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

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.

On the best choice of a branching variable in the subset sum problem

2018 ◽  
Vol 28 (1) ◽  
pp. 29-34 ◽  
Author(s):  
Roman M. Kolpakov ◽  
Mikhail A. Posypkin
Keyword(s):  
Computational Complexity ◽  
Branch And Bound ◽  
Branch And Bound Method ◽  
Subset Sum Problem ◽  
Variables Selection ◽  
Subset Sum ◽  
The Relationship ◽  
The Way

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.

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

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.

scholarly journals Branch and Bound Method to Solve Multi Objectives Function

2016 ◽  
Vol 12 (3) ◽  
pp. 5964-5974
Author(s):  
Tahani Jabbar Kahribt ◽  
Mohammed Kadhim Al- Zuwaini
Keyword(s):  
Lower Bounds ◽  
Branch And Bound ◽  
Single Machine ◽  
Upper Bound ◽  
Completion Time ◽  
Heuristic Method ◽  
Total Weighted Completion Time ◽  
Branch And Bound Method ◽  
Special Cases ◽  
Weighted Completion Time

This paper  presents  a  branch  and  bound  algorithm  for  sequencing  a  set  of  n independent  jobs  on  a single  machine  to  minimize sum of the discounted total weighted completion time and maximum lateness,  this problems is NP-hard. Two lower bounds were proposed and heuristic method to get an upper bound. Some special cases were  proved and some dominance rules were suggested and proved, the problem solved with up to 50 jobs.

scholarly journals Linear Integer Optimization Model for Two-Stage Guillotine Cutting Stock Problem Using Branch and Bound Method in the Garment Industry

2020 ◽  
Vol 1 (1) ◽  
pp. 40-52
Author(s):  
Eman Lesmana ◽  
Julita Nahar ◽  
Annisa D. P
Keyword(s):  
Branch And Bound ◽  
Optimal Solution ◽  
Garment Industry ◽  
Cutting Stock Problem ◽  
Cutting Stock ◽  
Branch And Bound Method ◽  
Integer Optimization ◽  
Two Stage ◽  
Stock Cutting ◽  
Guillotine Cutting

This paper discusses the Two-Stage Guillotine Cutting Stock Problem (2GCSP) in the garment industry, namely how to determine the two-stage guillotine pattern that is used to cut fabric stocks into several certain size t-shirt materials that are produced based on the demand for each size of the shirt. 2GCSP is modeled in the form of Linear Integer Optimization and finding solutions using the Branch and Bound method. In this paper also presented a Graphical User Interface with Maple software as an interactive tool to find the best fabric stock cutting patterns. The results show that the optimal solution can be determined by solving numerically using the Branch and Bound method and Maple optimization packages. The solution is shown with an illustration of the pattern and the amount of fabric cut based on the pattern.

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