Jonah Brown-Cohen: The matching problem has no small symmetric SDP

PRACTICALITIES
Lunch is served at 12:00 noon (register at the latest the Saturday before, but preferably earlier, at this doodle . The presentation starts at 12:10 pm and ends at 1 pm. Those of us who wish reconvene after a short break for 1½-2 hours of more technical discussions.
 
ABSTRACT
Yannakakis showed that the matching problem does not have a small symmetric linear program. Rothvos 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 nk.
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.