An asynchronous P system with branch and bound for solving the subset sum problem

Abstract The paper is concerned with estimating the computational complexity of the branch-and-bound method for the subset sum problem. We study the relationship between the way of decomposition of subproblems and the number of the method steps. The standard variant of the branch-and-bound method for the subset sum problem with binary branching is considered: any subproblem is decomposed into two more simple subproblems by assigning values 0 and 1 to a selected branching variable. It is shown that for any set of parameters of the problem the procedure of branching variables selection in the descending order of their weights is optimal.

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Optimality and Complexity Analysis of a Branch-and-Bound Method in Solving Some Instances of the Subset Sum Problem

Open Computer Science ◽

10.1515/comp-2020-0212 ◽

2020 ◽

Vol 11 (1) ◽

pp. 116-126

Author(s):

Roman Kolpakov ◽

Mikhail Posypkin

Keyword(s):

Branch And Bound ◽

Complexity Analysis ◽

Optimal Solution ◽

Branch And Bound Method ◽

Subset Sum Problem ◽

Natural Approach ◽

Second Stage ◽

Special Cases ◽

Subset Sum ◽

Special Case

AbstractIn this paper we study the question of parallelization of a variant of Branch-and-Bound method for solving of the subset sum problem which is a special case of the Boolean knapsack problem. The following natural approach to the solution of this question is considered. At the first stage one of the processors (control processor) performs some number of algorithm steps of solving a given problem with generating some number of subproblems of the problem. In the second stage the generated subproblems are sent to other processors for solving (one subproblem per processor). Processors solve completely the received subproblems and return their solutions to the control processor which chooses the optimal solution of the initial problem from these solutions. For this approach we define formally a model of parallel computing (frontal parallelization scheme) and the notion of complexity of the frontal scheme. We study the asymptotic behavior of the complexity of the frontal scheme for two special cases of the subset sum problem.

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Effective parallelization strategy for the solution of subset sum problems by the branch-and-bound method

Discrete Mathematics and Applications ◽

10.1515/dma-2020-0028 ◽

2020 ◽

Vol 30 (5) ◽

pp. 313-325

Author(s):

Roman M. Kolpakov ◽

Mikhail A. Posypkin

Keyword(s):

Branch And Bound ◽

Branch And Bound Method ◽

Subset Sum Problem ◽

Parallelization Strategy ◽

Subset Sum ◽

Time Optimal ◽

Subset Sum Problems

AbstractAn easily implementable recursive parallelization strategy for solving the subset sum problem by the branch-and-bound method is proposed. Two different frontal and balanced variants of this strategy are compared. On an example of a particular case of the subset sum problem we show that the balanced variant is more effective than the frontal one. Moreover, we show that, for the considered particular case of the subset sum problem, the balanced variant is also time optimal.

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