Symmetry Matters for Sizes of Extended Formulations

In 1991, Yannakakis (J. Comput. System Sci., 1991) proved that no symmetric extended formulation for the matching polytope of the complete graph K_n with n nodes has a number of variables and constraints that is bounded subexponentially in n. Here, symmetric means that the formulation remains invariant under all permutations of the nodes of K_n. It was also conjectured in the paper mentioned above that "asymmetry does not help much," but no corresponding result for general extended formulations has been found so far. In this paper we show that for the polytopes associated with the matchings in K_n with log(n) (rounded down) edges there are non-symmetric extended formulations of polynomial size, while nevertheless no symmetric extended formulations of polynomial size exist. We furthermore prove similar statements for the polytopes associated with cycles of length log(n) (rounded down). Thus, with respect to the question for smallest possible extended formulations, in general symmetry requirements may matter a lot. Compared to the extended abtract that has appeared in the Proceedings of IPCO XIV at Lausanne, this paper does not only contain proofs that had been ommitted there, but it also presents slightly generalized and sharpened lower bounds.Comment: 24 pages; incorporated referees' comments; to appear in: SIAM Journal on Discrete Mathematic

Tight Lower Bounds on the Sizes of Symmetric Extensions of Permutahedra and Similar Results

It is well known that the permutahedron Pi_n has 2^n-2 facets. The Birkhoff polytope provides a symmetric extended formulation of Pi_n of size Theta(n^2). Recently, Goemans described a non-symmetric extended formulation of Pi_n of size Theta(n log(n)). In this paper, we prove that Omega(n^2) is a lower bound for the size of symmetric extended formulations of Pi_n.Comment: corrected an error in the linear description of the permutahedron in introductio

Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix

The positive semidefinite rank of a nonnegative $(m\times n)$-matrix~$S$ is the minimum number~$q$ such that there exist positive semidefinite $(q\times q)$-matrices $A_1,\dots,A_m$, $B_1,\dots,B_n$ such that S(k,\ell) = \mbox{tr}(A_k^* B_\ell). The most important, lower bound technique for nonnegative rank is solely based on the support of the matrix S, i.e., its zero/non-zero pattern. In this paper, we characterize the power of lower bounds on positive semidefinite rank based on solely on the support.Comment: 9 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

Green's functions for multiply connected domains via conformal mapping

A method is described for the computation of the Green's function in the complex plane corresponding to a set of K symmetrically placed polygons along the real axis. An important special case is a set of K real intervals. The method is based on a Schwarz-Christoffel conformal map of the part of the upper half-plane exterior to the problem domain onto a semi-infinite strip whose end contains K-1 slits. From the Green's function one can obtain a great deal of information about polynomial approximations, with applications in digital filters and matrix iteration. By making the end of the strip jagged, the method can be generalised to weighted Green's functions and weighted approximations

Screening correlators with chiral Fermions

We study screening correlators of quark-antiquark composites at T=2T_c, where T_c is the QCD phase transition temperature, using overlap quarks in the quenched approximation of lattice QCD. As the lattice spacing is changed from 1/4T to a=1/6T and 1/8T, we find that screening correlators change little, in contrast with the situation for other types of lattice fermions. All correlators are close to the ideal gas prediction at small separations. The long distance falloff is clearly exponential, showing that a parametrization by a single screening length is possible at distances z > 1/T. The correlator corresponding to the thermal vector is close to the ideal gas value at all distances, whereas that for the thermal scalar deviates at large distances. This is examined through the screening lengths and momentum space correlators. There is strong evidence that the screening transfer matrix does not have reflection positivity.Comment: 10 pages, 9 fig