Entropy and Face Dimension of the Perfect-Matching Polytope

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 $2m\ge 2$ vertices, the maximum entropy $H(x)$ of a matching law with marginals x satisfies $F(x)-(2-2/m)B(x)\le H(x)\le F(x),$ 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 $|\mathrm{supp}x|-\dim F_x\le 3m-2$, where F${}_x$ is the minimal face of the perfect-matching polytope containing x.