Two-way Number Partitioning Problem Research Articles

An improved memetic algebraic differential evolution for solving the multidimensional two-way number partitioning problem

In this article, we propose a novel and effective evolutionary algorithm for the challenging combinatorial optimization problem known as Multidimensional Two-Way Number Partitioning Problem (MDTWNPP). Since the MDTWNPP has been proven to be NP-hard, in the recent years, it has been increasingly addressed by means of meta-heuristic approaches. Nevertheless, previous proposals in literature do not make full use of critical problem information that may improve the effectiveness of the search. Here, we bridge this gap by designing an improved Memetic Algebraic Differential Evolution (iMADEB) algorithm that incorporates critical information about the problem. In particular, iMADEB evolves a population of candidate local optimal solutions by adopting three key design concepts: a novel non-redundant bit-string representation which maps population individuals one-to-one to MDTWNPP solutions, a smoother local search operator purposely designed for the MDTWNPP landscapes, and a self-adaptive algebraic differential mutation scheme built on the basis of the Lévy flight concept which automatically regulates the exploration-exploitation trade-off of the search. Computational experiments have been conducted on a widely accepted benchmark suite for the MDTWNPP with a twofold purpose: analyzing the robustness of iMADEB and compare its effectiveness with respect to the state-of-the-art approaches to date for the MDTWNPP. The experimental results provide important indications about iMADEB robustness and, most importantly, clearly show that iMADEB is the new state-of-the-art algorithm for the MDTWNPP.

A mixed-integer linear programming model to solve the Multidimensional Multi-Way Number Partitioning Problem

This paper addresses a mixed-integer linear programming model for solving the Multidimensional Multi-Way Number Partitioning Problem (MDMWNPP), the most general version of the family of Number Partitioning Problems. First, a contextualization concerning the Two-Way Number Partitioning Problem (TWNPP), the Multi-Way Number Partitioning Problem (MWNPP), the Multidimensional Two-Way Number Partitioning Problem (MDTWNPP), and the approached problem are presented. MDMWNPP is a generalization of MDTWNPP for the number k of subsets greater than two. After the proposed model is introduced, some properties are shown, and the main constraints are proved. To validate the consistency of the proposed mathematical formulation, a comparison, using instances from the literature, between the proposed model for k=2 and a model from the literature for MDTWNPP, is performed. The proposed model showed competitive results against this specialized model and, besides, solved instances of the problem for values of k equals to 3 and 4.

A Comparative Analysis of Metaheuristic Approaches for Multidimensional Two-Way Number Partitioning Problem

In this study, a novel usage of four metaheuristic approaches Genetic algorithm (GA), Simulated annealing (SA), migrating bird optimization algorithm (MBO) and clonal selection algorithm (CSA) are applied to multidimensional two-way number partitioning problem (MDTWNPP). MDTWNPP is a classical combinatorial NP-hard optimization problem where a set of vectors have more than one coordinate is partitioned into two subsets. The main objective function of MDTWNPP is to minimize the maximum absolute difference between the sums per coordinate of elements. In order to solve this problem, GA is applied with greedy crossover and mutation operators. SA is improved with dual local search mechanism. MBO is specialized as multiple flock migrating birds optimization algorithms. CSA is applied with problem specific hyper mutation process. Furthermore, all instances are solved using an integer linear programming model which was previously presented in the literature. In the experiments, four metaheuristic approaches and integer linear programming model are used to solve 126 datasets with different sizes and coordinates. As a brief result, the GA and SA approaches designed for this problem outperformed all other heuristics and the integer programming model. Both the performance of GA and SA approaches are in a competitive manner where GA and SA yielded the best solution for 56 and 65 out of 125 datasets, respectively.

GRASP with exterior path-relinking and restricted local search for the multidimensional two-way number partitioning problem

In this work, we tackle multidimensional two-way number partitioning (MDTWNP) problem by combining GRASP with Exterior Path Relinking. In the last few years, the combination of GRASP with path relinking (PR) has emerged as a highly effective tool for finding high-quality solutions for several difficult problems in reasonable computational time. However, in most of the cases, this hybridisation is limited to the variant known as interior PR. Here, we couple GRASP with the “exterior form” of path relinking and perform extensive experimentation to evaluate this variant. In addition, we enhance our GRASP with PR method with a novel local search method specially designed for the MDTWNP problem. Our computational experiments show the superiority of this approach compared with the previous best method for MDTWNP and with alternative methods for this problem that use other forms of PR.

Two metaheuristic approaches for solving multidimensional two-way number partitioning problem

In this paper, we address two metaheuristic approaches, a Variable Neighborhood Search (VNS) and an Electromagnetism-like metaheuristic (EM), on an NP-hard optimization problem: Multi-dimensional Two-way Number Partitioning Problem (MDTWNPP). MDTWNPP is a generalization of a Two-way Number Partitioning Problem (TWNPP), where a set of vectors is partitioned rather than a set of numbers. The simple k-swap neighborhoods allow an effective shaking procedure in the VNS search. The attraction–repulsion mechanism of EM is extended with a scaling procedure, which additionally moves EM points closer to local optima. Both VNS and EM use the same local search procedure based on 1-swap improvements. Computational results were obtained on 210 standard instances. Direct comparison with results from the literature confirm the significance of applying these methods to MDTWNPP.

A memetic algorithm approach for solving the multidimensional multi-way number partitioning problem

In this paper, we describe a generalization of the multidimensional two-way number partitioning problem (MDTWNPP) where a set of vectors has to be partitioned into p sets (parts) such that the sums per every coordinate should be exactly or approximately equal. We will call this generalization the multidimensional multi-way number partitioning problem (MDMWNPP). Also, an efficient memetic algorithm (MA) heuristic is developed to solve the multidimensional multi-way number partitioning problem obtained by combining a genetic algorithm (GA) with a powerful local search (LS) procedure. The performances of our memetic algorithm have been compared with the existing numerical results obtained by CPLEX based on an integer linear programming formulation of the problem. The solution reveals that our proposed methodology performs very well in terms of both quality of the solutions obtained and the computational time compared with the previous method of solving the multidimensional two-way number partitioning problem.

Integer linear programming model for multidimensional two-way number partitioning problem

This paper introduces a multidimensional generalization of the two-way number partitioning problem, as well as an integer linear programming formulation of the problem. There are n binary variables and 2p constraints. The numerical experiments are made on a randomly generated set. In view of its integer linear programming formulation, tests are run in CPLEX. This NP-hard problem uses a set of vectors rather than a set of numbers. The presented experimental results indicate that the generalized problem is much harder than the initial problem.

A complete anytime algorithm for number partitioning

Given a set of numbers, the two-way number partitioning problem is to divide them into two subsets, so that the sum of the numbers in each subset are as nearly equal as possible. The problem is NP-complete. Based on a polynomial-time heuristic due to Karmarkar and Karp, we present a new algorithm, called Complete Karmarkar-Karp (CKK), that optimally solves the general number-partitioning problem, and significantly outperforms the best previously-known algorithms for large problem instances. For numbers with twelve significant digits or less, CKK can optimally solve two-way partitioning problems of arbitrary size in practice. For numbers with greater precision, CKK first returns the Karmarkar-Karp solution, then continues to find better solutions as time allows. Over seven orders of magnitude improvement in solution quality is obtained in less than an hour of running time. Rather than building a single solution one element at a time, or modifying a complete solution, CKK constructs subsolutions, and combines them together in all possible ways. This approach may be effective for other NP-hard problems as well.