Algorithms for the generalized quadratic assignment problem combining Lagrangean decomposition and the Reformulation-Linearization Technique

In this paper, we propose two exact algorithms for the GQAP (generalized quadratic assignment problem). In this problem, given M facilities and N locations, the facility space requirements, the location available space, the facility installation costs, the flows between facilities, and the distance costs between locations, one must assign each facility to exactly one location so that each location has sufficient space for all facilities assigned to it and the sum of the products of the facility flows by the corresponding distance costs plus the sum of the installation costs is minimized. This problem generalizes the well-known quadratic assignment problem (QAP). Both exact algorithms combine a previously proposed branch-and-bound scheme with a new Lagrangean relaxation procedure over a known RLT (Reformulation-Linearization Technique) formulation. We also apply transformational lower bounding techniques to improve the performance of the new procedure. We report detailed experimental results where 19 out of 21 instances with up to 35 facilities are solved in up to a few days of running time. Six of these instances were open.     

Reducing the elastic generalized assignment problem to the standard generalized assignment problem

The elastic generalized assignment problem (eGAP) is a natural extension of the generalized assignment problem (GAP) where the capacities are not fixed but can be adjusted; this adjustment can be expressed by continuous variables. These variables might be unbounded or restricted by a lower or upper bound, respectively. This paper concerns techniques aiming at reducing several variants of eGAP to GAP, which enables us to employ standard approaches for the GAP. This results in a heuristic, which can be customized in order to provide solutions having an objective value arbitrarily close to the optimal.     

A new algorithm for the assignment problem

We propose a new algorithm for the classical assignment problem. The algorithm resembles in some ways the Hungarian method but differs substantially in other respects. The average computational complexity of an efficient implementation of the algorithm seems to be considerably better than the one of the Hungarian method. In a large number of randomly generated problems the algorithm has consistently outperformed an efficiently coded version of the Hungarian method by a broad margin. The factor of improvement increases with the problem dimensionN and reaches an order of magnitude forN equal to several hundreds.Work supported by Grant NSF ENG-7906332.     

Fast separation for the three-index assignment problem

A critical step in a cutting plane algorithm is separation, i.e., establishing whether a given vector x violates an inequality belonging to a specific class. It is customary to express the time complexity of a separation algorithm in the number of variables n. Here, we argue that a separation algorithm may instead process the vector containing the positive components of x,  denoted as supp(x),  which offers a more compact representation, especially if x is sparse; we also propose to express the time complexity in terms of |supp(x)|. Although several well-known separation algorithms exploit the sparsity of x,  we revisit this idea in order to take sparsity explicitly into account in the time-complexity of separation and also design faster algorithms. We apply this approach to two classes of facet-defining inequalities for the three-index assignment problem, and obtain separation algorithms whose time complexity is linear in |supp(x)| instead of n. We indicate that this can be generalized to the axial k-index assignment problem and we show empirically how the separation algorithms exploiting sparsity improve on existing ones by running them on the largest instances reported in the literature.     

Improving heuristics for the frequency assignment problem

Lower bounds for the frequency assignment problem can be found from maximal cliques and subgraphs related to cliques. In this paper we show that for many types of problem optimal assignments can be found by a process of assigning these subgraphs first, fixing the assignment and then extending the assignment to the full problem. We demonstrate the advantages of the method for some typical examples. In particular we give the first optimal assignments of several variants of the “Philadelphia” problems. These problems have been used by several authors to assess assignment methods and lower bounds.     

Parallel hyperheuristics for the frequency assignment problem

This work presents a set of approaches used to deal with the frequency assignment problem (FAP), which is one of the key issues in the design of GSM networks. The used formulation of FAP is focused on aspects which are relevant for real-world GSM networks. A memetic algorithm, together with the specifically designed local search and variation operators, are presented. The memetic algorithm obtains good quality solutions but it must be adapted for each instance to be solved. A parallel hyperheuristic-based model was used to parallelize the approach and to avoid the requirement of the adaptation step of the memetic algorithm. The model is a hybrid algorithm which combines a parallel island-based scheme with a hyperheuristic approach. The main operation of the island-based model is kept, but the configurations of the memetic algorithms executed on each island are dynamically mapped. The model grants more computational resources to those configurations that show a more promising behavior. For this purpose two different criteria have been used in order to select the configurations. The first one is based on the improvements that each configuration is able to achieve along the executions. The second one tries to detect synergies among the configurations, i.e., detect which configurations obtain better solutions when they are cooperating. Computational results obtained for two different real-world instances of the FAP demonstrate the validity of the proposed model. The new designed schemes have made possible to improve the previously known best frequency plans for a real-world network.     

Algorithms for the vector maximization problem

We consider a convex setB inR ⁿ described as the intersection of halfspacesa _i ^T x ≦b _i (i ∈ I) and a set of linear objective functionsf _j =c _j ^T x (j ∈ J). The index setsI andJ are allowed to be infinite in one of the algorithms. We give the definition of theefficient points ofB (also called functionally efficient or Pareto optimal points) and present the mathematical theory which is needed in the algorithms. In the last section of the paper, we present algorithms that solve the following problems:

1. To decide if a given point inB is efficient.
2. To find an efficient point inB.
3. To decide if a given efficient point is the only one that exists, and if not, find other ones.
4. The solutions of the above problems do not depend on the absolute magnitudes of thec _j. They only describe the relative importance of the different activitiesx _i. Therefore we also consider $$\begin{gathered} \max G^T x \hfill \\ x efficient \hfill \\ \end{gathered} $$ for some vectorG.

