The Matching Problem Has No Fully Polynomial Size Linear Programming Relaxation Schemes

Recently, Rothvoß established that every linear program (LP) expressing the matching polytope has an exponential number of inequalities (formally, the matching polytope has exponential extension complexity). We generalize this result by deriving strong bounds on the LP inapproximability of the matching problem: for fixed 0 <; ε <; 1, every (1 - ε/n)-approximating LP requires an exponential number of inequalities, where n is the number of vertices. This is sharp, given the well-known ρ-approximation of size O(1/(1-ρn)) provided by the odd-sets of size up to 1/(1-ρ). Thus, matching is the first problem in F, which does not admit a fully polynomial-size LP relaxation scheme (the LP equivalent of an Fully Polynomial-Time Approximation Scheme), which provides a sharp separation from the polynomial-size LP relaxation scheme obtained, e.g., through constant-sized odd-sets mentioned above. Analyzing the size of LP formulations is equivalent to examining the nonnegative rank of matrices. We study the nonnegative rank through an information-theoretic approach; while it reuses key ideas from Rothvoß, the main lower bounding technique is different: we employ the information-theoretic notion of Wyner's common information used for studying LP formulations. This allows us to analyze the nonnegative rank of perturbations of slack matrices, e.g., the approximations of the matching polytope. It turns out that the high extension complexity for the matching problem stems from the same source of hardness as in the case of the correlation polytope: a direct sum structure.