The Matching Polytope has Exponential Extension Complexity

A popular method in combinatorial optimization is to express polytopes P, which may potentially have exponentially many facets, as solutions of linear programs that use few extra variables to reduce the number of constraints down to a polynomial. After two decades of standstill, recent years have brought amazing progress in showing lower bounds for the so-called extension complexity, which for a polytope P denotes the smallest number of inequalities necessary to describe a higher-dimensional polytope Q that can be linearly projected on P. However, the central question in this field remained wide open: can the perfect matching polytope be written as an LP with polynomially many constraints? We answer this question negatively. In fact, the extension complexity of the perfect matching polytope in a complete n-node graph is 2Ω (n). By a known reduction, this also improves the lower bound on the extension complexity for the TSP polytope from 2Ω (√ n) to 2Ω (n).