A Multi-Branch-and-Bound Binary Parallel Algorithm to Solve the Knapsack Problem 0–1 in a Multicore Cluster

Abstract

Featured Application: An uncorrelated instance is equivalent to solving any problem where the benefit is independent of the weight. A weakly correlated instance has a high correlation between the benefit and the weight of each element. Typically, the benefit differs from the weight by a small percentage. Such instances are the most practical in administration, such as with a return on an investment, which is generally proportional to the sum of the amount invested. Abstract: This paper presents a process that is based on sets of parts, where elements are fixed and removed to form different binary branch-and-bound (BB) trees, which in turn are used to build a parallel algorithm called "multi-BB". These sequential and parallel algorithms calculate the exact solution for the 0-1 knapsack problem. The sequential algorithm solves the instances published by other researchers (and the proposals by Pisinger) to solve the not-so-complex (uncorrelated) class and some problems of the medium-complex (weakly correlated) class. The parallel algorithm solves the problems that cannot be solved with the sequential algorithm of the weakly correlated class in a cluster of multicore processors. The multi-branch-and-bound algorithms obtained parallel efficiencies of approximately 75%, but in some cases, it was possible to obtain a superlinear speedup.The characteristic of this problem is that its items cannot be split among themselves and is classified as an NP-hard problem, where NP, indicates non-polynomial behavior problems. For certain sizes of the instances of these problems, it is possible to calculate their optimal solution by means of algorithms and parallel computers. Therefore, a parallel BB (Branch and Bound) algorithm is proposed to calculate the exact solution of KP 0-1 for instances of medium and low complexity (weak and uncorrelated [2], respectively).The novelty of this work is that using a different approach to a binary tree (from a formulation of a set of parts), we propose generating several different trees to find the optimal solution [4-10]. Each of these trees represents a decision tree that is generated by fixing or removing elements, so the roots of these trees are located in different search spaces with respect to the initial tree. Each binary tree forms new search spaces, and consequently there are a greater number of feasible solutions available. Because each decision tree is independent of the others, each of them is assigned in a processing unit for execution, and therefore it is possible to calculate the optimal solution more quickly and thus reduce the computation time, sometimes determining superlinear speedup as well as solving the instances that cannot be resolved with a sequential approach.The factors involved in efficient parallel implementation are diverse: those corresponding to the algorithm and those inherent in the use of parallel computers. It is important to consider algorithms that are efficiently parallelizable, i.e., algorithms whose execution times are polylogarithmic and use a number of...

Optimal solving of a binary knapsack problem on a D-Wave quantum machine and its implementation in production systems

“…polynomial with respect to the knapsack payload, are successively developed and published. An overview of the methods and algorithms are presented, among others, in the following works: Coniglio et al (2021), Zavala-Diaz et al (2019Rizk-Allah and Hassanien (2018) and Shen et al (2019). Many construction algorithms have also been published, mainly based on the greedy method.…”

“…The possibility of multi-processor computing has resulted in a significant shift in the size of instances that can be solved exactly in an acceptable time. In addition to the exact and approximation algorithms that have been known for years (Zavala-Diaz et al, 2019;Vu & Derbel, 2016;Vasilchikov, 2018), parallel versions of algorithms inspired by nature have also been published (He et al, 2024), as well as: neural networks, artificial intelligence (Zhang, 2011;Ji et al, 2017) and learning techniques (Refaei et al, 2020).…”

Modified Election Algorithm in Accelerating the Performance of Hopfield Neural Network for Random kSatisfiability

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