We give a fully polynomial randomized approximation scheme (FPRAS) for counting perfect matchings in arbitrary finite simple undirected graphs, resolving the general-graph perfect-matching approximation problem. The algorithm returns zero with certainty when no perfect matching exists. Otherwise, it achieves relative error ε with failure probability at most δ in worst-case bit time polynomial in the input length, $\varepsilon ^{-1}$, and $\log\delta^{-1}$.

We prove the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant. For every feasible vector x of perfect-matching edge marginals in a loopless labelled multigraph on $2m\ge2$ vertices, the maximum entropy $H(x)$ of a matching law with marginals x satisfies $\displaystyle F(x)-(2-2/m)B(x)\le H(x)\le F(x),$ where $F(x)=-\sum_e x_e\log x_e$ and $B(x)=-\sum_e(1-x_e)\log(1-x_e)$. This bound holds throughout the polytope, including its boundary. We also prove the sharp bound $|\mathop{\mathrm{supp}}\nolimits x|-\dim F_x\le3m-2$, where F_x is the minimal face of the perfect-matching polytope containing x.