-\delta$, returns 0 with certainty when $Z(G)=0$, and runs in worst-case bit time polynomial in the input length, $\varepsilon^{-1}$ and $\log\delta^{-1}$. Corollaries give counting schemes for edge sets with prescribed degrees and for matchings of a given size. It is for unweighted simple graphs: it does not claim an FPRAS for hafnians with arbitrary nonnegative weights, and the polynomial's degree is not optimised.