A Positive Semidefinite Approximation of the Symmetric Traveling Salesman Polytope

Abstract

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 \(k \leq \lfloor n/2 \rfloor\). 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.