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Yannakakis (Proceedings of the STOC, pp 223–228, 1988; J Comput Syst Sci 43(3):441–466, 1991. doi: 10.1016/0022-0000(91)90024-Y) showed that the matching problem does not have a small symmetric linear program. Rothvoß (Proceedings of the STOC, pp 263–272, 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.

The matching problem has no small symmetric SDP