The Matching Polytope has Exponential Extension Complexity

Abstract

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 2 Ω ( n ) . By a known reduction, this also improves the lower bound on the extension complexity for the TSP polytope from 2 Ω (√ n ) to 2 Ω ( n ) .