Approximating the Expected Values for Combinatorial Optimization Problems over Stochastic Points

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 #P-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.

Approximating the Distribution of the Median and other Robust Estimators on Uncertain Data

“…The algorithms and computational geometry communities have recently generated a large amount of research in trying to understand how to efficiently process and represent uncertain data [14,24,20,5,18,21,4,2,3,1], not to mention some motivating systems and other progress from the database community [27,16,15,34,6]. Some work in this area considers other models, with either worst-case representations of the data uncertainty [31] which do not naturally allow probabilistic models, or when the data may not exist with some probability [18,21,5]. The second model can often be handled as a special case of the locationally uncertain model we study.…”

“…Although algorithms for locationally uncertain points have been studied in quite a few contexts over the last decade [14,24,20,5,18,4,2,3,34] (see more through discussion in full version [10]), few have directly addressed the problem of noise in the data. As the uncertainty is often the direct consequence of noise in the data collection process, this is a pressing concern.…”

On the Expected Diameter, Width, and Complexity of a Stochastic Convex-Hull

“…The main difficulty is that one has to deal with exponentially many realizations of a stochastic dataset. For this reason, many similar problems of this type were known to be #P-hard [7], while other ones usually require much higher time costs than their non-stochastic versions. Our polynomial-time solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest.…”

Stochastic Closest-Pair Problem and Most-Likely Nearest-Neighbor Search in Tree Spaces

“…Our results imply the existence of an LVD data structure which answers k-LNN queries in O(log n + k) time using average-case O(t + k 2 n) space, and worst-case O(t + kn 2 ) space if the existence probabilities are constant-far from 0. Finally, we also give an O(t + n 2 log n + n 2 k)-time algorithm to construct the LVD data structure.to be NP-hard or #P-hard for general metric, and some of them remain #P-hard even when X = R d for d ≥ 2 [9,11]. Due to the hardness of the stochastic closest-pair problems in general and Euclidean space, it is then natural to ask whether these problems are easier in other kinds of metric spaces such as tree space (or tree network).…”

“…to be NP-hard or #P-hard for general metric, and some of them remain #P-hard even when X = R d for d ≥ 2 [9,11]. Due to the hardness of the stochastic closest-pair problems in general and Euclidean space, it is then natural to ask whether these problems are easier in other kinds of metric spaces such as tree space (or tree network).…”