An exact algorithm for the fixed-charge multiple knapsack problem

We formulate the fixed-charge multiple knapsack problem (FCMKP) as an extension of the multiple knapsack problem (MKP). The Lagrangian relaxation problem is easily solved, and together with a greedy heuristic we obtain a pair of upper and lower bounds quickly. We make use of these bounds in the pegging test to reduce the problem size. We also present a branch-and-bound (B&B) algorithm to solve FCMKP to optimality. This algorithm exploits the Lagrangian upper bound as well as the pegging result for pruning, and at each terminal subproblem solve MKP exactly by invoking MULKNAP code developed by Pisinger [Pisinger, D., 1999. An exact algorithm for large multiple knapsack problems. European Journal of Operational Research 114, 528–541]. As a result, we are able to solve almost all test problems with up to 32,000 items and 50 knapsacks within a few seconds on an ordinary computing environment, although the algorithm remains some weakness for small instances with relatively many knapsacks.     

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.     

Lagrangian relaxation guided problem space search heuristics for generalized assignment problems

We develop and test a heuristic based on Lagrangian relaxation and problem space search to solve the generalized assignment problem (GAP). The heuristic combines the iterative search capability of subgradient optimization used to solve the Lagrangian relaxation of the GAP formulation and the perturbation scheme of problem space search to obtain high-quality solutions to the GAP. We test the heuristic using different upper bound generation routines developed within the overall mechanism. Using the existing problem data sets of various levels of difficulty and sizes, including the challenging largest instances, we observe that the heuristic with a specific version of the upper bound routine works well on most of the benchmark instances known and provides high-quality solutions quickly. An advantage of the approach is its generic nature, simplicity, and implementation flexibility.     

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.     

多背包问题的半定松驰算法

The multiple knapsack problem denoted by MKP (B,S,rn,n) can be defined as follows. A set B of n items and a set S of rn knapsacks are given such that each item j has a profit pi and weight wj,and each knapsack i has a capacity Ci. The goal is to find a subset of items of maximum profit such that they have a feasible packing in the knapsacks. MKP (B,S,m,n) is strongly NP-Complete and no polynomial time approximation algorithm can have an approximation ratio better than 0.5. In the last ten years,semi-definite programming has been empolyed to solve some combinatorial problems successfully. This paper firstly presents a semi-definite relaxation algorithm (MKPS) for MKP (B,S,rn,n). It is proved that MKPS have a approximation ratio better than 0. 5 for a subclass of MKP (B,S,m,n) with n≤100, m≤5 and max^nj=1{wj}/min^mi=1={Ci}≤2/3.     

A two-phase algorithm for the partial accessibility constrained vehicle routing problem

In the partial accessibility constrained vehicle routing problem, a route can be covered by two types of vehicles, i.e. truck or truck + trailer. Some customers are accessible by both vehicle types, whereas others solely by trucks. After introducing an integer programming formulation for the problem, we describe a two-phase heuristic method which extends a classical vehicle routing algorithm. Since it is necessary to solve a combinatorial problem that has some similarities with the generalized assignment problem, we propose an enumerative procedure in which bounds are obtained from a Lagrangian relaxation. The routine provides very encouraging results on a set of test problems.     

An optimization algorithm for a penalized knapsack problem

We study a variation of the knapsack problem in which each item has a profit, a weight and a penalty; the sum of profits of the selected items minus the largest penalty associated with the selected items must be maximized. We present an ILP formulation and an exact optimization algorithm.     

The zero-one knapsack problem with equality constraint

The zero-one knapsack problem is a linear zero-one programming problem with a single inequality constraint. This problem has been extensively studied and many applications and efficient algorithms have been published. In this paper we consider a similar problem, one with an equality instead of the inequality constraint. By replacing the equality by two inequalities one of which is placed in the economic function, a Lagrangean relaxation of the problem is obtained. The relation between the relaxed problem and the original problem is examined and it is shown how the optimal value of the relaxed problem varies with increasing values of the Lagrangean multiplier. Using these results an algorithm for solving the problem is proposed.The paper concludes with a discussion of computational experience.     

Lower bounding and tabu search procedures for the frequency assignment problem with polarization constraints

The problem retained for the ROADEF’2001 international challenge was a Frequency Assignment Problem with polarization constraints (FAPP). This NP-hard problem was proposed by the CELAR of the French Department of Defense, within the context of the CALMA project. Twenty seven competitors took part to this contest, and we present in this paper the contribution of our team that allowed us to be selected as one of the six finalists qualified for the final round of the competition.There is typically no solution satisfying all constraints of the FAPP. For this reason, some electromagnetic compatibility constraints can be progressively relaxed, and the objective is to find a feasible solution with the lowest possible level of relaxation. We have developed a procedure that computes a lower bound on the best possible level of relaxation, as well as two tabu search algorithms for the FAPP, one for the frequency assignment, and one for the polarization assignment.Received: July 2003, Revised: October 2004, AMS classification: 90C27, 90C35, 90C59Alain Hertz: Correspondence to     

Greedy algorithm for the general multidimensional knapsack problem

In this paper, we propose a new greedy-like heuristic method, which is primarily intended for the general MDKP, but proves itself effective also for the 0-1 MDKP. Our heuristic differs from the existing greedy-like heuristics in two aspects. First, existing heuristics rely on each item’s aggregate consumption of resources to make item selection decisions, whereas our heuristic uses the effective capacity, defined as the maximum number of copies of an item that can be accepted if the entire knapsack were to be used for that item alone, as the criterion to make item selection decisions. Second, other methods increment the value of each decision variable only by one unit, whereas our heuristic adds decision variables to the solution in batches and consequently improves computational efficiency significantly for large-scale problems. We demonstrate that the new heuristic significantly improves computational efficiency of the existing methods and generates robust and near-optimal solutions. The new heuristic proves especially efficient for high dimensional knapsack problems with small-to-moderate numbers of decision variables, usually considered as “hard” MDKP and no computationally efficient heuristic is available to treat such problems. Supported in part by the NSF grant DMI 9812994.     

