A Positive Semidefinite Approximation of the Symmetric Traveling Salesman Polytope

For a convex body B in a vector space V , we construct its approximation Pk, k = 1, 2, . . . using an intersection of a cone of positive semidefinite quadratic forms with an affine subspace. We show that Pk is contained in B for each k. When B is the Symmetric Traveling Salesman Polytope on n cities Tn, we show that the scaling of Pk by n k + O ` 1 n ́ contains Tn for k ≤ b 2 c. Membership for Pk is computable in time polynomial in n (of degree linear in k). We also discuss facets of Tn that lie on the boundary of Pk and we use eigenvalues to evaluate our bounds.