Lower bounds for approximating the matching polytope

We prove that any linear program that approximates the matching polytope on n-vertex graphs up to a factor of (1 + ϵ) for any 2/n ≤ ϵ ≤ 1 must have at least n α= inequalities where 0 ≤ < 1 is an absolute constant. This is tight as exhibited by the (1 + ") approximating linear program obtained by dropping the odd set constraints of size larger than (1 + ϵ) from the description of the matching polytope. Previously, a tight lower bound of 2(n) was only known for " = O -1 n [22, 5] whereas for 2/n ≤ ϵ ≤ 1, the best lower bound was 2 Ω(1/ϵ) [22]. 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.