Lower Bounds for Approximating the Matching Polytope

Abstract

We prove that any extended formulation that approximates the matching polytope on $n$-vertex graphs up to a factor of $(1+\epsilon )$ for any $\frac{2}{n}\le \epsilon \le 1$ must have at least $\left(\genfrac{}{}{0ex}{}{n}{\alpha /\epsilon }\right)$ defining inequalities where $0<\alpha <1$ is an absolute constant. This is tight as exhibited by the $(1+\epsilon )$ approximating linear program obtained by dropping the odd set constraints of size larger than $(1+\epsilon )/\epsilon$ from the description of the matching polytope. Previously, a tight lower bound of $2^{\Omega (n)}$ was only known for $\epsilon =O\left(\frac{1}{n}\right)$ [Rothvoss, STOC '14; Braun and Pokutta, IEEE Trans. Information Theory '15] whereas for $\frac{2}{n}\le \epsilon \le 1$, the best lower bound was $2^{\Omega \left(1/\epsilon \right)}$ [Rothvoss, STOC '14]. The key new ingredient in our proof is a close connection to the non-negative rank of a lopsided version of the unique disjointness matrix.

To appear in proceedings of SODA '18