We study several stochastic combinatorial problems, including the expected utility maximization problem, the stochastic knapsack problem and the stochastic bin packing problem. A common technical challenge in these problems is to optimize some function (other than the expectation) of the sum of a set of random variables. The difficulty is mainly due to the fact that the probability distribution of the sum is the convolution of a set of distributions, which is not an easy objective function to work with. To tackle this difficulty, we introduce the Poisson approximation technique. The technique is based on the Poisson approximation theorem discovered by Le Cam, which enables us to approximate the distribution of the sum of a set of random variables using a compound Poisson distribution. Using the technique, we can reduce a variety of stochastic problems to the corresponding deterministic multiple-objective problems, which either can be solved by standard dynamic programming or have known solutions in the literature. For the problems mentioned above, we obtain the following results: 1. We first study the expected utility maximization problem introduced recently [Li and Despande, FOCS11]. For monotone and Lipschitz utility functions, we obtain an additive PTAS if there is a multidimensional PTAS for the multi-objective version of the problem, strictly generalizing the previous result. The result implies the first additive PTAS for maximizing threshold probability for the stochastic versions of global mincut, matroid base and matroid intersection. 2. For the stochastic bin packing problem (introduced in [Kleinberg, Rabani and Tardos, STOC97]), we show there is a polynomial time algorithm which uses at most the optimal number of bins, if we relax the size of each bin and the overflow probability by ǫ for any constant ǫ > 0. Based on this result, we obtain a 3-approximation if only the size of each bin can be relaxed by ǫ, improving the known O(1/ǫ) factor for constant overflow probability. 3. For stochastic knapsack, we show a (1+ǫ)-approximation using ǫ extra capacity for any ǫ > 0, even when the size and reward of each item may be correlated and canPermission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. celations of items are allowed. This generalizes the previous work [Balghat, Goel and Khanna, SODA11] for the case without correlation and cancelation. Our algorithm is also simpler. We also present a factor 2 + ǫ approximation algorithm for stochastic knapsack with cancelations, for any constant ǫ > 0, improving the current known approximation factor of 8 [Gupta, Krishnaswamy, Molinaro and Ravi, FOCS11]. 4. We also study an interesting variant of the stochastic knapsack problem, where the size and the profit of each item are revealed before the decision is made. The problem falls into the framework of Bayesian online selection problems, which has been studied a lot recently.
Recently, Li and Yuan REF showed that an additive PTAS for µ ∈ C bounded can be obtained using a completely different approach, called the Poisson approximation technique.
Stochastic combinatorial optimization via poisson approximation
STOC '13
Mathematics Computer Science