Preference Order Stochastic Knapsack Problems: Methodological Issues

Sniedovich, Moshe

doi:10.1057/jors.1980.189

Cited by 41 publications

(18 citation statements)

References 5 publications

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“…Variants of the stochastic knapsack problem have been studied in the literature such as knapsack problems with deterministic item volumes and random item profits [9,36,70,72]. Another somewhat related variant, known as the stochastic and dynamic knapsack problem, involves items that arrive online according to certain stochastic process [50,62].…”

Section: A1 Stochastic Ordered Adaptive Knapsack Problemmentioning

confidence: 99%

Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

Halman¹,

Klabjan

Li³

et al. 2014

SIAM J. Discrete Math.

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Abstract. We present a framework for obtaining fully polynomial time approximation schemes (FPTASs) for stochastic univariate dynamic programs with either convex or monotone single-period cost functions. This framework is developed through the establishment of two sets of computational rules, namely, the calculus of K-approximation functions and the calculus of K-approximation sets. Using our framework, we provide the first FPTASs for several NP-hard problems in various fields of research such as knapsack models, logistics, operations management, economics, and mathematical finance. Extensions of our framework via the use of the newly established computational rules are also discussed.Key words. fully polynomial time approximation schemes, stochastic dynamic programming, K-approximation AMS subject classifications. 68Q25, 68W25, 90B05, 90B06, 90C15, 90C39, 90C40, 90C56, 90C59

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Section: A1 Stochastic Ordered Adaptive Knapsack Problemmentioning

confidence: 99%

Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

Halman¹,

Klabjan

Li³

et al. 2014

SIAM J. Discrete Math.

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“…The stochastic knapsack problem with deterministic sizes and random profits has been studied in [CSW93], [Hen90], [Sne80], [SP79]; the objective is to find a set which maximizes the probability of achieving some threshold profit value. These works primarily focus on heuristic approaches for this problem.…”

Section: Other Related Workmentioning

confidence: 99%

Improved Approximation Results for Stochastic Knapsack Problems

Bhalgat¹,

Goel²,

Khanna³

2011

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

127

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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 nonadaptive 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. Disciplines Computer Sciences AbstractIn 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 schedulin...

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“…In [12] and [10], a version of the problem with fixed item values and random sizes is considered as well, but there the goal is to choose a valuable set of items whose probability of exceeding the knapsack capacity is small. Other authors [5,11,18,19] have studied versions with fixed item sizes but random values. If viewed as a version of stochastic knapsack, our problem is different from all of these in several respects.…”

Section: Related Workmentioning

confidence: 99%

Stochastic Models for Budget Optimization in Search-Based Advertising

2009

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Internet search companies sell advertisement slots based on users' search queries via an auction. Advertisers have to determine how to place bids on the keywords of their interest in order to maximize their return for a given budget: this is the budget optimization problem. The solution depends on the distribution of future queries. In this paper, we formulate stochastic versions of the budget optimization problem based on natural probabilistic models of distribution over future queries, and address two questions that arise.Evaluation Given a solution, can we evaluate the expected value of the objective function?Optimization Can we find a solution that maximizes the objective function in expectation?Our main results are approximation and complexity results for these two problems in our three stochastic models. In particular, our algorithmic results show that simple prefix strategies that bid on all cheap keywords up to some level are either optimal or good approximations for many cases; we show other cases to be NP-hard.

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Preference Order Stochastic Knapsack Problems: Methodological Issues

Cited by 41 publications

References 5 publications

Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

Improved Approximation Results for Stochastic Knapsack Problems

Stochastic Models for Budget Optimization in Search-Based Advertising

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