Improved convergent heuristics for the 0-1 multidimensional knapsack problem

At the end of the seventies, Soyster et al. (Eur. J. Oper. Res. 2:195–201, 1978) proposed a convergent algorithm that solves a series of small sub-problems generated by exploiting information obtained through a series of linear programming relaxations. This process is suitable for the 0-1 mixed integer programming problems when the number of constraints is relatively smaller when compared to the number of variables. In this paper, we first revisit this algorithm, once again presenting it and some of its properties, including new proofs of finite convergence. This algorithm can, in practice, be used as a heuristic if the number of iterations is limited. We propose some improvements in which dominance properties are emphasized in order to reduce the number of sub problems to be solved optimally. We also add constraints to these sub-problems to speed up the process and integrate adaptive memory. Our results show the efficiency of the proposed improvements for the 0-1 multidimensional knapsack problem.     

Lagrangean heuristics combined with reoptimization for the 0-1 bidimensional knapsack problem

First, this paper deals with lagrangean heuristics for the 0-1 bidimensional knapsack problem. A projected subgradient algorithm is performed for solving a lagrangean dual of the problem, to improve the convergence of the classical subgradient algorithm. Secondly, a local search is introduced to improve the lower bound on the value of the biknapsack produced by lagrangean heuristics. Thirdly, a variable fixing phase is embedded in the process. Finally, the sequence of 0-1 one-dimensional knapsack instances obtained from the algorithm are solved by using reoptimization techniques in order to reduce the total computational time effort. Computational results are presented.     

Dynamic programming algorithms for the bi-objective integer knapsack problem

This paper presents two new dynamic programming (DP) algorithms to find the exact Pareto frontier for the bi-objective integer knapsack problem. First, a property of the traditional DP algorithm for the multi-objective integer knapsack problem is identified. The first algorithm is developed by directly using the property. The second algorithm is a hybrid DP approach using the concept of the bound sets. The property is used in conjunction with the bound sets. Next, the numerical experiments showed that a promising partial solution can be sometimes discarded if the solutions of the linear relaxation for the subproblem associated with the partial solution are directly used to estimate an upper bound set. It means that the upper bound set is underestimated. Then, an extended upper bound set is proposed on the basis of the set of linear relaxation solutions. The efficiency of the hybrid algorithm is improved by tightening the proposed upper bound set. The numerical results obtained from different types of bi-objective instances show the effectiveness of the proposed approach.     

On the 0/1 knapsack polytope

This paper deals with the 0/1 knapsack polytope. In particular, we introduce the class ofweight inequalities. This class of inequalities is needed to describe the knapsack polyhedron when the weights of the items lie in certain intervals. A generalization of weight inequalities yields the so-called “weight-reduction principle” and the class of extended weight inequalities. The latter class of inequalities includes minimal cover and (l,k)-configuration inequalities. The properties of lifted minimal cover inequalities are extended to this general class of inequalities.     

Reoptimizing the 0-1 knapsack problem

In this paper we study the problem where an optimal solution of a knapsack problem on n items is known and a very small number k of new items arrive. The objective is to find an optimal solution of the knapsack problem with n+k items, given an optimal solution on the n items (reoptimization of the knapsack problem). We show that this problem, even in the case k=1, is NP-hard and that, in order to have effective heuristics, it is necessary to consider not only the items included in the previously optimal solution and the new items, but also the discarded items. Then, we design a general algorithm that makes use, for the solution of a subproblem, of an α-approximation algorithm known for the knapsack problem. We prove that this algorithm has a worst-case performance bound of , which is always greater than α, and therefore that this algorithm always outperforms the corresponding α-approximation algorithm applied from scratch on the n+k items. We show that this bound is tight when the classical Ext-Greedy algorithm and the algorithm are used to solve the subproblem. We also show that there exist classes of instances on which the running time of the reoptimization algorithm is smaller than the running time of an equivalent PTAS and FPTAS.     

