A (2 + )-Approximation Algorithm for the Stochastic Knapsack Problem

The stochastic knapsack problem is a natural generalization of classical knapsack problem. In this problem, we are given a set of items each associated with a probability distribution on sizes and a profit, and a knapsack of unit capacity. The size of an item is unknown before inserting it into the knapsack, and it is revealed as soon as the item inserted into the knapsack. The objective is to design a policy that maximizes the expected profit of items that are successfully inserted into the knapsack. The stochastic knapsack problem is a fundamental stochastic packing problem and it arises in many applications, such as bandwidth allocation, budgeted learning, and scheduling. The optimal policy for stochastic knapsack problem can have an exponentially large explicit description and is known to be PSPACE-hard to compute. The best known approximation for this problem is a ( 8 3 + ) approximation for any > 0 given by Bhalgat et al [1]. They also give a (1+ )-approximation when the knapsack capacity is relaxed to (1+ ). In this paper, we develop a (2+ )-approximation algorithm for the stochastic knapsack problem. Our technique is based on a novel two knapsack experiment where we show that an optimal policy can be simulated by a pair of poly-time computable policies that use two knapsacks. The experiment thus yields a poly-time computable single knapsack policy that has an expected profit of at least (1− )OPT 2 .