The matching problem has no small symmetric SDP

Abstract

Yannakakis [1991, 1988] showed that the matching problem does not have a small symmetric linear program. Rothvoß [2014] recently proved that any, not necessarily symmetric, linear program also has exponential size. In light of this, 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 asymmetric 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.

Affine reductions for LPs and SDPs

“…Also, approximations of the slack matrix provide approximate programs for the optimization problem, as shown in Theorem 7.2. Our approach also immediately carries over to symmetric formulations (see Braun et al [2015a] for SDPs) and other conic programming paradigms; the details are left to the interested reader.…”

Perfect Matching in Random Graphs is as Hard as Tseitin

“…Rather than linear programs, i.e., polytopes, we consider semide nite programs which are more expressive. The extension complexity in the semide nite setting has also been studied before [46,12] but these results are incomparable for the same reason just mentioned. While these results are incomparable, it is worth mentioning that there is a connection between Sherali-Adams (a proof system weaker than SoS) and extended formulations [17,40].…”