Network-based formulations of the quadratic assignment problem

Reduced costs obtained from the optimal simplex tableau are not necessarily correct under degeneracy in network flow problems with side constraints. In one application, simplex reduced costs were shown not to be good when applied to actual operations, while reoptimization to find all true reduced costs was too time-consuming. An efficient algorithm to obtain approximate reduced costs was developed in the research to offset these disadvantages. The algorithm combines the uses of Lagrangian relaxation, a minimum-cost network flow algorithm, and a shortest path algorithm. In theory, the approximate reduced costs prove to be a better lower bound than are simplex reduced costs. Numerical experiments have been performed to show its potential usefulness in practice. © 1996 John Wiley & Sons, Inc.

The problem of optimally (re)allocating Navy personnel to combat units is compounded by several considerations: availability of trained personnel, staffing of positions by occupation groups or ranks, and maintaining an acceptable level of readiness. In this paper we model this problem as a nonlinear nondifferentiable optimization problem. A reformulation of the nonlinear optimization problem as a network flow problem is then developed. The formulation results in a network flow problem with side constraints. An additional, nonnetwork, variable measures the readiness level. This new formulation permits the use of network optimization tools in order to solve effectively very large problems. We then develop two numerical methods for solving this problem. One method is based on a heuristic that solves (approximately) the nondifferentiable problem. The second method is based on a Linear-Quadratic Penalty (LQP) algorithm, and it exploits the embedded network structure by placing the side constraints into the objective function. The resulting nonlinear network program is solved using a simplicial decomposition of the network constraint set. Numerical results indicate the viability of this approach on problems with up to 36,000 arcs and 17,000 nodes with 3,700 side constraints.

Report of the Session on Network Flow, Assignment and Travelling Salesman Problems

-Quadratic assignment problems (QAPs) are NP-hard combinatorial optimization problems often used to compare the performance of meta-heuristics. In this paper, we propose a QAP problem instance generator whose instances can be used as benchmark for meta-heuristics. Our generator aggregates various QAP instances into a larger size QAP instance in order to obtain an large size, additively decomposable QAP. We assume that a QAP instance that is difficult for local search has many local optima from which local search needs to escape from. We call the resulting QAPs the composite QAP instances (cQAPs). We use numerical and empirical techniques for the landscape analysis of generated composite QAPs. The comparison between our QAP instances with the other QAPs from the literature classify cQAPs as difficult. We show that heuristic algorithms that exploit the particularities of the cQAP search space, like iterated local search, can outperform other heuristics that do not consider the structure of this search space, like the multi-restart local search.

Quadratic assignment problems (QAPs) is a NP-hard combinatorial optimization problem. QAPs are often used to compare the performance of meta-heuristics. In this paper, we propose a QAP problem instance generator that can be used for benchmarking for heuristic algorithms. Our QAP generator combines small size QAPs with known optimum solution into a larger size QAP instance. We call these instances composite QAPs (cQAPs), and we show that the cost function of cQAPs is additively decomposable. We give mild conditions for which a cQAP instance has known optimum solution. We generate cQAP instances using uniform distributions with different bounds for the component QAPs and for the rest of the cQAP elements. Numerical and analytical techniques that measure the difficulty of the cQAP instances in comparison with other QAPs from the literature are introduced. These methods point out that some cQAP instances are difficult for local search with many local optimum of various values, low epistasis and non-trivial asymptotic behaviour.

