A Positive Semidefinite Approximation of the Symmetric Traveling Salesman Polytope

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