## 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 X_{i} 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 c_{i} is deterministic and the value X_{i} is an independent random variable with a known (discrete) distribution πi. Opening a box i incurs a cost of c_{i}. 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 X_{i} 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 X_{i}. Each item i ∈ [n] has a size X_{i} 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 language | English (US) |
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Title of host publication | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 |

Editors | Christos Kaklamanis, Daniel Marx, Ioannis Chatzigiannakis, Donald Sannella |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770767 |

DOIs | |

State | Published - Jul 1 2018 |

Externally published | Yes |

Event | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 - Prague, Czech Republic Duration: Jul 9 2018 → Jul 13 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 107 |

ISSN (Print) | 1868-8969 |

### Other

Other | 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 |
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Country/Territory | Czech Republic |

City | Prague |

Period | 7/9/18 → 7/13/18 |

## All Science Journal Classification (ASJC) codes

- Software

## Keywords

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