Topic

# Continuous knapsack problem

About: Continuous knapsack problem is a research topic. Over the lifetime, 2126 publications have been published within this topic receiving 57727 citations.

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01 Nov 1990TL;DR: This paper focuses on the part of the knapsack problem where the problem of bin packing is concerned and investigates the role of computer codes in the solution of this problem.

Abstract: Introduction knapsack problem bounded knapsack problem subset-sum problem change-making problem multiple knapsack problem generalized assignment problem bin packing problem. Appendix: computer codes.

3,694 citations

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TL;DR: In this paper, the authors propose an approach that attempts to make this trade-off more attractive by flexibly adjusting the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations.

Abstract: A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library.

3,364 citations

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TL;DR: An algorithm is presented which finds for any 0 < e < 1 an approximate solution P satisfying (P* P)/P* < ~, where P* is the desired optimal sum.

Abstract: Given a positive integer M and n pairs of positive integers (p~, cD, , (p. , c.), maximize the s u m ~ ~p~ subject to the cons t ramts~ ~c, < M and ~, = 0 or 1 This is the well-known 0/1 knapsack problem An algorithm is presented which finds for any 0 < e < 1 an approximate solution P satisfying (P* P)/P* < ~, where P* is the desired optimal sum Moreover, for any fixed e, the algorithm has time complexity 0(n log n) and space complexity O(n) Modification of the algorithm for the unbounded knapsack problem where the ~,'s can be any nonnegative integer results in a O(n) computing time A hnear-time algorithm is also obtained for a special class of 0/1 knapsack problems having the property that p,/c, is the same for all 1 < z < n

999 citations

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TL;DR: Specific instances of the knapsack problem that appear very difficult to solve unless one possesses "trapdoor information" used in the design of the problem are demonstrated.

Abstract: The knapsack problem is an NP-complete combinatorial problem that is strongly believed to be computationally difficult to solve in general. Specific instances of this problem that appear very difficult to solve unless one possesses "trapdoor information" used in the design of the problem are demonstrated. Because only the designer can easily solve problems, others can send him information hidden in the solution to the problems without fear that an eavesdropper will be able to extract the information. This approach differs from usual cryptographic systems in that a secret key is not needed. Conversely, only the designer can generate signatures for messages, but anyone can easily check their authenticity.

828 citations

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TL;DR: A heuristic operator which utilises problem-specific knowledge is incorporated into the standard genetic algorithm approach and is capable of obtaining high-quality solutions for problems of various characteristics.

Abstract: In this paper we present a heuristic based upon genetic algorithms for the multidimensional knapsack problem. A heuristic operator which utilises problem-specific knowledge is incorporated into the standard genetic algorithm approach. Computational results show that the genetic algorithm heuristic is capable of obtaining high-quality solutions for problems of various characteristics, whilst requiring only a modest amount of computational effort. Computational results also show that the genetic algorithm heuristic gives superior quality solutions to a number of other heuristics.

822 citations