Improved Approximation Results for Stochastic Knapsack Problems
In the stochastic knapsack 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 revealed as soon as it is inserted into the knapsack, and the goal 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 natural generalization of the classical knapsack problem, and arises in many applications, including bandwidth allocation, budgeted learning, and scheduling. An adaptive policy for stochastic knapsack specifies the next item to be inserted based on observed sizes of the items inserted thus far. The adaptive policy 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 (3 + є)-approximation for any є > 0. Our first main result is a relaxed PTAS (Polynomial Time Approximation Scheme) for the adaptive policy, that is, for any є > 0, we present a poly-time computable (1+є)-approximate adaptive policy when knapsack capacity is relaxed to 1+є. At a high-level, the proof is based on transforming an arbitrary collection of item size distributions to canonical item size distributions that admit a compact description. We then establish a coupling that shows a (1+є)- approximation can be achieved for the original problem by a canonical policy that makes decisions at each step by observing events drawn from the sample space of canonical size distributions. Finally, we give a mechanism for approximating the optimal canonical policy. Our second main result is an (8/3 + є)-approximate adaptive policy for any є > 0 without relaxing the knapsack capacity, improving the earlier (3+є)-approximation result. Interestingly, we obtain this result by using the PTAS described above. We establish an existential result that the optimal policy for the knapsack with capacity 1 can be folded to get a policy with expected profit 3OPT/8 for a knapsack with capacity (1-є), with capacity relaxed to 1 only for the first item inserted. We then use our PTAS result to compute the (1 + є)-approximation to such policy. Our techniques also yield a relaxed PTAS for non- adaptive policies. Finally, we also show that our ideas can be extended to yield improved approximation guarantees for multidimensional and fixed set variants of the stochastic knapsack problem.