Symmetry Matters for Sizes of Extended Formulations

Abstract

In 1991, Yannakakis [J. Comput. System Sci., 43 (1991), pp. 441--466] proved that no symmetric extended formulation for the matching polytope of the complete graph $K_n$ with $n$ nodes has a number of variables and constraints that is bounded subexponentially in $n$. Here, symmetric means that the formulation remains invariant under all permutations of the nodes of $K_n$. It was also conjectured by Yannakakis that “asymmetry does not help much,” but no corresponding result for general extended formulations has been found so far. In this paper we show that for the polytopes associated with the matchings in $K_n$ with $\lfloor\log n\rfloor$ edges there are nonsymmetric extended formulations of polynomial size, while nevertheless no symmetric extended formulations of polynomial size exist. We furthermore prove similar statements for the polytopes associated with cycles of length $\lfloor\log n\rfloor$. Thus, with respect to the question for smallest possible extended formulations, in general symmetry requirements may matter a lot. Compared to the extended abstract [Integer Pgrogramming and Combinatiorial Optimization, Lecture Notes in Comput. Sci. 6080, Springer, New York, 2010, pp. 135--148], this paper not only contains proofs that had been omitted there but also presents slightly generalized and sharpened lower bounds.