The quadratic assignment problem (QAP) has attracted a surpassing algorithmic research interest since its introduction in 1957 by Koopmans and Beckmann. A wide variety of algorithms and heuristics have been developed to solve the QAP exactly or approximately. Moreover, since all the problems described in Chapter 2 are closely related to the QAP, one could modify the available exact and approximate techniques for the QAP and utilize them to “solve” every one of these problems. While this is conceptually correct, we do not recommend to solve e.g. traveling salesman problems this way, because the largest size QAP solved to optimality, so far, has n = 30; see Clausen [1994], Mans et al. [1992], Pardalos et al. [1994], and Resende et al. [1994]. More to the point, this means that existing algorithms for QAPs are nowhere close to solving practical problems arising from real-life applications to optimality. This state of affairs is unsatisfactory, but not surprising since very little is known about the mathematical properties of QAPs. A straight-forward application of the appropriately modified QAP algorithms to solve its variants can thus not be expected to solve large-scale instances of these problems. While many authors propose (different) mixed zero-one formulations of QAPs, they are hardly exploited in the numerical computations and the facial structure of the associated integer polyhedra has not been studied in any detail.KeywordsAssignment ProblemTravel Salesman ProblemMaster ProblemTask GraphQuadratic Assignment ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Signed graphs are graphs with an assignment of a positive or a negative sign to each edge. These graphs are helpful to represent different types of networks. For instance, they have been used in social networks, where a positive sign in an edge represents friendship between the two endpoints of that edge, while a negative sign represents enmity. Given a signed graph, an important question is how to embed such a graph in a metric space so that in the embedding every vertex is closer to its positive neighbors than to its negative ones. This problem is known as Sitting Arrangement (SA) problem and it was introduced by Kermarrec et al. (Proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS), pp. 388---399, 2011). Cygan et al. (Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS), 2012) proved that the decision version of SA problem is NP-Complete when the signed graph has to be embedded into the Euclidean line. In this work, we study the minimization version of SA (MinSA) problem in the Euclidean line. We relate MinSA problem to the well known quadratic assignment (QA) problem. We establish such a relation by proving that local minimums in MinSA problem are equivalent to local minimums in a particular case of QA problem. In this document, we design two heuristics based on the combinatorial structure of MinSA problem. We experimentally compare their performances against heuristics designed for QA problem. This comparison favors the proposed heuristics.

Introduction. The quadratic assignment problem is a well-established NP-hard problem in combinatorial optimization with applications in diverse fields like economics, archaeology, and chemistry. Due to its complexity, research on efficient solution methods remains active, including efforts for parallelization on multiprocessor computing systems. However, effective parallel algorithms are crucial to fully leverage these computational resources. In this context, algorithm unions (portfolios and teams) play a significant role in achieving parallel execution for solving such problems. Research objectives. This work investigates the application of portfolios constructed from modifications of the repeated iterated tabu search algorithm to the quadratic assignment problem. The effectiveness of these portfolios was evaluated through experimental computations. Results. The portfolios, derived from modifications of the repeated iterated tabu search algorithm, were applied to the quadratic assignment problem. For the most demanding test instances, the proposed algorithms were evaluated on the SCIT-4 supercomputer, alongside previously published results from the authors, confirming their competitive performance. Additionally, we assessed the parallel efficiency of these portfolios in solving instances of the quadratic assignment problem. The results demonstrate their ability to accelerate the optimization process (with speedup dependent on problem size and utilized processors), enabling the solution of large-scale problems. Conclusions. The conducted studies demonstrate that employing algorithm portfolios significantly accelerates the solution process for the quadratic assignment problem. Analysis of the results reveals a near-linear speedup factor achieved by the portfolio. For the challenging test instance tai100a, a new best solution value of 21040996 was obtained using a portfolio of 16 algorithms. Keywords: quadratic assignment problem, algorithm portfolios, experimental research, algorithm portfolios efficiency.

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

The Quadratic Assignment Problem (QAP) is a difficult and important problem studied in the domain of combinatorial optimisation. It is possible to solve QAP instances with 10--20 facilities using exhaustive parallel algorithms within a few days on a cluster machine. However, large QAP instances with more than 100 facilities are not solvable using exhaustive techniques. We have explored a variety of Genetic Algorithm crossover operators for this problem and verified its performance experimentally using well-known instances from the QAPLIB library. By increasing the number of processors, generations and population sizes we have been able to find solutions that are the same as (or very close to) the best reported solutions for large QAP instances in QAPLIB. In order to parallelise the Genetic Algorithm we generate and evolve separate solution pools on each cluster processor, using an island model. This model exchanges 10% of each processor’s solutions at the initial stages of optimisation. We show experimentally that both execution times and solution qualities are improved for large QAP instances by using our Island Parallel Genetic Algorithm.

