Complexity indices for the multidimensional knapsack problem

Solving the bi-objective multi-dimensional knapsack problem exploiting the concept of core

We study the multidimensional knapsack problem, present some theoretical and empirical results about its structure, and evaluate different integer linear programming (ILP)-based, metaheuristic, and collaborative approaches for it. We start by considering the distances between optimal solutions to the LP relaxation and the original problem and then introduce a new core concept for the multidimensional knapsack problem (MKP), which we study extensively. The empirical analysis is then used to develop new concepts for solving the MKP using ILP-based and memetic algorithms. Different collaborative combinations of the presented methods are discussed and evaluated. Further computational experiments with longer run times are also performed to compare the solutions of our approaches to the best-known solutions of another so-far leading approach for common MKP benchmark instances. The extensive computational experiments show the effectiveness of the proposed methods, which yield highly competitive results in significantly shorter run times than do previously described approaches.

Knapsack problems are among the most common problems in literature tackled with evolutionary algorithms (EA). Their major advantage lies in the fact that they are relatively simple to implement while they allow generalizations for a wide range of real world problems. The multi-dimensional knapsack problem (MKP), which belongs to the class of NP-complete combinatorial optimization problems, is one of the variations of the knapsack problem. The MKP has a wide range of real world applications such as cargo loading, selecting projects to fund, budget management, cutting stock, etc. The MKP has been studied quite extensively in the EA community. Due to the constrained nature of the problem, constraint handling techniques gain great importance in the performance of the proposed EA approaches. In this study, the applicability of a generational EA that uses a penalty-based constraint handling technique and a gene locus based, asymmetric, adaptive mutation scheme is explored for the MKP. The effects of the parameters of the explored approach is determined through tests. Further experiments, using large MKP instances from commonly used benchmarks available through the Internet are performed. Comparison tables are given for the performance of the explored approach and other good performing EAs found in literature for the MKP. Results show that performance improves greatly when compared with other penalty-based techniques, but the explored approach is still not the best performer among all. However, unlike the explored technique, the EAs using the other constraint handling techniques require a great amount of extra computational effort and need heuristic information specific to the optimization problem. Based on these observations, and the fact that the performance difference between the explored scheme and the better performers is not too high, research on improving the explored approach is still in progress.

The pigeon-inspired optimization algorithm is a category of a newly proposed swarm intelligence-based algorithm that belongs to the population-based solution technique. The MKP is a class of complex optimization problems that have many practical applications in the fields of engineering and sciences. Due to the practical applications of MKP, numerous algorithmic-based methods like local search and population-based search algorithms have been proposed to solve the MKP in the past few decades. This paper proposes a modified binary pigeon-inspired optimization algorithm named (Modified-BPIO) for the 0 - 1 multidimensional knapsack problem (MKP). The utilization of the binary pigeon-inspired optimization (BPIO) for solving the multidimensional knapsack problem came with huge success. However, it can be observed that the BPIO converges prematurely due to lost diversity during the search activities. Given the above, the crossover operator is integrated with the landmark component of the BPIO to improve the diversity of the solution space. The MKP benchmarks from the Operations Research (OR) library are utilized to test the performance of the proposed binary method. Experimentally, it is concluded that the proposed Modified-BPIO has a better performance when compared with the BPIO and existing state-of-the-arts that worked on the same MKP benchmarks.

This paper proposes a novel approach for the multidimensional knapsack problem (MDKP) using differential evolution. Firstly, the principle and the pseudo-code of binary differential evolution with hybrid encoding (HBDE) are presented. On the basis of the existing repair operator 2 (RO2), an improved repair operator 3 (RO3) for handling the infeasible solutions of MDKP is developed. Then, combine HBDE with RO3, an efficient algorithm (HBDE-RO3) for MDKP is proposed. Finally, the experiment results of the 138 well-known MDKP benchmarks show that RO3 is advantageous to deal with the infeasible solutions than RO2, and the proposed algorithm HBDE-RO3 has superior performance for solving MDKP than the state-of-the-art algorithms.

There has been recent extensive research on integrating cellular networks and ad hoc networks to overcome the limitations of cellular networks. Although several schemes have been proposed to use such hybrid networks to improve the performance of individual multicast groups, they do not address quality of service (QoS) issues when multiple groups are present. This paper, on the other hand, considers an interesting scenario of hybrid networks when an ad hoc network cannot accommodate all the groups and a base station has to select a subset of groups to optimize its bandwidth savings and maximize the utilization of the ad hoc network while providing QoS support for multicast users. In this paper, a network model for multicast admission control that takes wireless interference into account is developed, the group selection problem is formulated as a multidimensional knapsack problem, and an integer linear programming (ILP) formulation and a polynomial-time dynamic algorithm are proposed. A distributed implementation of the dynamic algorithm in real systems is also examined. Simulation studies demonstrate that the dynamic algorithm is able to achieve very competitive performance under various conditions, in comparison with the optimal solution computed by the ILP approach

Binary metaheuristic algorithms for 0–1 knapsack problems: Performance analysis, hybrid variants, and real-world application

This paper introduces solutions to deal with the Multidimensional Knapsack Problem (MKP), which is a NP-hard combinatorial optimisation problem. Two hybrid heuristics based on Genetic Algorithms (GA) are proposed: the Memetic Search Algorithm (MSA) and the Genetic Algorithm Guided by Pretreatment information (GAGP). MSA combines sequentially GA with the Stochastic Local Search-Simulated Annealing algorithm (SLSA). GAGP is composed of two steps, in the first, a ratio-based greedy algorithm extracts useful information and the core concept is utilised to decompose items according to their ratios; In the second, these information are integrated to the operators of a GA allowing to reach the best solutions faster. An operator is added to the GA to dynamically update the ratio values of the items. Two groups of data were used to examine the proposed approaches. A group of simple instances of MKP has been used to examine MSA and a group of complex MKP has been used to examine GAGP. The obtained results indicate that MSA and GAGP have the capability to give solutions of high quality.

