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.
Highlights
The KP 0–1 has a wide variety of applications in everyday life, including production planning, financial modeling, project selection, the allocation of data in distributed systems, and facility capacity planning [1].Due to the diversity of real problems, it is necessary to consider them through problem models, such as those proposed by Reference [2]
Shared knowledge: this consists of the list or lists of the subproblems or nodes that are stored for later calculation and divided in two (the type of storage of the subproblems and how that list is updated); Knowledge use: This is divided in two and is updated immediately; Division of work: This is formed by the active processes
The following conclusions could be made: It is possible to make binary branch-and-bound trees by fixing and removing subset elements to build a parallel algorithm, which calculates the optimal solution of the 0–1 knapsack problem
Summary
The KP 0–1 (knapsack problem 0–1) has a wide variety of applications in everyday life, including production planning, financial modeling, project selection, the allocation of data in distributed systems, and facility capacity planning [1]. The weakly correlated problem model offers many practical applications, such as capital budgeting, project selection, resource allocation, cutting stock, and investment decision-making [3] In this way, if an algorithm performs well in solving these instances, it is likely to solve a problem in everyday life. 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 [4,5,6,7,8,9,10] 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.
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