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

Roman Kolpakov,Mikhail Posypkin

doi:10.1515/comp-2020-0212

Abstract

Abstract In 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.

Highlights

In 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
We study the asymptotic behavior of the complexity of the frontal scheme for two special cases of the subset sum problem
We consider one of the possible parallel realizations of a variant of Branch-and-Bound method for solving the subset sum problem which is a particular case of knapsack problem [1]

Summary

Introduction

We consider one of the possible parallel realizations of a variant of Branch-and-Bound method for solving the subset sum problem which is a particular case of knapsack problem [1]. Theoretical bounds on the complexity of parallel solving of SSP are obtained in [7,8,9,10,11] for dynamic programming method and in [12] for two-list method. Empirical parallel algorithms of solving SSP by dynamic programming. Empirical parallelizations of two-list subproblem (2) is called optimal if for any other feasible somethod of solving of SSP for GPU and hybrid CPU/GPU lutiony of this subproblem the inequality f ( ̃y) ≤ f ( ̃x) holds computational models are proposed in [15,16,17].

Preliminaries

Branch-and-Bound algorithm

The case for big values of k

The case for small values of k

Findings

Conclusion