An FPTAS for Stochastic Unbounded Min-Knapsack Problem
In this paper, we study the stochastic unbounded min-knapsack problem (Min-SUKP). The ordinary unbounded min-knapsack problem states that: There are n types of items, and there is an infinite number of items of each type. The items of the same type have the same cost and weight. We want to choose a set of items such that the total weight is at least W and the total cost is minimized. The Min-SUKP generalizes the ordinary unbounded min-knapsack problem to the stochastic setting, where the weight of each item is a random variable following a known distribution and the items of the same type follow the same weight distribution. In Min-SUKP, different types of items may have different cost and weight distributions. In this paper, we provide an FPTAS for Min-SUKP, i.e., the approximate value our algorithm computes is at most \((1+\epsilon )\) times the optimum, and our algorithm runs in \(poly(1/\epsilon ,n,\log W)\) time.
KeywordsStochastic Knapsack Renewal decision problem Approximation algorithms
The authors would like to thank Jian Li for several useful discussions and the help with polishing the paper. The research is supported in part by the National Basic Research Program of China Grant 2015CB358700, the National Natural Science Foundation of China Grant 61822203, 61772297, 61632016, 61761146003, and a grant from Microsoft Research Asia.
- 2.Bhalgat, A., Goel, A., Khanna, S.: Improved approximation results for stochastic knapsack problems. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, Philadelphia (2011)Google Scholar
- 3.Bhalgat, A.: A (2 + \(\epsilon \))-approximation algorithm for the stochastic knapsack problem. Manuscript (2012)Google Scholar
- 6.Dean, B.C., Goemans, M.X., Vondrak, J.: Approximating the stochastic knapsack problem: the benefit of adaptivity. In: Annual IEEE Symposium on Foundations of Computer Science, pp. 208–217. IEEE Computer Society, Los Alamitos (2004)Google Scholar
- 9.Garey, M.R., Johnson, D.S.: Computers and Intractability, vol. 29. W. H. Freeman, New York (2002)Google Scholar
- 11.Gupta, A., Krishnaswamy, R., Molinaro, M., Ravi, R.: Approximation algorithms for correlated knapsacks and non-martingale bandits. In: IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, 22–25 October 2011, pp. 827–836 (2011)Google Scholar
- 17.Li, J., Yuan, W.: Stochastic combinatorial optimization via poisson approximation. In: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, STOC 2013, pp. 971–980. ACM, New York (2013)Google Scholar
- 18.Ma, W.: Improvements and generalizations of stochastic knapsack and multi-armed bandit approximation algorithms. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (2014)Google Scholar