Exact and greedy solutions of the knapsack problem: the ratio of values of objective functions

Ratios δ of the values of objective functions of optimal Boolean (or integer) to the values of greedy solutions for the knapsack problem are considered. The relationship of the parameter δ with the ratio Δ of the values of objective functions for the optimal solution of linear relaxation to the values of optimal integer solution was found. Two-sided estimates for δ and Δ were obtained. A computational experiment was conducted to investigate the ratio of δ of problems of one- and two-dimensional knapsack problems with Boolean variables. A hypothesis on asymptotic behavior of the ratio δ with growth of the number of problem variables was formulated.     

A computational study on the quadratic knapsack problem with multiple constraints

The quadratic knapsack problem (QKP) has been the subject of considerable research in recent years. Despite notable advances in special purpose solution methodologies for QKP, this problem class remains very difficult to solve. With the exception of special cases, the state-of-the-art is limited to addressing problems of a few hundred variables and a single knapsack constraint.In this paper we provide a comparison of quadratic and linear representations of QKP based on test problems with multiple knapsack constraints and up to eight hundred variables. For the linear representations, three standard linearizations are investigated. Both the quadratic and linear models are solved by standard branch-and-cut optimizers available via CPLEX. Our results show that the linear models perform well on small problem instances but for larger problems the quadratic model outperforms the linear models tested both in terms of solution quality and solution time by a wide margin. Moreover, our results demonstrate that QKP instances larger than those previously addressed in the literature as well as instances with multiple constraints can be successfully and efficiently solved by branch and cut methodologies.     

Reoptimization in Lagrangian methods for the 0-1 quadratic knapsack problem

The 0-1 quadratic knapsack problem consists of maximizing a quadratic objective function subject to a linear capacity constraint. To exactly solve large instances of this problem with a tree search algorithm (e.g., a branch and bound method), the knowledge of good lower and upper bounds is crucial for pruning the tree but also for fixing as many variables as possible in a preprocessing phase. The upper bounds used in the best known exact approaches are based on Lagrangian relaxation and decomposition. It appears that the computation of these Lagrangian dual bounds involves the resolution of numerous 0-1 linear knapsack subproblems. Thus, taking this huge number of resolutions into account, we propose to embed reoptimization techniques for improving the efficiency of the preprocessing phase of the 0-1 quadratic knapsack resolution. Namely, reoptimization is introduced to accelerate each independent sequence of 0-1 linear knapsack problems induced by the Lagrangian relaxation as well as the Lagrangian decomposition. Numerous numerical experiments validate the relevance of our approach.     

Labeling algorithms for multiple objective integer knapsack problems

The paper presents a generic labeling algorithm for finding all nondominated outcomes of the multiple objective integer knapsack problem (MOIKP). The algorithm is based on solving the multiple objective shortest path problem on an underlying network. Algorithms for constructing four network models, all representing the MOIKP, are also presented. Each network is composed of layers and each network algorithm, working forward layer by layer, identifies the set of all permanent nondominated labels for each layer. The effectiveness of the algorithms is supported with numerical results obtained for randomly generated problems for up to seven objectives while exact algorithms reported in the literature solve the multiple objective binary knapsack problem with up to three objectives. Extensions of the approach to other classes of problems including binary variables, bounded variables, multiple constraints, and time-dependent objective functions are possible.     

Problem reduction heuristic for the 0-1 multidimensional knapsack problem

This paper introduces new problem-size reduction heuristics for the multidimensional knapsack problem. These heuristics are based on solving a relaxed version of the problem, using the dual variables to formulate a Lagrangian relaxation of the original problem, and then solving an estimated core problem to achieve a heuristic solution to the original problem. We demonstrate the performance of these heuristics as compared to legacy heuristics and two other problem reduction heuristics for the multi-dimensional knapsack problem. We discuss problems with existing test problems and discuss the use of an improved test problem generation approach. We use a competitive test to highlight the performance of our heuristics versus the legacy heuristic approaches. We also introduce the concept of computational versus competitive problem test data sets as a means to focus the empirical analysis of heuristic performance.     

A variable-grouping based genetic algorithm for large-scale integer programming

To effectively reduce the dimensionality of search space, this paper proposes a variable-grouping based genetic algorithm (VGGA) for large-scale integer programming problems (IPs). The VGGA first groups IP’s decision variables based on the optimal solution to the IP’s continuous relaxation problem, and then applies a standard genetic algorithm (GA) to the subproblem for each group of variables. We compare the VGGA with the standard GA and GAs based on even variable-grouping by applying them to solve randomly generated convex quadratic knapsack problems and integer knapsack problems. Numerical results suggest that the VGGA is superior to the standard GA and GAs based on even variable-grouping both on computation time and solution quality.     

