A PTAS for a Class of Stochastic Dynamic Programs
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We develop a framework for obtaining polynomial time approximation schemes (PTAS) for a class of stochastic dynamic programs. Using our framework, we obtain the first PTAS for the following stochastic combinatorial optimization problems: : We are given a set of n items, each item i∈ [n] has a value X_i which is an independent random variable with a known (discrete) distribution π_i. We can probe a subset P⊆ [n] of items sequentially. Each time after probing an item i, we observe its value realization, which follows the distribution π_i. We can adaptively probe at most m items and each item can be probed at most once. The reward is the maximum among the m realized values. Our goal is to design an adaptive probing policy such that the expected value of the reward is maximized. To the best of our knowledge, the best known approximation ratio is 1-1/e, due to Asadpour asadpour2015maximizing. We also obtain PTAS for some generalizations and variants of the problem and some other problems.
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