45th Annual IEEE Symposium on Foundations of Computer Science (2004)
Oct. 17, 2004 to Oct. 19, 2004
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2004.15
Brian C. Dean , Massachusetts Institute of Technology
Michel X. Goemans , Massachusetts Institute of Technology
Jan Vondrák , Massachusetts Institute of Technology
We consider a stochastic variant of the NP-hard 0/1 knapsack problem in which item values are deterministic and item sizes are independent random variables with known, arbitrary distributions. Items are placed in the knapsack sequentially, and the act of placing an item in the knapsack instantiates its size. Our goal is to compute a solution "policy" that maximizes the expected value of items placed in the knapsack, and we consider both non-adaptive policies (that designate a priori a fixed sequence of items to insert) and adaptive policies (that can make dynamic choices based on the instantiated sizes of items placed in the knapsack thus far). We show that adaptivity provides only a constant-factor improvement by demonstrating a greedy non-adaptive algorithm that approximates the optimal adaptive policy within a factor of 7. We also design an adaptive polynomial-time algorithm which approximates the optimal adaptive policy within a factor of 5 + ε, for any constant ε > 0.
J. Vondrák, M. X. Goemans and B. C. Dean, "Approximating the Stochastic Knapsack Problem: The Benefit of Adaptivity," 45th Annual IEEE Symposium on Foundations of Computer Science(FOCS), Rome, Italy, 2004, pp. 208-217.