A simple dual ascent algorithm for the multilevel facility location problem

We present a simple dual ascent method for the multilevel facility location problem which finds a solution within $6$ times the optimum for the uncapacitated case and within $12$ times the optimum for the capacitated one. The algorithm is deterministic and based on the primal-dual technique. \u

An average case analysis of the minimum spanning tree heuristic for the range assignment problem

We present an average case analysis of the minimum spanning tree heuristic for the range assignment problem on a graph with power weighted edges. It is well-known that the worst-case approximation ratio of this heuristic is 2. Our analysis yields the following results: (1) In the one dimensional case ($d = 1$), where the weights of the edges are 1 with probability $p$ and 0 otherwise, the average-case approximation ratio is bounded from above by $2-p$. (2) When $d =1$ and the distance between neighboring vertices is drawn from a uniform $[0,1]$-distribution, the average approximation ratio is bounded from above by $2-2^{-\alpha}$ where $\alpha$ denotes the distance power radient. (3) In Euclidean 2-dimensional space, with distance power gradient $\alpha = 2$, the average performance ratio is bounded from above by $1 + \log 2$

Integration and Optimization of Multivariate Polynomials by Restriction onto a Random Subspace

We consider the problem of efficient integration of an n-variate polynomial with respect to the Gaussian measure in R^n and related problems of complex integration and optimization of a polynomial on the unit sphere. We identify a class of n-variate polynomials f for which the integral of any positive integer power f^p over the whole space is well-approximated by a properly scaled integral over a random subspace of dimension O(log n). Consequently, the maximum of f on the unit sphere is well-approximated by a properly scaled maximum on the unit sphere in a random subspace of dimension O(log n). We discuss connections with problems of combinatorial counting and applications to efficient approximation of a hafnian of a positive matrix.Comment: 15 page