Aurora and Mehta (J Comb Optim 36(3):965–1006, 2018) show that two graphs $$G_1,G_2$$ , on n vertices each, are isomorphic if and only if the feasible region of a certain linear program, LP-GI, intersects with the Quadratic Assignment Problem (QAP)-polytope in $${\mathbb {R}}^{(n^4+n^2)/2}$$ . The linear program LP-GI in Aurora and Mehta (J Comb Optim 36(3):965–1006, 2018) is obtained by relaxing an integer linear program whose feasible points correspond to the isomorphisms between $$G_1,G_2$$ . In this paper we take an analogous approach with the linear programs replaced with conic programs. A completely positive description of the QAP-polytope was obtained in Povh and Rendl (Discrete Optim 6(3):231–241, 2009). By adding the graph conditions to this description we get a completely positive formulation of the graph isomorphism problem. However, analogous to integer linear programs, it is NP-hard to optimize over the cone of completely positive matrices. So we relax this formulation by replacing the cone of completely positive matrices with the cone of positive semidefinite matrices. We observe that the resulting SDP is the Lovasz Theta function (Lovasz in IEEE Trans Inf Theory 25(1):1–7, 1979) of a graph product of $$G_1,G_2$$ and can be efficiently computed. We provide a natural heuristic that uses the SDP to solve the graph isomorphism problem. We run our heuristic on several pairs of non-isomorphic strongly regular graphs and find the results to be encouraging. Further, by adding the non-negativity constraints to the SDP, we obtain a doubly non-negative formulation, DNN-GI. We show that if the set of optimal points in DNN-GI contains a point of rank at most 3, then the given pair of graphs must be isomorphic.

Graph matching problem that incorporates pair-wise constraints can be formulated as Quadratic Assignment Problem(QAP). The optimal solution of QAP is discrete and combinational, which makes QAP problem NP-hard. Thus, many algorithms have been proposed to find approximate solutions. In this paper, we propose a new algorithm, called Nonnegative Orthogonal Graph Matching (NOGM), for QAP matching problem. NOGM is motivated by our new observation that the discrete mapping constraint of QAP can be equivalently encoded by a nonnegative orthogonal constraint which is much easier to implement computationally. Based on this observation, we develop an effective multiplicative update algorithm to solve NOGM and thus can find an effective approximate solution for QAP problem. Comparing with many traditional continuous methods which usually obtain continuous solutions and should be further discretized, NOGM can obtain a sparse solution and thus incorporates the desirable discrete constraint naturally in its optimization. Promising experimental results demonstrate benefits of NOGM algorithm.

The 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.

The facility layout design problem is an extensively studied research problem and belongs to nonpolynomial hard (NP-hard) combinatorial optimization problem. Quadratic assignment problem (QAP) is one of the formulations that is investigated for facility layout design because of its wide applicability. Ant colony optimization (ACO), a biologically inspired heuristic has centered on solving the QAP by achieving approximation as good as possible. This paper presents a population-based hybrid ant system (PHAS), which is an extension of the hybrid ant system (HAS) in which the size of the ant colony has been fixed. The performance of the proposed ant algorithm for QAP is compared with the existing metaheuristic implementations such as tabu search, reactive tabu search, simulated annealing, genetic hybrid method, HAS, and max–min ant system. The experimental results show that the proposed PHAS perform significantly better than the other existing algorithms of QAP.

The power of genetic algorithms (GAs) has been demonstrated for various domains of the computer science, including combinatorial optimization. In this paper, we propose a new conceptual modification of the genetic algorithm entitled a “restart-based genetic algorithm” (RGA). An effective implementation of RGA for a well-known combinatorial optimization problem, the quadratic assignment problem (QAP), is discussed. The results obtained from the computational experiments on the QAP instances from the publicly available library QAPLIB show excellent performance of RGA. This is especially true for the real-life like QAPs.