Quadratic Assignment Problem Instances Research Articles

Learning Individualized Hyperparameter Settings

The performance of optimization algorithms, and consequently of AI/machine learning solutions, is strongly influenced by the setting of their hyperparameters. Over the last decades, a rich literature has developed proposing methods to automatically determine the parameter setting for a problem of interest, aiming at either robust or instance-specific settings. Robust setting optimization is already a mature area of research, while instance-level setting is still in its infancy, with contributions mainly dealing with algorithm selection. The work reported in this paper belongs to the latter category, exploiting the learning and generalization capabilities of artificial neural networks to adapt a general setting generated by state-of-the-art automatic configurators. Our approach differs significantly from analogous ones in the literature, both because we rely on neural systems to suggest the settings, and because we propose a novel learning scheme in which different outputs are proposed for each input, in order to support generalization from examples. The approach was validated on two different algorithms that optimized instances of two different problems. We used an algorithm that is very sensitive to parameter settings, applied to generalized assignment problem instances, and a robust tabu search that is purportedly little sensitive to its settings, applied to quadratic assignment problem instances. The computational results in both cases attest to the effectiveness of the approach, especially when applied to instances that are structurally very different from those previously encountered.

A new hybrid-heuristic for large-scale combinatorial optimization: A case of quadratic assignment problem

Real world combinatorial optimization problems arise in very-large-scale. Solving such problems to optimality is impractical. In the past several decades, different heuristics and meta-heuristics have been proposed as solution procedure. In particular, to tackle the large-scale real applications, researchers have intelligently combined key components of competing methodologies to create superior solution procedures. In this study, we design a hybrid-heuristic (HH) using high-level solution algorithm framework that blends key components of three well-known meta-heuristics, scatter search, critical even tabu search, and genetic algorithm random keys, and applies to large-scale quadratic assignment problem (QAP). The HH has the potential to go beyond the goals of usual hybrid-heuristics, it potentially provides a framework for a self-adapted heuristic that can be applied to a wide range of problems with minimal inclusion of problem specific information. Extensive computational experiments applied to large-scale QAP instances from database of benchmark problems e.g., QAPLIB and taixxeyy collections, show that the proposed hybrid-heuristic (HH) provides an efficient approach to the large-scale QAP. The algorithm provides new best solutions for 61 out of 70 taixxeyy instances of large-scale QAP and finds the same best known solutions for QAPLIB and the 9 remaining instances from taixxeyy collections.

A new tool for automated transformation of Quadratic Assignment Problem instances to Quadratic Unconstrained Binary Optimisation models

Growing class of optimisation problems are reported to attract increasing interest even though they are very challenging and difficult to solve exactly. Quadratic Assignment Problem (QAP), Travelling salesman, weapon target assignment, and query optimisation in distributed databases are some of these well known problems. QAP is considered to be one of the most prominent combinatorial optimisation problems with various application areas. Economic aspects of design and location of factories, hospitals, industrial plants, network models, transportation infrastructures, and military bases can all be addressed by QAP. There is no known polynomial time algorithm reported for the solution of the problem since it is NP-hard. Even small problem instances take long times to be solved exhaustively. Exact solution techniques and parallelisation of these algorithms end up with only little or negligible increase in performance. Recent meta-heuristics based on genetic algorithms, simulated annealing, and tabu search provide promising results even for problem instances up to instance sizes of 256 which is currently the largest instance in the QAPLIB. The Quadratic Unconstrained Binary Optimisation (QUBO) model has a critical role for experiments conducted with quantum computers of D-Wave Systems and IBM. Quantum Annealing is a technique which exploits quantum fluctuations to reach the global minimum of a given cost function. The QUBO model and its connection with Ising problem constitute the foundations for quantum annealing. Advances in quantum computing highlight the fact that QUBO models can be an efficient alternative to traditional models. Even though QAP can be formulated by QUBO using penalty functions, there is no known automated transformation method to map a QAP instance to a QUBO model. In this paper, we propose an automated model for QAP to QUBO mappings which can generate these formulations using the pre-defined penalty functions. We also present the benchmark results for a group of QAPLIB instances using exact solver for our QUBO models.