Genetic algorithms with decomposition procedures for multidimensional 0-1 knapsack problems with block angular structures

This paper presents a detailed treatment of genetic algorithms with decomposition procedures as developed for large scale multidimensional 0-1 knapsack problems with block angular structures. Through the introduction of a triple string representation and the corresponding decoding algorithm, it is shown that a potential solution satisfying not only block constraints but also coupling constraints can be obtained for each individual. Then genetic algorithms with decomposition procedures are presented as an approximate solution method for multidimensional 0-1 knapsack problems with block angular structures. Many computational experiments on numerical examples with 30, 50, 70, 100, 150, 200, 300, 500, and 1000 variables demonstrate the feasibility and efficiency of the proposed method.     

A new exact algorithm for concave knapsack problems with integer variables

Concave knapsack problems with integer variables have many applications in real life, and they are NP-hard. In this paper, an exact and efficient algorithm is presented for concave knapsack problems. The algorithm combines the contour cut with a special cut to improve the lower bound and reduce the duality gap gradually in the iterative process. The lower bound of the problem is obtained by solving a linear underestimation problem. A special cut is performed by exploiting the structures of the objective function and the feasible region of the primal problem. The optimal solution can be found in a finite number of iterations, and numerical experiments are also reported for two different types of concave objective functions. The computational results show the algorithm is efficient.     

Network optimization and problems with coupling variables

Network optimization problems are considered. Their statements include numerous variables and equipments. Decomposition methods are used for their solution. In specific situations, intermediate problems solved by the algorithms have the form of the knapsack problem. In the case when the constraints have a staircase structure, an efficient algorithm can be constructed. The procedure for the sequential recalculation of the coefficients of the objective function in this algorithm can be applied to other problems with unimodular matrices, in particular, to various transportation problems.     

Bi-dimensional knapsack problems with one soft constraint

In this paper, we consider bi-dimensional knapsack problems with a soft constraint, i.e., a constraint for which the right-hand side is not precisely fixed or uncertain. We reformulate these problems as bi-objective knapsack problems, where the soft constraint is relaxed and interpreted as an additional objective function. In this way, a sensitivity analysis for the bi-dimensional knapsack problem can be performed: The trade-off between constraint satisfaction, on the one hand, and the original objective value, on the other hand, can be analyzed. It is shown that a dynamic programming based solution approach for the bi-objective knapsack problem can be adapted in such a way that a representation of the nondominated set is obtained at moderate extra cost. In this context, we are particularly interested in representations of that part of the nondominated set that is in a certain sense close to the constrained optimum in the objective space. We discuss strategies for bound computations and for handling negative cost coefficients, which occur through the transformation. Numerical results comparing the bi-dimensional and bi-objective approaches are presented.     

Determining the K-best solutions of knapsack problems

It is well-known that knapsack problems arise as subproblems of a number of large-scale integer optimization problems. In order to solve these large problems, it is necessary to solve the subproblems efficiently, and for many of them it can be useful to determine the K-best solutions. In this paper, a branch-and-bound method for the unbounded knapsack problem described in the literature is extended to determine the K-best solutions of unbounded and bounded knapsack problems. We show that the proposed extension determines exactly the K-best solutions and we solve important classical instances using high values of K.     

Estimating the computational complexity of one variant of parallel realization of the branch-and-bound method for the knapsack problem

One of the possible realizations of the branch-and-bound method on multiprocessor systems with distributed memory, the front-end algorithm is addressed. The complexity of the front-end algorithm is studied for a family of Boolean knapsack problems with one constraint under the assumption that the number of processors is not limited. Formulas for the order of complexity growth with an increase in dimension of the problems from the addressed family are obtained for the front-end algorithm. The asymptotic acceleration behavior and efficiency of resource use with an increase in the number of variables are studied.     

On complexity of optimal recombination for binary representations of solutions

We consider the optimization problem of finding the best possible offspring as a result of a recombination operator in an evolutionary algorithm, given two parent solutions. The optimal recombination is studied in the case where a vector of binary variables is used as a solution encoding. By means of efficient reductions of the optimal recombination problems (ORPs) we show the polynomial solvability of the ORPs for the maximum weight set packing problem, the minimum weight set partition problem, and for linear Boolean programming problems with at most two variables per inequality, and some other problems. We also identify several NP-hard cases of optimal recombination: the Boolean linear programming problems with three variables per inequality, the knapsack, the set covering, the p-median, and some other problems.     

