Computing Partitions with Applications to the Knapsack Problem

Horowitz, Ellis; Sahni, Sartaj

doi:10.1145/321812.321823

Cited by 475 publications

(299 citation statements)

References 3 publications

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“…The algorithm used the best bound selection rule and branching was done on the fractional variable. The large computer memory requirements of this algorithm led to the development of other Branch-and-Bound algorithms by Horowitz and Sahni [7], Nauss [14], Fayard and Plateau [6] and Martello and Toth [11], to name but a few.…”

Section: Historical Notes On Exact Algorithmsmentioning

confidence: 99%

A partitioning scheme for solving the 0-1 knapsack problem

Kruger¹,

Hattingh²

2014

ORiON

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The application of valid inequalities to provide relaxations which can produce tight bounds, is now common practice in Combinatorial Optimisation. This paper attempts to complement current theoretical investigations in this regard. We experimentally search for "valid" equalities which have the potential of strengthening the problem's formulation.Recently, Martello and Toth [13] included cardinality constraints to derive tight upper bounds for the 0-1 Knapsack Problem. Encouraged by their results, we partition the search space by using equality cardinality constraints. Instead of solving the original problem, an equivalent problem, which consists of one or more 0-1 Knapsack Problem with an exact cardinality bound, is solved.By explicitly including a bound on the cardinality, one is able to reduce the size of each subproblem and compute tight upper bounds. Good feasible solutions found along the way are employed to reduce the computational effort by reducing the number of trees searched and the size of the subsequent search trees.We give a brief description of two Lagrangian-based Branch-and-Bound algorithms proposed in Kruger [9] for solving the exact cardinality bounded subproblems and report on results of numerical experiments with a sequential implementation. Implications for and strategies towards parallel implementation are also given.

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Section: Historical Notes On Exact Algorithmsmentioning

confidence: 99%

A partitioning scheme for solving the 0-1 knapsack problem

Kruger¹,

Hattingh²

2014

ORiON

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“…It allows to solve a generic integer knapsack problem on n-elements in timeÕ(2 n/2 ) using a memory of sizeÕ(2 n/4 ). It improves on the birthday algorithm of Horowitz and Sahni [10] that can be applied on such a knapsack. We first recall this basic birthday algorithm, which is based on the rewriting of a knapsack solution as an equality:…”

Section: The Algorithm Of Schroeppel and Shamirmentioning

confidence: 99%

“…For hard knapsacks, the state-of-the-art algorithm is due to Schroeppel and Shamir [21,22] and runs in time O(n · 2 n/2 ) using O(n · 2 n/4 ) bits of memory. This algorithm has the same running time as the basic birthday based algorithm on the knapsack problem introduced by Horowitz and Sahni [10], but much lower memory requirements. To simplify the notation of the complexities in the sequel, we extensively use the soft-Oh notation.…”

Section: Introductionmentioning

confidence: 99%

New Generic Algorithms for Hard Knapsacks

Howgrave-Graham¹,

Joux

2010

Lecture Notes in Computer Science

105

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Abstract. In this paper, we study the complexity of solving hard knapsack problems, i.e., knapsacks with a density close to 1 where latticebased low density attacks are not an option. For such knapsacks, the current state-of-the-art is a 31-year old algorithm by Schroeppel and Shamir which is based on birthday paradox techniques and yields a running time ofÕ(2 n/2 ) for knapsacks of n elements and usesÕ(2 n/4 ) storage. We propose here two new algorithms which improve on this bound, finally lowering the running time down to eitherÕ(2 0.385 n ) orÕ(2 0.3113 n ) under a reasonable heuristic. We also demonstrate the practicality of these algorithms with an implementation.

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“…The (0-1) knapsack dynamic programming approach has been intensively studied and used for a long time for optimal allocation of resources, [2][3][4][5][6][7][8][9] and in particular, as a resource allocation method among competing projects. More recently, the 0-1 knapsack problem has also been used and discussed.…”

Section: Introduction and Related Workmentioning

confidence: 99%

Optimal allocation of limitted resources among discrete risk-reduction options

Todinov

2014

AIR

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This study exposes a critical weakness of the (0-1) knapsack dynamic programming approach, widely used for optimal allocation of resources. The (0-1) knapsack dynamic programming approach could waste resources on insignificant improvements and prevent the more efficient use of the resources to achieve maximum benefit. Despite the numerous extensive studies, this critical shortcoming of the classical formulation has been overlooked. The main reason is that the standard (0-1) knapsack dynamic programming approach has been devised to maximise the benefit derived from items filling a space with no intrinsic value. While this is an appropriate formulation for packing and cargo loading problems, in applications involving capital budgeting, this formulation is deeply flawed. The reason is that budgets do have intrinsic value and their efficient utilisation is just as important as the maximisation of the benefit derived from the budget allocation.Accordingly, a new formulation of the (0-1) knapsack resource allocation model is proposed where the weighted sum of the benefit and the remaining budget is maximised instead of the total benefit. The proposed optimisation model produces solutions superior to both -the standard (0-1) dynamic programming approach and the cost-benefit approach.On the basis of common parallel-series systems, the paper also demonstrates that because of synergistic effects, sets including the same number of identical options could remove different amount of total risk. The existence of synergistic effects does not permit the application of the (0-1) dynamic programming approach. In this case, specific methods for optimal resource allocation should be applied. Accordingly, the paper formulates and proves a theorem stating that the maximum amount of removed total risk from operations and systems with parallel-series logical arrangement is achieved by using preferentially the available budget on improving the reliability of operations/components belonging to the same parallel branch. Improving the reliability of randomly selected operations/components not forming a parallel branch leads to a sub-optimal risk reduction. The theorem is a solid basis for achieving a significant risk reduction for systems and processes with parallel-series logical arrangement.

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Computing Partitions with Applications to the Knapsack Problem

Cited by 475 publications

References 3 publications

A partitioning scheme for solving the 0-1 knapsack problem

A partitioning scheme for solving the 0-1 knapsack problem

New Generic Algorithms for Hard Knapsacks

Optimal allocation of limitted resources among discrete risk-reduction options

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