Multi-spin-flip engineering in an Ising machine

A merge process is proposed to engineer a multi-spin flip in an Ising machine. The merge process deforms the Hamiltonian (energy function) of the Ising model. We prove a theorem for the merge process and show that a single-spin flip in the deformed Hamiltonian is equivalent to a multi-spin flip in the original Hamiltonian. A merge process induces a transition within the subspace of feasible solutions. We propose a hybrid simulated annealing (SA) algorithm with the merge process. The hybrid algorithm outperforms the conventional SA algorithm, genetic algorithm, and tabu search in the binary quadratic knapsack problems (QKP) and the quadratic assignment problems (QAP). Finally, the hybrid merge process is used in a real Ising machine. The performance is improved in QKP and QAP instances. The merge process is generally applicable to existing Ising machine hardware because the deformed Hamiltonian keeps the format of the Ising model.

A performance study of meta‐heuristic approaches for quadratic assignment problem

AbstractThe quadratic assignment problem (QAP) is a well‐known challenging combinatorial optimization problem that has received many researchers' attention with varied real‐world and industrial applications areas. It is noteworthy to mention that a plethora of nature‐inspired optimization algorithms have successfully been used to solve various optimization problems, including several variants of the QAPs. In this article, a comprehensive literature review is presented to show the most relevant nature‐inspired algorithms that have been used in solving the QAP. More so, extensive experiments are conducted and analyzed to show the performance of the well‐known state‐of‐the‐art nature‐inspired meta‐heuristic optimization algorithms in solving the QAP, including the ant colony optimization (ACO), bat algorithm, genetic algorithm (GA), particle swarm optimization (PSO), and tabu search algorithm. Besides, a modified variant of the discrete PSO algorithm is implemented and compared with existing approaches. The six selected algorithms' performances, including the modified PSO, are validated on eight commonly used QAP instances of varying complexity and size, considering the quality of solutions achieved and computational time consumed by the representative algorithms. The numerical results revealed that the most competitive algorithm was ACO, while the GA appeared to be the worst performed algorithm among the six compared meta‐heuristic algorithms. However, based on the extensive analysis conducted on the tested algorithms, further improvements are suggested, including implementing new modified versions of the tested algorithms to tackle the QAP and its variant instances.

A proximal DC approach for quadratic assignment problem

In this paper, we show that the quadratic assignment problem (QAP) can be reformulated to an equivalent rank constrained doubly nonnegative (DNN) problem. Under the framework of the difference of convex functions (DC) approach, a semi-proximal DC algorithm is proposed for solving the relaxation of the rank constrained DNN problem whose subproblems can be solved by the semi-proximal augmented Lagrangian method. We show that the generated sequence converges to a stationary point of the corresponding DC problem, which is feasible to the rank constrained DNN problem under some suitable assumptions. Moreover, numerical experiments demonstrate that for most QAP instances, the proposed approach can find the global optimal solutions efficiently, and for others, the proposed algorithm is able to provide good feasible solutions in a reasonable time.

A modified single and multi-objective bacteria foraging optimisation for the solution of quadratic assignment problem

Non-polynomial hard (NP-hard) problems are challenging due to time-constraint. The bacteria foraging optimisation (BFO) algorithm is a metaheuristics algorithm that is used for NP-hard problems. BFO is inspired by the behaviour of the bacteria foraging such as E. coli. The aim of BFO is to eliminate weak foraging properties bacteria and maintain breakthrough foraging properties bacteria toward the optimum. However, reaching to optimal solutions are time-demanding. In this paper, we modified single objective and multi-objective BFO (MOBFO) by adding mutation and crossover from genetic algorithm operators to update the solutions in each generation, and local tabu search algorithm to reach the local optimum solution. Additionally, we used fast non-dominated sort algorithm in MOBFO to find the best non-dominated solutions. We evaluated the performance of the proposed algorithms with quadratic assignment problem instances. The experimental results show that our approaches outperform some previous optimisation algorithms in both convergent and divergent aspects.

A 2-Level Iterated Tabu Search Algorithm for the Quadratic Assignment Problem

In this paper, a 2-level iterated tabu search (ITS) algorithm for the solution of the quadratic assignment problem (QAP) is considered. The novelty of the proposed ITS algorithm is that the solution mutation procedures are incorporated within the algorithm, which enable to diversify the search process and eliminate the search stagnation, thus increasing the algorithm’s efficiency. In the computational experiments, the algorithm is examined with various implemented variants of the mutation procedures using the QAP test (sample) instances from the library of the QAP instances – QAPLIB. The results of these experiments demonstrate how the different mutation procedures affect and possibly improve the overall performance of the ITS algorithm.

