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 k no more than n/2. Membership for P_k is computable in time polynomial in n (of degree linear in k). We discuss facets of T_n that lie on the boundary of P_k. We introduce a new measure on each facet defining inequality for T_n in terms of the eigenvalues of a quadratic form. Using these eigenvalues of facets, we show that the scaling of P_1 by n^(1/2) has all of the facets of T_n defined by the subtour elimination constraints either in its interior or lying on its boundary.Comment: 25 page

The computational complexity of convex bodies

We discuss how well a given convex body B in a real d-dimensional vector space V can be approximated by a set X for which the membership question: ``given an x in V, does x belong to X?'' can be answered efficiently (in time polynomial in d). We discuss approximations of a convex body by an ellipsoid, by an algebraic hypersurface, by a projection of a polytope with a controlled number of facets, and by a section of the cone of positive semidefinite quadratic forms. We illustrate some of the results on the Traveling Salesman Polytope, an example of a complicated convex body studied in combinatorial optimization.Comment: 24 page

Some upper and lower bounds on PSD-rank

Positive semidefinite rank (PSD-rank) is a relatively new quantity with applications to combinatorial optimization and communication complexity. We first study several basic properties of PSD-rank, and then develop new techniques for showing lower bounds on the PSD-rank. All of these bounds are based on viewing a positive semidefinite factorization of a matrix $M$ as a quantum communication protocol. These lower bounds depend on the entries of the matrix and not only on its support (the zero/nonzero pattern), overcoming a limitation of some previous techniques. We compare these new lower bounds with known bounds, and give examples where the new ones are better. As an application we determine the PSD-rank of (approximations of) some common matrices.Comment: 21 page

Lower bounds on the size of semidefinite programming relaxations

We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on $n$-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than $2^{n^c}$, for some constant $c > 0$. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes. Our results follow from a general technique for proving lower bounds on the positive semidefinite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-$O(1)$ sum-of-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7/8-approximation for MAX-3-SAT

Exponential Lower Bounds for Polytopes in Combinatorial Optimization

We solve a 20-year old problem posed by Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.Comment: 19 pages, 4 figures. This version of the paper will appear in the Journal of the ACM. The earlier conference version in STOC'12 had the title "Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds

On Semidefinite Programming Relaxations of the Travelling Salesman Problem (Replaced by DP 2008-96)

AMS classification: 90C22, 20Cxx, 70-08