Local and global lifted cover inequalities for the 0–1 multidimensional knapsack problem

A two-phase tabu-evolutionary algorithm for the 0–1 multidimensional knapsack problem

The multi-dimensional knapsack problem (MDKP) is a well-known NP-hard problem in combinatorial optimization. As it has various real-life applications, the MDKP has been intensively studied in the literature. On the other hand, far too little attention has been paid to the multi-objective version of the MDKP. In this chapter, we consider the bi-objective multi-dimensional knapsack problem (BOMDKP). We propose a Binary Genetic Algorithm (BGA) with an external archive for the problem. Our proposed BGA algorithm also employs a binary local search. The non-dominated solution sets are obtained for various bi-objective benchmark instances with 100, 250, 500 and 750 items, by employing the proposed BGA. Then, the performance of the BGA is compared with other multi-objective algorithms from the literature, i.e., MOEA/D and MOFPA. Furthermore, it is observed that the Pareto-optimal solution set provided by Zitzler and Laumans for 500 items and 2 knapsacks includes 30 dominated solutions. Also, the Pareto-optimal solutions for the scenario with 750 items are not reported in Zitzler and Thiele [43]. Hence, the true Pareto-optimal solution sets are found for all benchmark problem instances using Improved Augmented Epsilon Constraint (AUGMECON2) method. The non-dominated solution sets of the BGA, MOEA/D and MOFPA are compared with the Pareto-optimal solution sets for all test instances. The computational results indicate that the proposed BGA is more effective to solve the BOMDKP than the best-performing algorithms from the literature.

In this paper, we have proposed a novel algorithm based on Ant Colony Optimization (ACO) for finding near-optimal solutions for the Multi-dimensional Multi-choice Knapsack Problem (MMKP). MMKP is a discrete optimization problem, which is a variant of the classical 0-1 Knapsack Problem and is also an NP-hard problem. Due to its high computational complexity, exact solutions of MMKP are not suitable for most real-time decision-making applications e.g. QoS and Admission Control for Adaptive Multimedia Systems, Service Level Agreement (SLA) etc. Although ACO algorithms are known to have scalability and slow convergence issues, here we have augmented the traditional ACO algorithm with a unique random local search, which not only produces near-optimal solutions but also greatly enhances convergence speed. A comparative analysis with other state-of-the-art heuristic algorithms based on public MMKP dataset shows that, in all cases our approaches outperform others. We have also shown that our algorithms find near optimal (within 3% of the optimal value) solutions within milliseconds, which makes our approach very attractive for large scale real time systems.

Empirical orthogonal constraint generation for Multidimensional 0/1 Knapsack Problems

Finding and exploring promising search space for The 0–1 Multidimensional Knapsack Problem

We study stochastic variants of Packing Integer Programs (PIP) --- the problems of finding a maximum-value 0/1 vector x satisfying Ax ≤ b, with A and b nonnegative. Many combinatorial problems belong to this broad class, including the knapsack problem, maximum clique, stable set, matching, hypergraph matching (a.k.a. set packing), b-matching, and others. PIP can also be seen as a multidimensional knapsack problem where we wish to pack a maximum-value collection of items with vector-valued sizes. In our stochastic setting, the vector-valued sizes of each item is known to us apriori only as a probability distribution, and the size of an item is instantiated once we commit to including the item in our solution.Following the framework of [3], we consider both adaptive and non-adaptive policies for solving such problems, adaptive policies having the flexibility of being able to make decisions based on the instantiated sizes of items already included in the solution. We investigate the adaptivity gap for these problems: the maximum ratio between the expected values achieved by optimal adaptive and non-adaptive policies. We show tight bounds on the adaptivity gap for set packing and b-matching, and we also show how to find efficiently non-adaptive policies approximating the adaptive optimum. For instance, we can approximate the adaptive optimum for stochastic set packing to within O(d1/2), which is not only optimal with respect to the adaptivity gap, but it is also the best known approximation factor in the deterministic case. It is known that there is no polynomial-time d1/2-e approximation for set packing, unless NP = ZPP. Similarly, for b-matching, we obtain algorithmically a tight bound on the adaptivity gap of O(λ) where λ satisfies Σ λbj+1 = 1.For general Stochastic Packing, we prove that a simple greedy algorithm provides an O(d)-approximation to the adaptive optimum. For A ∈ [0, 1]dxn, we provide an O(λ) approximation where Σ 1/λbj = 1. (For b = (B, B,..., B), we get λ = d1/B.) We also improve the hardness results for deterministic PIP: in the general case, we prove that a polynomial-time d1-e-approximation algorithm would imply NP = ZPP. In the special case when A ∈ [0,1]dxn and b = (B,B,...,B), we show that a d1/B-∈-approximation would imply NP = ZPP. Finally, we prove that it is PSPACE-hard to find the optimal adaptive policy for Stochastic Packing in any fixed dimension d ≥ 2.