Maximizing expected utility over a knapsack constraint

The expected utility knapsack problem is to pick a set of items whose values are described by random variables so as to maximize the expected utility of the total value of the items picked while satisfying a constraint on the total weight of items picked. We consider the following solution approach for this problem: (i) use the sample average approximation framework to approximate the stochastic problem as a deterministic knapsack-constrained submodular maximization problem, and then (ii) use an approximation algorithm on the deterministic counterpart. We show that a polynomial number of samples is enough for a deterministic approximation that is close in relative error. Then, exploiting the strict monotonicity of typical utility functions, we present an algorithm that maximizes an increasing submodular function over a knapsack constraint with approximation ratio better than 1 − 1/e. For power utility functions we provide explicit approximation ratios leading to a polynomial time approximation algorithm. Assuming that the random values are completely described by a fixed and finite set of realizations, we also give a fully polynomial approximation scheme (FPTAS) for the expected utility knapsack problem with power utilities.