The upper bound on the complexity of branch-and-bound with cardinality bound for subset sum problem

2016 ◽  
Author(s):  
Si Thu Thant Sin ◽  
Mikhail Posypkin ◽  
Roman Kolpakov
Keyword(s):  
Branch And Bound ◽  
Upper Bound ◽  
Subset Sum Problem ◽  
Subset Sum

A Mixture of Dynamic Programming and Branch-and-Bound for the Subset-Sum Problem

1984 ◽  
Vol 30 (6) ◽  
pp. 765-771 ◽  
Author(s):  
Silvano Martello ◽  
Paolo Toth
Keyword(s):  
Dynamic Programming ◽  
Branch And Bound ◽  
Subset Sum Problem ◽  
Subset Sum

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.

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.

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 Revisiting Modified Greedy Algorithm for Monotone Submodular Maximization with a Knapsack Constraint

2021 ◽  
Vol 5 (1) ◽  
pp. 1-22
Author(s):  
Jing Tang ◽  
Xueyan Tang ◽  
Andrew Lim ◽  
Kai Han ◽  
Chongshou Li ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Greedy Algorithm ◽  
Branch And Bound ◽  
Upper Bound ◽  
Optimization Problem ◽  
Approximation Factor ◽  
Real World Application ◽  
Efficiency Of Algorithms ◽  
Knapsack Constraint ◽  
Submodular Maximization

Monotone submodular maximization with a knapsack constraint is NP-hard. Various approximation algorithms have been devised to address this optimization problem. In this paper, we revisit the widely known modified greedy algorithm. First, we show that this algorithm can achieve an approximation factor of 0.405, which significantly improves the known factors of 0.357 given by Wolsey and (1-1/e)/2\approx 0.316 given by Khuller et al. More importantly, our analysis closes a gap in Khuller et al.'s proof for the extensively mentioned approximation factor of (1-1/\sqrte )\approx 0.393 in the literature to clarify a long-standing misconception on this issue. Second, we enhance the modified greedy algorithm to derive a data-dependent upper bound on the optimum. We empirically demonstrate the tightness of our upper bound with a real-world application. The bound enables us to obtain a data-dependent ratio typically much higher than 0.405 between the solution value of the modified greedy algorithm and the optimum. It can also be used to significantly improve the efficiency of algorithms such as branch and bound.

An Algorithm Using Modular Arithmetic for the Subset Sum Problem

1990 ◽  
Vol 21 (2) ◽  
pp. 1-10
Author(s):  
Toshiro Tachibana ◽  
Hideo Nakano ◽  
Yoshiro Nakanishi ◽  
Mitsuru Nakao
Keyword(s):  
Modular Arithmetic ◽  
Subset Sum Problem ◽  
Subset Sum
Sign in / Sign up

Export Citation Format

Share Document