Discrete random variables are essential ingredients in various artificial intelligence problems. These include the estimation of the probability of missing the deadline in a series-parallel schedule and the assignment of suppliers to tasks in a project in a manner that maximizes the probability of meeting the overall project deadline. The solving of such problems involves repetitive operations, such as summation, over random variables. However, these computations are NP-hard. Therefore, we explore techniques and methods for approximating random variables with a given support size and minimal Kolmogorov distance. We examine both the general problem of approximating a random variable and a one-sided version in which over-approximation is allowed but not under-approximation. We propose several algorithms and evaluate their performance through computational complexity analysis and empirical evaluation. All the presented algorithms are optimal in the sense that given an input random variable and a requested support size, they return a new approximated random variable with the requested support size and minimal Kolmogorov distance from the input random variable. Our approximation algorithms offer useful estimations of probabilities in situations where exact computations are not feasible due to NP-hardness complexity.

中文翻译：

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随机变量的高效最优柯尔莫哥洛夫近似

离散随机变量是各种人工智能问题的重要组成部分。其中包括估计串联并行计划中错过最后期限的概率，以及以最大化满足整个项目最后期限的概率的方式将供应商分配给项目中的任务。解决此类问题涉及对随机变量进行重复运算，例如求和。然而，这些计算是 NP 困难的。因此，我们探索在给定支持大小和最小柯尔莫哥洛夫距离的情况下逼近随机变量的技术和方法。我们研究了近似随机变量的一般问题和允许过度近似但不允许欠近似的片面版本。我们提出了几种算法，并通过计算复杂性分析和实证评估来评估它们的性能。所有提出的算法在给定输入随机变量和请求的支持大小的意义上都是最优的，它们返回具有请求的支持大小和与输入随机变量的最小柯尔莫哥洛夫距离的新近似随机变量。在由于 NP 难度复杂性而无法进行精确计算的情况下，我们的近似算法提供了有用的概率估计。