Lower Bounds for Approximating the Matching Polytope

We prove that any extended formulation that approximates the matching polytope on nvertex graphs up to a factor of (1 + ε) for any 2 n ≤ ε ≤ 1 must have at least ( n α/ε ) defining inequalities where 0 < α < 1 is an absolute constant. This is tight as exhibited by the (1 + ε) approximating linear program obtained by dropping the odd set constraints of size larger than (1 + ε)/ε from the description of the matching polytope. Previously, a tight lower bound of 2Ω(n) was only known for ε = O ( 1 n ) [Rot14, BP15] whereas for 2 n ≤ ε ≤ 1, the best lower bound was 2Ω(1/ε) [Rot14]. 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.