A variable neighbourhood search enhanced estimation of distribution algorithm for quadratic assignment problems

Quadratic Assignment Problem (QAP) is one of the most complex combinatorial optimization problems. Many real-world problems such as printed circuit board design, facility location problems, assigning gates to airplanes can be modelled as QAP. Problems of size greater than 35 is not able to solve optimally using conventional optimization methods. This warrants the use of evolutionary optimization methods for obtaining optimal or near optimal solutions for QAPs. This work proposes a hybridization on a univariate Estimation of Distribution Algorithm, namely the Population Based Incremental Learning Algorithm (PBILA), with Variable Neighbourhood Search (VNS) for solving QAPs. The proposed algorithm is employed to solve benchmark instances of QAP and the results are reported. The results of this work reveals that PBILA on its own is not efficient for solving the QAPs. However, when hybridised with VNS, the algorithm performs well providing best known solutions for 95 test instances out of the 101 instances considered. For most of the test instances, the percentage deviation is less than one percentage. The overall average percentage deviation of the obtained solutions from the best-known solutions is 0.037%, which is a significant improvement when compared with state-of-the-art algorithms.

Cantor set based neighbor generation method for permutation solution representation

Metaheuristics gained world-wide popularity and researchers have been studying them vigorously in the last two decades. A relatively less explored approach in the improvement of metaheuristics is to design new neighbor generation mechanisms. Neighbor generation mechanisms are very important in the success of any single solution-based heuristic since they directly guide the search. In this study, a neighbor generation mechanism called cantor-set based (CB) method for single solution-based heuristics which use permutation solution representation is described. The inspiration for CB method stems from the recursive algorithm that constructs a cantor set which is a fractal set. Three variations of CB method are discussed (CB-1, CB-2, CB-3) considering the presented design possibilities. The computational experiments are conducted by embedding the mechanisms into the classical local search and simulated annealing algorithms, separately, to test their efficiency and effectiveness by comparing them to classical swap and insertion mechanisms. The traveling salesman problem (TSP) and quadratic assignment problem (QAP) which are very different problems that have incompatible characteristics have been chosen to test the mechanisms and sets of benchmark instances with varying sizes are chosen for the comparisons. The computational tests show that CB-2 gives very favorable results for TSP and CB-1 gives favorable results for QAP which means that CB-2 is suitable for problems that have steep landscapes and CB-1 is suitable for the problems that have flat landscapes. It is observed that CB-3 is a more generalized mechanism because it gives consistently good results for both TSP and QAP instances. The best mechanism for a given instance of the both problem types outperforms the classical neighbor generation of swap and insertion in terms of effectiveness.

Dviejų lygių iteracinis tabu paieškos algoritmas kvadratinio paskirstymo uždaviniui

In this paper, a 2-level iterated tabu search (ITS) algorithm for the solution of the quadratic assignment problem (QAP) is considered. The novelty of the proposed ITS algorithm is that the solution mutation procedures are incorporated within the algorithm, which enable to diversify the search process and eliminate the search stagnation, thus increasing the algorithm’s efficiency. In the computational experiments, the algorithm is examined with various implemented variants of the mutation procedures using the QAP test (sample) instances from the library of the QAP instances – QAPLIB. The results of these experiments demonstrate how the different mutation procedures affect and possibly improve the overall performance of the ITS algorithm.

On the use of fitness landscape features in meta-learning based algorithm selection for the quadratic assignment problem

Meta-heuristics perform differently depending on the problem instance they are solving, meaning that manually choosing an algorithm is not trivial and an automatic selection is desirable. This task can be addressed using a meta-learning approach, which relates the characteristics of the problem instances to the performance of a set of solving algorithms. Therefore, the success of such approach is based on the quality of the extracted set of features. Some studies have proposed the use of features based on Fitness Landscape Analysis (FLA) to characterize optimization problems. However, extracting measures based on FLA usually requires a high computational effort. In a previous work, we have employed meta-heuristic selection on the Quadratic Assignment Problem (QAP) using some FLA measures. This research extends our study by including additional FLA meta-features, by using a less costly extraction method, and by considering more QAP instances. In total, we built five multi-label datasets, each composed of meta-features that were extracted by different sampling sizes, and then we used them to train Random Forest classifiers. Besides presenting satisfactory classification performance, and a decrease in time consumption in relation to our previous work, this selection approach was able to achieve better solution costs over the set of QAP instances if compared to running the meta-heuristics individually.

