The matching problem has no small symmetric SDP

TL;DR

This paper proves that the matching problem cannot be efficiently represented by small symmetric semidefinite programs, extending known limitations of linear programming formulations and analyzing SDP relaxations for related problems.

Contribution

It demonstrates that symmetric SDPs for the matching problem must be exponentially large, and relates SDP relaxations for TSP to symmetric SDP limitations.

Findings

Symmetric SDPs for matching are exponentially large.

An O(k)-round Lasserre SDP matches the approximation power of size n^k symmetric SDPs.

Upper bounds on polynomial identities over matchings and TSP tours.

Abstract

Yannakakis showed that the matching problem does not have a small symmetric linear program. Rothvo{\ss} recently proved that any, not necessarily symmetric, linear program also has exponential size. It is natural to ask whether the matching problem can be expressed compactly in a framework such as semidefinite programming (SDP) that is more powerful than linear programming but still allows efficient optimization. We answer this question negatively for symmetric SDPs: any symmetric SDP for the matching problem has exponential size. We also show that an O(k)-round Lasserre SDP relaxation for the metric traveling salesperson problem yields at least as good an approximation as any symmetric SDP relaxation of size $n^k$. The key technical ingredient underlying both these results is an upper bound on the degree needed to derive polynomial identities that hold over the space of matchings or traveling salesperson tours.