A reduction dynamic programming algorithm for the bi-objective integer knapsack problem

This paper presents a backward state reduction dynamic programming algorithm for generating the exact Pareto frontier for the bi-objective integer knapsack problem. The algorithm is developed addressing a reduced problem built after applying variable fixing techniques based on the core concept. First, an approximate core is obtained by eliminating dominated items. Second, the items included in the approximate core are subject to the reduction of the upper bounds by applying a set of weighted-sum functions associated with the efficient extreme solutions of the linear relaxation of the multi-objective integer knapsack problem. Third, the items are classified according to the values of their upper bounds; items with zero upper bounds can be eliminated. Finally, the remaining items are used to form a mixed network with different upper bounds. The numerical results obtained from different types of bi-objective instances show the effectiveness of the mixed network and associated dynamic programming algorithm.     

LP based heuristics for the multiple knapsack problem with assignment restrictions

Starting with a problem in wireless telecommunication, we are led to study the multiple knapsack problem with assignment restrictions. This problem is NP-hard. We consider special cases and their computational complexity. We present both randomized and deterministic LP based algorithms, and show both theoretically and computationally their usefulness for large-scale problems.     

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

This paper deals with the bi-objective multi-dimensional knapsack problem. We propose the adaptation of the core concept that is effectively used in single-objective multi-dimensional knapsack problems. The main idea of the core concept is based on the “divide and conquer” principle. Namely, instead of solving one problem with n variables we solve several sub-problems with a fraction of n variables (core variables). The quality of the obtained solution can be adjusted according to the size of the core and there is always a trade off between the solution time and the quality of solution. In the specific study we define the core problem for the multi-objective multi-dimensional knapsack problem. After defining the core we solve the bi-objective integer programming that comprises only the core variables using the Multicriteria Branch and Bound algorithm that can generate the complete Pareto set in small and medium size multi-objective integer programming problems. A small example is used to illustrate the method while computational and economy issues are also discussed. Computational experiments are also presented using available or appropriately modified benchmarks in order to examine the quality of Pareto set approximation with respect to the solution time. Extensions to the general multi-objective case as well as to the computation of the exact solution are also mentioned.     

Heuristics for the 0–1 multidimensional knapsack problem

Two heuristics for the 0–1 multidimensional knapsack problem (MKP) are presented. The first one uses surrogate relaxation, and the relaxed problem is solved via a modified dynamic-programming algorithm. The heuristics provides a feasible solution for (MKP). The second one combines a limited-branch-and-cut-procedure with the previous approach, and tries to improve the bound obtained by exploring some nodes that have been rejected by the modified dynamic-programming algorithm. Computational experiences show that our approaches give better results than the existing heuristics, and thus permit one to obtain a smaller gap between the solution provided and an optimal solution.     

Algorithmic improvements on dynamic programming for the bi-objective {0,1} knapsack problem

This paper presents several methodological and algorithmic improvements over a state-of-the-art dynamic programming algorithm for solving the bi-objective {0,1} knapsack problem. The variants proposed make use of new definitions of lower and upper bounds, which allow a large number of states to be discarded. The computation of these bounds are based on the application of dichotomic search, definition of new bound sets, and bi-objective simplex algorithms to solve the relaxed problem. Although these new techniques are not of a common application for dynamic programming, we show that the best variants tested in this work can lead to an average improvement of 10 to 30 % in CPU-time and significant less memory usage than the original approach in a wide benchmark set of instances, even for the most difficult ones in the literature.     

A new upper bound for the 0-1 quadratic knapsack problem

The 0-1 quadratic knapsack problem (QKP) consists in maximizing a positive quadratic pseudo-Boolean function subject to a linear capacity constraint. We present in this paper a new method, based on Lagrangian decomposition, for computing an upper bound of QKP. We report computational experiments which demonstrate the sharpness of the bound (relative error very often less than 1%) for large size instances (up to 500 variables).     