Level 2 Reformulation Linearization Technique–Based Parallel Algorithms for Solving Large Quadratic Assignment Problems on Graphics Processing Unit Clusters

This paper discusses efficient parallel algorithms for obtaining strong lower bounds and exact solutions for large instances of the quadratic assignment problem (QAP). Our parallel architecture is comprised of both multicore processors and compute unified device architecture–enabled NVIDIA graphics processing units (GPUs) on the Blue Waters Supercomputing Facility at the University of Illinois at Urbana–Champaign. We propose novel parallelization of the Lagrangian dual ascent algorithm on the GPUs, which is used for solving a QAP formulation based on the level-2 reformulation linearization technique. The linear assignment subproblems in this procedure are solved using our accelerated Hungarian algorithm [Date K, Rakesh N (2016) GPU-accelerated Hungarian algorithms for the linear assignment problem. Parallel Computing 57:52–72.]. We embed this accelerated dual-ascent algorithm in a parallel branch-and-bound scheme and conduct extensive computational experiments on single and multiple GPUs, using problem instances with up to 42 facilities from the quadratic assignment problem library (QAPLIB). The experiments suggest that our GPU-based approach is scalable, and it can be used to obtain tight lower bounds on large QAP instances. Our accelerated branch-and-bound scheme is able to comfortably solve Nugent and Taillard instances (up to 30 facilities) from the QAPLIB, using a modest number of GPUs.

Tournament selection based antlion optimization algorithm for solving quadratic assignment problem

We propose and develop an improved version of antlion optimizer (ALO), namely tournament selection based antlion optimization algorithm for quadratic assignment problem (QAP). ALO algorithm has some handicaps, such as long run time, local optima stagnation and premature convergence for some problems. The literature describes different methods that improve the performance of antlion optimizer, but most of these are about specific optimization problems. In this paper, we introduce the tournament selection method instead of the roulette wheel method on random walking mechanism of ALO and we update some equations used in ALO algorithm. To compare the proposed tournament selection based ALO (TALO) algorithm with classic ALO, we deal with ten benchmark functions from literature. The comparison results are evaluated according to the different metrics, such as mean best, standard deviation, optimality, accuracy, CPU time, number of function evaluations (NFE). The detailed analyzes of the proposed TALO algorithm are performed. These are the convergence analysis, statistical analysis, search history analysis, trajectory analysis, average distance analysis, computational complexity analysis. The proposed TALO algorithm is compared with the other ALO versions (binary ALO and chaotic ALO variants) for same ten benchmark functions. As last, TALO algorithm has been also implemented for the quadratic assignment problem (QAP). The QAP results has been compared with several well-known meta-heuristic algorithms. TALO’s performance has been evaluated with those of binary ALO and chaotic ALO variants for same QAP instance. Finally, the solution quality of proposed TALO algorithm has been analyzed for solving QAP using some instances presented in QAPLIB site. The results provide the proposed TALO algorithm has the best performance in comparison with those of the other meta-heuristic algorithms. Due to the performance of TALO algorithm, we expect this version to be applied for different optimization problems.

Estimating the number of basins of attraction of multi-objective combinatorial problems

The efficiency of local search is proportional to the number and the distribution of basins of attraction. Often combinatorial optimisation problems have a large number of local optima, uncountable with available computational resources. Approximating the number of basins of attraction and the minimal number of samples for visiting all basins at least once are complex problems for which we assume specific distributions of basins of attraction. We define two types of basins of attraction of multi-objective combinatorial optimisation problems with complementary properties. Acknowledging that each local search run generates a Pareto front of solutions, either each Pareto local solution corresponds to a basin of attraction, or a Pareto basin matches an entire Pareto local front. Simulations compare parametric and non-parametric estimators on bi-objective quadratic assignment problem instances.

Equivalences and differences in conic relaxations of combinatorial quadratic optimization problems

Various conic relaxations of quadratic optimization problems in nonnegative variables for combinatorial optimization problems, such as the binary integer quadratic problem, quadratic assignment problem (QAP), and maximum stable set problem have been proposed over the years. The binary and complementarity conditions of the combinatorial optimization problems can be expressed in several ways, each of which results in different conic relaxations. For the completely positive, doubly nonnegative and semidefinite relaxations of the combinatorial optimization problems, we discuss the equivalences and differences among the relaxations by investigating the feasible regions obtained from different representations of the combinatorial condition which we propose as a generalization of the binary and complementarity condition. We also study theoretically the issue of the primal and dual nondegeneracy, the existence of an interior solution and the size of the relaxations, as a result of different representations of the combinatorial condition. These characteristics of the conic relaxations affect the numerical efficiency and stability of the solver used to solve them. We illustrate the theoretical results with numerical experiments on QAP instances solved by SDPT3, SDPNAL+ and the bisection and projection method.

