Optimal Approximation of Random Variables for Estimating the Probability of Meeting a Plan Deadline

Abstract

In planning algorithms and in other domains, there is often a need to run long computations that involve summations, maximizations and other operations on random variables, and to store intermediate results. In this paper, as a main motivating example, we elaborate on the case of estimating probabilities of meeting deadlines in hierarchical plans. A source of computational complexity, often neglected in the analysis of such algorithms, is that the support of the variables needed as intermediate results may grow exponentially along the computation. Therefore, to avoid exponential memory and time complexities, we need to trim these variables. This is similar, in a sense, to rounding intermediate results in numerical computations. Of course, to maintain the quality of algorithms, the trimming procedure should be efficient and it must maintain accuracy as much as possible. In this paper, we propose an optimal trimming algorithm with polynomial time and memory complexities for the purpose of estimating probabilities of deadlines in plans. More specifically, we show that our algorithm, given the needed size of the representation of the variable, provides the best possible approximation, where approximation accuracy is considered with a measure that fits the goal of estimating deadline meeting probabilities.

Efficient One Sided Kolmogorov Approximation

“…The most relevant work related to this paper is the papers on approximations of random variables in the context of estimating deadlines [3,2]. In these papers, X ′ is defined to be a good approximation of X if F X ′ (t) > F X (t) for any t and sup t F X ′ (t) − F X (t) is small.…”

“…Various approaches for approximation of probability distributions are studied in the literature [17,15,20,3,16,2]. These approaches vary in the types random variables considered, how they are represented, and in the criteria used for evaluation of the quality of the approximations.…”

“…Our main motivation for this work is estimation of the probability for missing deadlines, as described, e.g., in Cohen et al [3,2] and in [13]. Specifically, when X represents the probability distribution of the time to complete some complex schedule and we cannot afford to maintain the full table of its probability mass function, we pro-pose an algorithm for producing a smaller table, whose size can be specified, such that probabilities for missing deadlines are preserved as much as possible.…”

Assigning Suppliers to Meet a Deadline

“…The problem with computing the M value stems from the hardness of computation that lies in the exponential growth of the support size (2015). Consequently, after every addend in the computation of U (•) we apply the OPTAPPROX (X, m) operator of Cohen et al (2018), which for a given random variable X and a requested support size m returns a new random variable X with a support size of at most m in polynomial time. By restricting the support size to a given m, we bound the exponential growth.…”