Shuffled frog leaping algorithm and its application to 0/1 knapsack problem

This paper proposes a modified discrete shuffled frog leaping algorithm (MDSFL) to solve 01 knapsack problems. The proposed algorithm includes two important operations: the local search of the ‘particle swarm optimization’ technique; and the competitiveness mixing of information of the ‘shuffled complex evolution’ technique. Different types of knapsack problem instances are generated to test the convergence property of MDSFLA and the result shows that it is very effective in solving small to medium sized knapsack problems. Further, computational experiments with a set of large-scale instances show that MDSFL can be an efficient alternative for solving tightly constrained 01 knapsack problems.     

Gradient elements of the knapsack polytope

The knapsack problem is a classical NP-hard problem of special interest in combinatorial mathematics and complexity theory. Particularly interesting are the properties of the knapsack problem domain. In this paper we study the set of gradient elements of the knapsack polytope. They are related with the well-known greedy algorithm for finding approximate solutions to the optimization knapsack problem and constitute an important subset of the extremal elements (vertices) of the knapsack polytope. We obtain a tight bound for the number of distinct gradient elements, which is an exponential function of the problem dimension. We also use the greedy-algorithmic approach and the concept of gradient elements to devise a polynomial algorithm for a large subclass of a linear Diophantine problem which is a variant of the knapsack problem. Some optimization issues related to the problems considered are discussed as well. Received: September 1999 / Accepted: December 2000     

An improved monkey algorithm for a 0-1 knapsack problem

The 0-1 knapsack problem is a classic combinational optimization problem. However, many exiting algorithms have low precision and easily fall into local optimal solutions to solve the 0-1 knapsack problem. In order to overcome these problems, this paper proposes a binary version of the monkey algorithm where the greedy algorithm is used to strengthen the local search ability, the somersault process is modified to avoid falling into local optimal solutions, and the cooperation process is adopted to speed up the convergence rate of the algorithm. To validate the efficiency of the proposed algorithm, experiments are carried out with various data instances of 0-1 knapsack problems and the results are compared with those of five metaheuristic algorithms.     

Generalized quadratic multiple knapsack problem and two solution approaches

The Quadratic Knapsack Problem (QKP) is one of the well-known combinatorial optimization problems. If more than one knapsack exists, then the problem is called a Quadratic Multiple Knapsack Problem (QMKP). Recently, knapsack problems with setups have been considered in the literature. In these studies, when an item is assigned to a knapsack, its setup cost for the class also has to be accounted for in the knapsack. In this study, the QMKP with setups is generalized taking into account the setup constraint, assignment conditions and the knapsack preferences of the items. The developed model is called Generalized Quadratic Multiple Knapsack Problem (G-QMKP). Since the G-QMKP is an NP-hard problem, two different meta-heuristic solution approaches are offered for solving the G-QMKP. The first is a genetic algorithm (GA), and the second is a hybrid solution approach which combines a feasible value based modified subgradient (F-MSG) algorithm and GA. The performances of the proposed solution approaches are shown by using randomly generated test instances. In addition, a case study is realized in a plastic injection molding manufacturing company. It is shown that the proposed hybrid solution approach can be successfully used for assigning jobs to machines in production with plastic injection, and good solutions can be obtained in a reasonable time for a large scale real-life problem.     

混沌优化算法及其在组合优化问题中的应用

混沌优化方法(COA)是针对数值优化问题提出的，在解决数值优化问题上具有一定的普遍性，能够很快地搜索到全局最优解，而利用COA解决组合优化问题存在一定的难度，该文提出了混沌优化算法解决组合优化问题的方法，该方法先产生组合优化问题的初始解，再利用混沌变量产生新解或对原解进行混沌扰动，产生新解，然后在解空间中进行最优搜索。将该方法应用到2个典型的组合优化问题(TSP问题，0/1背包问题)的求解中，仿真实验表明了该方法的有效性。     

基于0-1背包问题的两种算法

0-1背包问题是计算机科学中一个经典问题,0-1背包问题是一个最优化问题。因其结构简单,可扩展性强,可作为其他问题的子问题,因此通过对其研究可以解决更为复杂的优化问题。本文概述了两种求解0-1背包问题的算法设计方法,并对两种算法进行了分析和比较。     

非超递增序列背包加密算法研究

为了提高背包加密体制的安全性,对基于超递增序列的背包加密算法进行了分析,指出了利用非超递增序列构造背包所存在的难题,提出一种无冲突非超递增序列的构造方法,并给出严格的证明。依据该方法提出了一种基于无冲突非超递增序列的背包公钥加密算法,有效地避免了利用非超递增序列构造背包的过程中出现的难题。理论分析和仿真实验结果表明,该算法具有高的安全性能,在抵抗Shamir攻击和低密度攻击方面都具有良好的性能。