Showing papers by "Alexander Barvinok published in 2001"

A Remark on the Rank of Positive Semidefinite Matrices Subject to Affine Constraints

Abstract: Let K n be the cone of positive semidefinite n X n matrices and let A be an affine subspace of the space of symmetric matrices such that the intersection K n ?A is nonempty and bounded. Suppose that n ? 3 and that \codim A = r+2 \choose 2 for some 1 ≤ r ≤ n-2 . Then there is a matrix X ? K n ?A such that rank X ≤ r . We give a short geometric proof of this result, use it to improve a bound on realizability of weighted graphs as graphs of distances between points in Euclidean space, and describe its relation to theorems of Bohnenblust, Friedland and Loewy, and Au-Yeung and Poon.

The distance approach to approximate combinatorial counting

Abstract: We develop general methods to obtain fast (polynomial time) estimates of the cardinality of a combinatorially defined set via solving some randomly generated optimization problems on the set. Examples include enumeration of perfect matchings in a graph, linearly independent subsets of a set of vectors and colored spanning subgraphs of a graph. Geometrically, we estimate the cardinality of a subset of the Boolean cube via the average distance from a point in the cube to the subset with respect to some distance function. We derive asymptotically sharp cardinality bounds in the case of the Hamming distance and show that for small subsets a suitably defined “randomized” Hamming distance allows one to get tighter estimates of the cardinality.

The distribution of values in the quadratic assignment problem

Abstract: We obtain a number of results regarding the distribution of values of a quadratic function f on the set of nxn permutation matrices (identified with the symmetric group S_n) around its optimum (minimum or maximum). In particular, we estimate the fraction of permutations sigma such that f(sigma) lies within a given neighborhood of the optimal value of f. We identify some ``extreme'' functions f (there are 4 of those for n even and 5 for n odd) such that the distribution of every quadratic function around its optimum is a certain ``mixture'' of the distributions of the extremes and describe a natural class of functions (which includes, for example, the objective function in the Traveling Salesman Problem) with a relative abundance of near-optimal permutations. In particular, we identify a large class of functions f with the property that permutations in the vicinity of the optimal permutation (in the Hamming metric of S_n) tend to produce near optimal values of f (such is, for example, the objective function in the symmetric Traveling Salesman Problem) and show that for general f, just the opposite behavior may take place: an average permutation in the vicinity of the optimal permutation may be much worse than an average permutation in the whole group S_n.