Polynomial size linear programs for problems in P

A perfect matching in an undirected graph G = (V,E) is a set of vertex disjoint edges from E that include all vertices in V . The perfect matching problem is to decide if G has such a matching. Recently Rothvoß proved the striking result that the Edmonds’ matching polytope has exponential extension complexity. Here we describe a perfect matching polytope that is different from Edmonds’ polytope and define a weaker notion of extended formulation. We show that the new polytope has a weak extended formulation (WEF) Q of polynomial size. For each graph G we can readily construct an objective function so that solving the resulting linear program over Q decides whether or not G has a perfect matching. With this construction, a straightforward O(n) implementation of Edmonds’ matching algorithm using O(n logn) bits of space would yield a WEF Q with O(n logn) inequalities and variables. The method described here can be easily adapted for other matching problems, or indeed, for any problem in P/poly which can be solved by a well defined circuit or algorithm. The main tool is a method that converts a given circuit, or algorithm written in our pseudocode, for a decision problem into a polytope. We also apply these results to obtain polynomial upper bounds on the non-negative rank of certain slack matrices related to membership testing of languages in P/Poly.