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

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

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

AMS classification: 90C22, 20Cxx, 70-08traveling salesman problem;semidefinite programming;quadratic as- signment problem

The linearization problem of a binary quadratic problem and its applications

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore-Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. Finally, we present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem

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

Uncapacitated Flow-based Extended Formulations

An extended formulation of a polytope is a linear description of this polytope using extra variables besides the variables in which the polytope is defined. The interest of extended formulations is due to the fact that many interesting polytopes have extended formulations with a lot fewer inequalities than any linear description in the original space. This motivates the development of methods for, on the one hand, constructing extended formulations and, on the other hand, proving lower bounds on the sizes of extended formulations. Network flows are a central paradigm in discrete optimization, and are widely used to design extended formulations. We prove exponential lower bounds on the sizes of uncapacitated flow-based extended formulations of several polytopes, such as the (bipartite and non-bipartite) perfect matching polytope and TSP polytope. We also give new examples of flow-based extended formulations, e.g., for 0/1-polytopes defined from regular languages. Finally, we state a few open problems

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

Semidefinite Programming Approach for the Quadratic Assignment Problem with a Sparse Graph

The matching problem between two adjacency matrices can be formulated as the NP-hard quadratic assignment problem (QAP). Previous work on semidefinite programming (SDP) relaxations to the QAP have produced solutions that are often tight in practice, but such SDPs typically scale badly, involving matrix variables of dimension $n^2$ where n is the number of nodes. To achieve a speed up, we propose a further relaxation of the SDP involving a number of positive semidefinite matrices of dimension $\mathcal{O}(n)$ no greater than the number of edges in one of the graphs. The relaxation can be further strengthened by considering cliques in the graph, instead of edges. The dual problem of this novel relaxation has a natural three-block structure that can be solved via a convergent Augmented Direction Method of Multipliers (ADMM) in a distributed manner, where the most expensive step per iteration is computing the eigendecomposition of matrices of dimension $\mathcal{O}(n)$. The new SDP relaxation produces strong bounds on quadratic assignment problems where one of the graphs is sparse with reduced computational complexity and running times, and can be used in the context of nuclear magnetic resonance spectroscopy (NMR) to tackle the assignment problem.Comment: 31 page