A Graphics Processing Unit Algorithm to Solve the Quadratic Assignment Problem Using Level-2 Reformulation-Linearization Technique

The quadratic assignment problem (QAP) is a combinatorial optimization problem that arises in many real-world applications, such as equipment allocation in industry. The QAP is NP-hard and, in practice, one of the hardest combinatorial optimization problems to solve to optimality. Exact solutions of QAP are typically obtained by the branch-and-bound method. This method, however, potentially requires a high computational effort, and the use of good lower bounds is essential to prune the search tree. Branch-and-bound algorithms that use the dual-ascent procedure based on the level-2 reformulation linearization technique (RLT2) belong to the state of the art on exactly solving QAP. In this work, we propose a parallel implementation of that branch-and-bound algorithm. Our approach uses the Auction Algorithm of Bertsekas and Castañon to solve the linear assignment problems of RLT2, which allows us to take advantage of the massive parallel environment of graphics processing units to speed up the lower bound computation and implement some memory optimization techniques to address large-size problems. We report experimental results that show significant execution time reductions compared to previous works and allow us to provide, for the first time, exact solutions for two instances of QAP: tai35b and tai40b.

Solution to the quadratic assignment problem using semi-lagrangian relaxation

The semi-Lagrangian relaxation (SLR), a new exact method for combinatorial optimization problems with equality constraints, is applied to the quadratic assignment problem (QAP). A dual ascent algorithm with finite convergence is developed for solving the semi-Lagrangian dual problem associated to the QAP. We perform computational experiments on 30 moderately difficult QAP instances by using the mixed integer programming solvers, Cplex, and SLR+Cplex, respectively. The numerical results not only further illustrate that the SLR and the developed dual ascent algorithm can be used to solve the QAP reasonably, but also disclose an interesting fact: comparing with solving the unreduced problem, the reduced oracle problem cannot be always effectively solved by using Cplex in terms of the CPU time.

Identifying Features of Fitness Landscapes and Relating Them to Problem Difficulty.

Complex combinatorial problems are most often optimised with heuristic solvers, which usually deliver acceptable results without any indication of the quality obtained. Recently, predictive diagnostic optimisation was proposed as a means of characterising the fitness landscape while optimising a combinatorial problem. The scalars produced by predictive diagnostic optimisation appear to describe the difficulty of the problem with relative reliability. In this study, we record more scalars that may be helpful in determining problem difficulty during the optimisation process and analyse these in combination with other well-known landscape descriptors by using exploratory factor analysis on four landscapes that arise from different search operators, applied to a varied set of quadratic assignment problem instances. Factors are designed to capture properties by combining the collinear variances of several variables. The extracted factors can be interpreted as the features of landscapes detected by the variables, but disappoint in their weak correlations with the result quality achieved by the optimiser, which we regard as the most reliable indicator of difficulty available. It appears that only the prediction error of predictive diagnostic optimisation has a strong correlation with the quality of the results produced, followed by a medium correlation of the fitness distance correlation of the local optima.

A novel multistart hyper-heuristic algorithm on the grid for the quadratic assignment problem

Hyper-heuristics introduce novel approaches for solving challenging combinatorial optimization problems by operating over a set of low level (meta)-heuristics. This is achieved by an evolutionary selection mechanism that controls and combines the strengths of the low level (meta)-heuristics. In this study, we propose a high-performance MultiStart Hyper-heuristic algorithm (MSH-QAP) on the grid for the solution of the Quadratic Assignment Problem (QAP). MSH-QAP algorithm makes use of state-of-the-art (meta)-heuristics, Simulated Annealing (SA), Robust Tabu Search (RTS), Ant Colony Optimization (FAnt), and Breakout Local Search (BLS) that have been reported among the best performing algorithms for the solution of difficult QAP instances in standard benchmark libraries. In the first phase of the algorithm, the most appropriate (meta)-heuristic with its near-optimal parameter settings is selected by using a genetic algorithm optimization layer that uses a self-adaptive parameter setting method for the given problem instance. In the second phase, if an optimal solution cannot be found, selected best performing (meta)-heuristic (with its finely adjusted parameter settings) is executed on the grid using parallel processing and performing several multistarts in order to increase the quality of the discovered solution. MSH-QAP algorithm is tested on 134 problem instances of the QAPLIB benchmark and is shown to be able to solve 122 of the instances exactly. The overall deviation for the problem instances is obtained as 0.013% on the average.