Lower Bounds for Approximating the Matching Polytope
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We prove that any extended formulation 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 nα/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 2^Ω(n) was only known for ε = O(1/n) [Rothvoss, STOC '14; Braun and Pokutta, IEEE Trans. Information Theory '15] whereas for 2/n<ε< 1, the best lower bound was 2^Ω(1/ε) [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.
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