     

A GRASP for the biquadratic assignment problem

The biquadratic assignment problem (BiQAP) is a generalization of the quadratic assignment problem (QAP). It is a nonlinear integer programming problem where the objective function is a fourth degree multivariable polynomial and the feasible domain is the assignment polytope. BiQAP problems appear in VLSI synthesis. Due to the difficulty of this problem, only heuristic solution approaches have been proposed. In this paper, we propose a new heuristic for the BiQAP, a greedy randomized adaptive search procedure (GRASP). Computational results on instances described in the literature indicate that this procedure consistently finds better solutions than previously described algorithms.     

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.     

Autocorrelation measures for the quadratic assignment problem

In this article we provide an exact expression for computing the autocorrelation coefficient ξ and the autocorrelation length ? of any arbitrary instance of the Quadratic Assignment Problem (QAP) in polynomial time using its elementary landscape decomposition. We also provide empirical evidence of the autocorrelation length conjecture in QAP and compute the parameters ξ and ? for the 137 instances of the QAPLIB. Our goal is to better characterize the difficulty of this important class of problems to ease the future definition of new optimization methods. Also, the advance that this represents helps to consolidate QAP as an interesting and now better understood problem.     

Ant colonies for the quadratic assignment problem

This paper presents HAS–QAP, a hybrid ant colony system coupled with a local search, applied to the quadratic assignment problem. HAS–QAP uses pheromone trail information to perform modifications on QAP solutions, unlike more traditional ant systems that use pheromone trail information to construct complete solutions. HAS–QAP is analysed and compared with some of the best heuristics available for the QAP: two versions of tabu search, namely, robust and reactive tabu search, hybrid genetic algorithm, and a simulated annealing method. Experimental results show that HAS–QAP and the hybrid genetic algorithm perform best on real world, irregular and structured problems due to their ability to find the structure of good solutions, while HAS–QAP performance is less competitive on random, regular and unstructured problems.     

Lower bounds for the quadratic assignment problem

We investigate the classical Gilmore-Lawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the Gilmore-Lawler bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new lower bounds have advantages over previous bounds and can be used in a branch-and-bound type algorithm for the quadratic assignment problem.     

Effective algorithm and heuristic for the generalized assignment problem

We present new Branch-and-Bound algorithm for the generalized assignment problem. A standard subgradient method (SM), used at each node of the decision tree to solve the Lagrangian dual, provides an upper bound. Our key contribution in this paper is a new heuristic, applied at each iteration of the SM, which tries to exploit the solution of the relaxed problem, by solving a smaller generalized assignment problem. The feasible solution found is then subjected to a solution improvement heuristic. We consider processing the root node as a Lagrangian heuristic. Computational comparisons are made with new existing methods.     

Algorithms for the quickest path problem and the reliable quickest path problem

The quickest path problem consists of finding a path in a directed network to transmit a given amount of items from an origin node to a destination node with minimal transmission time, when the transmission time depends on both the traversal times of the arcs, or lead time, and the rates of flow along arcs, or capacity. In telecommunications networks, arcs often also have an associated operational probability of the transmission being fault free. The reliability of a path is defined as the product of the operational probabilities of its arcs. The reliability as well as the transmission time are of interest. In this paper, algorithms are proposed to solve the quickest path problem as well as the problem of identifying the quickest path whose reliability is not lower than a given threshold. The algorithms rely on both the properties of a network which turns the computation of a quickest path into the computation of a shortest path and the fact that the reliability of a path can be evaluated through the reliability of the ordered sequence of its arcs. Other constraints on resources consumed, on the number of arcs of the path, etc. can also be managed with the same algorithms.     

Bees algorithm for generalized assignment problem

Bees algorithm (BA) is a new member of meta-heuristics. BA tries to model natural behavior of honey bees in food foraging. Honey bees use several mechanisms like waggle dance to optimally locate food sources and to search new ones. This makes them a good candidate for developing new algorithms for solving optimization problems. In this paper a brief review of BA is first given, afterwards development of a BA for solving generalized assignment problems (GAP) with an ejection chain neighborhood mechanism is presented. GAP is a NP-hard problem. Many meta-heuristic algorithms were proposed for its solution. So far BA is generally applied to continuous optimization. In order to investigate the performance of BA on a complex integer optimization problem, an attempt is made in this paper. An extensive computational study is carried out and the results are compared with several algorithms from the literature.     

A genetic algorithm for the generalised assignment problem

A new algorithm for the generalised assignment problem is described in this paper. The algorithm is adapted from a genetic algorithm which has been successfully used on set covering problems, but instead of genetically improving a set of feasible solutions it tries to genetically restore feasibility to a set of near-optimal ones. Thus it may be regarded as operating in a dual sense to the more familiar genetic approach. The algorithm has been tested on generalised assignment problems of substantial size and compared to an exact integer programming approach and a well-established heuristic approach.     

Iterated local search for the quadratic assignment problem

Iterated local search (ILS) is a simple and powerful stochastic local search method. This article presents and analyzes the application of ILS to the quadratic assignment problem (QAP). We justify the potential usefulness of an ILS approach to this problem by an analysis of the QAP search space. However, an analysis of the run-time behavior of a basic ILS algorithm reveals a stagnation behavior which strongly compromises its performance. To avoid this stagnation behavior, we enhance the ILS algorithm using acceptance criteria that allow moves to worse local optima and we propose population-based ILS extensions. An experimental evaluation of the enhanced ILS algorithms shows their excellent performance when compared to other state-of-the-art algorithms for the QAP.