A PTAS for a class of stochastic dynamic programs

Hao Fu, Jian Li, Pan Xu

Research output: Chapter in Book/Report/Conference proceedingConference contribution

11 Scopus citations

Abstract

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: 1. Probemax [19]: We are given a set of n items, each item i ∈ [n] has a value Xi which is an independent random variable with a known (discrete) distribution πi. 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 πi. 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 et al. [2]. We also obtain PTAS for some generalizations and variants of the problem. 2. Committed Pandora's Box [24, 22]: We are given a set of n boxes. For each box i ∈ [n], the cost ci is deterministic and the value Xi is an independent random variable with a known (discrete) distribution πi. Opening a box i incurs a cost of ci. We can adaptively choose to open the boxes (and observe their values) or stop. We want to maximize the expectation of the realized value of the last opened box minus the total opening cost. 3. Stochastic Target [15]: Given a predetermined target T and n items, we can adaptively insert the items into a knapsack and insert at most m items. Each item i has a value Xi which is an independent random variable with a known (discrete) distribution. Our goal is to design an adaptive policy such that the probability of the total values of all items inserted being larger than or equal to T is maximized. We provide the first bi-criteria PTAS for the problem. 4. Stochastic Blackjack Knapsack [16]: We are given a knapsack of capacity C and probability distributions of n independent random variables Xi. Each item i ∈ [n] has a size Xi and a profit pi. We can adaptively insert the items into a knapsack, as long as the capacity constraint is not violated. We want to maximize the expected total profit of all inserted items. If the capacity constraint is violated, we lose all the profit. We provide the first bi-criteria PTAS for the problem.

Original languageEnglish (US)
Title of host publication45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
EditorsChristos Kaklamanis, Daniel Marx, Ioannis Chatzigiannakis, Donald Sannella
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770767
DOIs
StatePublished - Jul 1 2018
Externally publishedYes
Event45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 - Prague, Czech Republic
Duration: Jul 9 2018Jul 13 2018

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume107
ISSN (Print)1868-8969

Other

Other45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
Country/TerritoryCzech Republic
CityPrague
Period7/9/187/13/18

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Approximation algorithm
  • Block policy
  • Dynamic program
  • Markov decision process
  • Stochastic optimization

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