Solving the 0–1 proportional knapsack problem by sampling

In this paper, we deal with the proportional knapsack problem that is a variation on the ordinary knapsack problem. In the proportional knapsack problem, we look at filling an urn with objects having two characteristics: color and weight. The colors of the objects in the urn should be proportional to the distribution of the colors in the object universe, and the total weight of the objects in the urn should be as close as possible to the capacity of the urn. The formulation of the problem was motivated by a real-life application from the area of finance, called a dollar roll. We show that the proportional knapsack problem is NP-hard, and then, using sampling, develop a heuristic procedure for solving the problem.Partial support from the Fund for the Promotion of Research and from the Alexander Goldberg Memorial Fund at Technion is gratefully acknowledged.     

A fully polynomial approximation algorithm for the 0–1 knapsack problem

A fully polynomial ?-approximation algorithm is developed for the 0–1 knapsack problem. The algorithm uses results of Lawler and Ibarra and Kim. A pseudo-polynomial dynamic programming algorithm is first suggested which solves the problem in O(nb log n) time and O(b) space.     

The 0-1 knapsack problem with fuzzy data

The 0-1 knapsack problem with imprecise profits and imprecise weights of items is considered. The imprecise parameters are modeled as fuzzy intervals. A method of choosing a solution under the uncertainty is proposed and two methods for solving the constructed models are provided.     

An iterative variable-based fixation heuristic for the 0-1 multidimensional knapsack problem

An iterative scheme which is based on a dynamic fixation of the variables is developed to solve the 0-1 multidimensional knapsack problem. Such a scheme has the advantage of generating memory information, which is used on the one hand to choose the variables to fix either permanently or temporarily and on the other hand to construct feasible solutions of the problem. Adaptations of this mechanism are proposed to explore different parts of the search space and to enhance the behaviour of the algorithm. Encouraging results are presented when tested on the correlated instances of the 0-1 multidimensional knapsack problem.     

A core approach to the 0–1 equality knapsack problem

For the 0–1 knapsack problem with equality constraint a partitioning procedure is introduced which focuses on the core of the problem. The purpose of the procedure is to reduce the required preliminary sorting for large problem instances. Computational results are presented for an improved heuristic as well as for a complete (exact) algorithm showing the success of the core approach. Test problems of size up to 15–000 objects are solved within 400–ms on a standard personal computer, that is, within the time that is needed for sorting the profit-weight ratios. The core algorithm reduces the solution times by a factor of up to four for large problem instances.     

On the supermodular knapsack problem

In this paper we introduce binary knapsack problems where the objective function is nonlinear, and investigate their Lagrangean and continuous relaxations. Some of our results generalize previously known theorems concerning linear and quadratic knapsack problems. We investigate in particular the case in which the objective function is supermodular. Under this hypothesis, although the problem remains NP-hard, we show that its Lagrangean dual and its continuous relaxation can be solved in polynomial time. We also comment on the complexity of recognizing supermodular functions. The particular case in which the knapsack constraint is of the cardinality type is also addressed and some properties of its optimal value as a function of the right hand side are derived.Work done while the authors were visiting Rutgers University.     

Approximation algorithms for the m-dimensional 0–1 knapsack problem: Worst-case and probabilistic analyses

We describe a polynomial approximation scheme for an m-constraint 0–1 integer programming problem (m fixed) based on the use of the dual simplex algorithm for linear programming.We also analyse the asymptotic properties of a particular random model.     

The 0–1 knapsack problem with multiple choice constraints

In this paper we consider the 0–1 knapsack problem with multiple choice constraints appended. Such a problem may arise in a capital budgeting context where only one project may be selected from a particular group of projects. Thus the problem is to choose one project from each group such that the budgetary constraint is satisfied and the maximum return is realized. We formulate two branch and bound algorithms which use two different relaxations as the primary bounding relaxations. In addition, theoretical results are given for a simple reduction in the number of variables in the problem.