A PTAS for a Class of Stochastic Dynamic Programs

TL;DR

This paper introduces a polynomial time approximation scheme (PTAS) for a class of stochastic dynamic programs, including a new PTAS for a stochastic maximization problem involving adaptive probing of items with known distributions.

Contribution

The paper develops a general framework for PTAS in stochastic dynamic programs and provides the first PTAS for a stochastic combinatorial optimization problem involving adaptive probing.

Findings

First PTAS for the stochastic max problem with adaptive probing

Achieved approximation ratios better than the previous 1-1/e bound

Extended framework to other variants and related problems

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: \probemax: We are given a set of $n$ items, each item $i\in [n]$ has a value $X_i$ which is an independent random variable with a known (discrete) distribution $\pi_i$. We can {\em probe} a subset $P\subseteq [n]$ of items sequentially. Each time after {probing} an item $i$, we observe its value realization, which follows the distribution $\pi_i$. We can {\em 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 \etal~\cite{asadpour2015maximizing}. We also obtain PTAS for some generalizations and variants of the problem and some other problems.