Sampling Sub-problems of Heterogeneous Max-cut Problems and Approximation Algorithms

Abstract

Recent work in the analysis of randomized approximation algorithms for NP-hard optimization problems has involved approximating the solution to a problem by the solution of a related sub-problem of constant size, where the sub-problem is constructed by sampling elements of the original problem uniformly at random. In light of interest in problems with a heterogeneous structure, for which uniform sampling might be expected to yield sub-optimal results, we investigate the use of nonuniform sampling probabilities. We develop and analyze an algorithm which uses a novel sampling method to obtain improved bounds for approximating the Max-Cut of a graph. In particular, we show that by judicious choice of sampling probabilities one can obtain error bounds that are superior to the ones obtained by uniform sampling, both for weighted and unweighted versions of Max-Cut. Of at least as much interest as the results we derive are the techniques we use. The first technique is a method to compute a compressed approximate decomposition of a matrix as the product of three smaller matrices, each of which has several appealing properties. The second technique is a method to approximate the feasibility or infeasibility of a large linear program by checking the feasibility or infeasibility of a nonuniformly randomly chosen sub-program of the original linear program. We expect that these and related techniques will prove fruitful for the future development of randomized approximation algorithms for problems whose input instances contain heterogeneities.