Symmetry Matters for Sizes of Extended Formulations

TL;DR

This paper demonstrates that for certain polytopes related to matchings and cycles in complete graphs, non-symmetric extended formulations can be polynomial in size, whereas symmetric ones cannot, highlighting the importance of symmetry in formulation size.

Contribution

It proves that non-symmetric extended formulations can be polynomial in size for specific polytopes, contrasting with symmetric formulations, thus emphasizing the significance of symmetry in optimization.

Findings

Non-symmetric formulations of polynomial size exist for certain matchings and cycles.

Symmetric formulations of polynomial size do not exist for these polytopes.

Symmetry can significantly impact the size of extended formulations.

Abstract

In 1991, Yannakakis (J. Comput. System Sci., 1991) 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 in the paper mentioned above 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 log(n) (rounded down) edges there are non-symmetric 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 log(n) (rounded down). Thus, with respect to the question for smallest possible extended formulations, in general symmetry requirements may matter a lot. Compared to the extended abtract that has appeared in the Proceedings of IPCO XIV at Lausanne, this paper does not only contain proofs that had been ommitted there, but it also presents slightly generalized and sharpened lower bounds.