Approximating the Expected Values for Combinatorial Optimization Problems over Stochastic Points

TL;DR

This paper develops approximation algorithms for computing expected values and probabilities of various combinatorial optimization problems over stochastic points, addressing computational hardness and improving existing methods.

Contribution

It introduces FPRAS algorithms for several stochastic geometric problems, including minimum spanning trees, and improves upon previous results for stochastic MSTs.

Findings

FPRAS algorithms for multiple stochastic optimization problems

Improved approximation for stochastic minimum spanning trees

First known algorithms for several stochastic geometric problems

Abstract

We consider the stochastic geometry model where the location of each node is a random point in a given metric space, or the existence of each node is uncertain. We study the problems of computing the expected lengths of several combinatorial or geometric optimization problems over stochastic points, including closest pair, minimum spanning tree, $k$-clustering, minimum perfect matching, and minimum cycle cover. We also consider the problem of estimating the probability that the length of closest pair, or the diameter, is at most, or at least, a given threshold. Most of the above problems are known to be $\sharpP$-hard. We obtain FPRAS (Fully Polynomial Randomized Approximation Scheme) for most of them in both the existential and locational uncertainty models. Our result for stochastic minimum spanning trees in the locational uncertain model improves upon the previously known constant factor approximation algorithm. Our results for other problems are the first known to the best of our knowledge.