Boolean function analysis meets stochastic optimization: An approximation scheme for stochastic knapsack.

The stochastic knapsack problem is the stochastic variant of the classical knapsack problem in which the algorithm designer is given a a knapsack with a given capacity and a collection of items where each item is associated with a profit and a probability distribution on its size. The goal is to select a subset of items with maximum profit and violate the capacity constraint with probability at most p (referred to as the overflow probability). While several approximation algorithms [27, 22, 4, 17, 30] have been developed for this problem, most of these algorithms relax the capacity constraint of the knapsack. In this paper, we design efficient approximation schemes for this problem without relaxing the capacity constraint. (i) Our first result is in the case when item sizes are Bernoulli random variables. In this case, we design a (nearly) fully polynomial time approximation scheme (FPTAS) which only relaxes the overflow probability. (ii) Our second result generalizes the first result to the case when all the item sizes are supported on a (common) set of constant size. In this case, we obtain a quasi-FPTAS. (iii) Our third result is in the case when item sizes are so-called "hypercontractive" random variables i.e., random variables whose second and fourth moments are within constant factors of each other. In other words, the kurtosis of the random variable is upper bounded by a constant. This class has been widely studied in probability theory and most natural random variables are hypercontractive including well-known families such as Poisson, Gaussian, exponential and Laplace distributions. In this case, we design a polynomial time approximation scheme which relaxes both the overflow probability and maximum profit. Crucially, all of our algorithms meet the capacity constraint exactly, a result which was previously known only when the item sizes were Poisson or Gaussian random variables [22, 24]. Our results rely on new connections between Boolean function analysis and stochastic optimization and are obtained by an adaption and extension of ideas such as (central) limit theorems, moment matching theorems and the inuential critical index machinery of Servedio [43] developed in the context of complexity theoretic analysis of halfspaces. We believe that these ideas and techniques may prove to be useful in other stochastic optimization problems as well.