A facet generation and relaxation technique applied to an assignment problem with side constraints

We consider a solution method for combinatorial optimization problems based on a combination of Lagrangean relaxation and constraint generation techniques. The procedure is applied to a constrained assignment problem, where subsets of variables are specified, and variables belonging to the same subset must have the same value. The model can be applied to solve constrained job assignment or classroom assignment problems. The procedure we suggest requires only the solution of classical assignment subproblems. An illustrative numerical example is given.     

A strongly polynomial FPTAS for the symmetric quadratic knapsack problem

The symmetric quadratic knapsack problem (SQKP), which has several applications in machine scheduling, is NP-hard. An approximation scheme for this problem is known to achieve an approximation ratio of (1 + ?) for any ? > 0. To ensure a polynomial time complexity, this approximation scheme needs an input of a lower bound and an upper bound on the optimal objective value, and requires the ratio of the bounds to be bounded by a polynomial in the size of the problem instance. However, such bounds are not mentioned in any previous literature. In this paper, we present the first such bounds and develop a polynomial time algorithm to compute them. The bounds are applied, so that we have obtained for problem (SQKP) a fully polynomial time approximation scheme (FPTAS) that is also strongly polynomial time, in the sense that the running time is bounded by a polynomial only in the number of integers in the problem instance.     

Upper and lower bounding procedures for the minimum caterpillar spanning problem

A spanning caterpillar in a graph is a tree composed by a path such that all vertices not in the path are leaves. In the Minimum Spanning Caterpillar Problem (MSCP) each edge has two costs: a path cost when it belongs to the path and a connection cost when it is incident to a leaf. The goal is to find a spanning caterpillar minimizing the sum of all path and connection costs. In this paper we formulate the as a minimum Steiner arborescence problem. This reduction is the basis for the development of an efficient branch-and-cut algorithm for the MSCP. We als developed a GRASP heuristic to generate primal bounds. Experiments carried out on instances adapted from TSPLIB 2.1 revealed that the exact algorithm is capable to solve to optimality instances with up to 300 vertices in reasonable time. They also showed that our heuristic yields very high quality solutions.     

A dynamic programming based reduction procedure for the multidimensional 0–1 knapsack problem

This paper presents a preprocessing procedure for the 0–1 multidimensional knapsack problem. First, a non-increasing sequence of upper bounds is generated by solving LP-relaxations. Then, a non-decreasing sequence of lower bounds is built using dynamic programming. The comparison of the two sequences allows either to prove that the best feasible solution obtained is optimal, or to fix a subset of variables to their optimal values. In addition, a heuristic solution is obtained. Computational experiments with a set of large-scale instances show the efficiency of our reduction scheme. Particularly, it is shown that our approach allows to reduce the CPU time of a leading commercial software.     

A theoretical and empirical investigation on the Lagrangian capacities of the 0-1 multidimensional knapsack problem

We present a novel Lagrangian method to find good feasible solutions in theoretical and empirical aspects. After investigating the concept of Lagrangian capacity, which is the value of the capacity constraint that Lagrangian relaxation can find an optimal solution, we formally reintroduce Lagrangian capacity suitable to the 0-1 multidimensional knapsack problem and present its new geometric equivalent condition including a new associated property. Based on the property, we propose a new Lagrangian heuristic that finds high-quality feasible solutions of the 0-1 multidimensional knapsack problem. We verify the advantage of the proposed heuristic by experiments. We make comparisons with existing Lagrangian approaches on benchmark data and show that the proposed method performs well on large-scale data.     

A survey for the quadratic assignment problem

The quadratic assignment problem (QAP), one of the most difficult problems in the NP-hard class, models many real-life problems in several areas such as facilities location, parallel and distributed computing, and combinatorial data analysis. Combinatorial optimization problems, such as the traveling salesman problem, maximal clique and graph partitioning can be formulated as a QAP. In this paper, we present some of the most important QAP formulations and classify them according to their mathematical sources. We also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms. We then give a detailed discussion of the progress made in both exact and heuristic solution methods, including those formulated according to metaheuristic strategies. Finally, we analyze the contributions brought about by the study of different approaches.     

A primal-proximal heuristic applied to the French Unit-commitment problem

This paper is devoted to the numerical resolution of unit-commitment problems, with emphasis on the French model optimizing the daily production of electricity. The solution process has two phases. First a Lagrangian relaxation solves the dual to find a lower bound; it also gives a primal relaxed solution. We then propose to use the latter in the second phase, for a heuristic resolution based on a primal proximal algorithm. This second step comes as an alternative to an earlier approach, based on augmented Lagrangian (i.e. a dual proximal algorithm). We illustrate the method with some real-life numerical results. A companion paper is devoted to a theoretical study of the heuristic in the second phase.     

A dual approach for the continuous collapsing knapsack problem

We formulate and solve a dual version of the Continuous Collapsing Knapsack Problem using a geometric approach. Optimality conditions are found and an algorithm is presented. Computational experience shows that this procedure is efficient.