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In this paper, we study the quadratic assignment problem with a rank-one cost matrix (QAP-R1). Four integer-programming formulations are introduced of which three are assumed to have partial integer data. Unlike the standard quadratic assignment problem, some of our formulations can solve reasonably large instances of QAP-R1 with impressive running times and are faster than some metaheuristics. Pairwise relative strength of the LP relaxations of these formulations are also analyzed from theoretical and experimental points of view. Finally, we present a new metaheuristic algorithm to solve QAP-R1 along with its computational analysis. Our study offers the first systematic experimental analysis of integer-programming models and heuristics for QAP-R1. The benchmark instances with various characteristics generated for our study are made available to the public for future research work. Some new polynomially solvable special cases are also introduced. Summary of Contribution: This paper aims to advance our knowledge and ability in solving an important special case of the quadratic assignment problem. It shows how to exploit inherent properties of an optimization problem to achieve computational advantages, a strategy that was followed by researchers in model building and algorithm developments for decades. Our computational results attest to this time-tested general philosophy. The paper presents the first systematic computational study of the rank one quadratic assignment problem, along with new mathematical programming models and complexity analysis. We believe the theoretical and computational results of this paper will inspire further research on the topic and will be of significant value to practitioners using rank one quadratic assignment models.

This paper studies the cross-correlation of two real-valued functions whose common domain is a set on which a finite group acts transitively. Such a group action generalizes, e.g., circular time shifts for discrete periodic time signals or spatial translations in the context of image processing. Cross-correlations can be reformulated as convolutions in group algebras and are thus transformable into the spectral domain via Fast Fourier Transforms. The resulting general discrete cross-correlation theorem will serve as the basis for a unified FFT-based approach to maximum assignment problems related to transitive finite group actions. Our main focus is on those assignment problems arising from the actions of symmetric groups on the cosets of Young subgroups. The “simplest” nontrivial representatives of the latter class of problems are the NP-hard symmetric and asymmetric maximum quadratic assignment problem. We present a systematic spectral approach in terms of the representation theory of symmetric groups to solve such assignment problems. This generalizes and algorithmically improves Kondor's spectral branch-and-bound approach (SODA, 2010) to the exact solution of the asymmetric maximum quadratic assignment problem.

We commence an algorithmic study of Bulk-Robustness, a new model of robustness in combinatorial optimization. Unlike most existing models, Bulk-Robust combinatorial optimization features a highly nonuniform failure model. Instead of an interdiction budget, Bulk-Robust counterparts provide an explicit list of interdiction sets, comprising the admissible set of scenarios, thus allowing to model correlations between failures of different components in the system, interdiction sets of variable cardinality and more. The resulting model is suitable for capturing failures of complex structures in the system. We provide complexity results and approximation algorithms for Bulk-Robust counterparts of the Minimum Matroid Basis problems and the Shortest Path problem. Our results rely on various techniques, and outline the rich and heterogeneous combinatorial structure of Bulk-Robust optimization., Mathematical Programming, 149 (1-2), ISSN:0025-5610, ISSN:1436-4646

The random walk’ correlation matrix of multi-objective combinatorial optimization problems utilizes both local structure and general statistics of the objective functions. Reckoning time of correlation, or the random walk of lag 0, is quadratic in problem size L and number of objectives D. The computational complexity of the correlation coefficients of mNK is $$O(D^2 K^2 L)$$ , and of mQAP is $$O(D^2 L^2)$$ , where K is the number of interacting bits. To compute the random walk of a lag larger than 0, we employ a weighted graph Laplacian that associates a mutation operator with the difference in the objective function. We calculate the expected objective vector of a neighbourhood function and the eigenvalues of the corresponding transition matrix. The computational complexity of random walk’s correlation coefficients is polynomial with the problem size L and the number of objectives D. The computational effort of the random walks correlation coefficients of mNK is $$O(2^K L D^2)$$ , whereas of mQAP is $$O(L^6 D^2)$$ . Numerical examples demonstrate the utilization of these techniques.

We introduce and study the combinatorial optimization problem with interaction costs (COPIC). COPIC is the problem of finding two combinatorial structures, one from each of two given families, such that the sum of their independent linear costs and the interaction costs between elements of the two selected structures is minimized. COPIC generalizes the quadratic assignment problem and many other well studied combinatorial optimization problems, and hence covers many real world applications. We show how various topics from different areas in the literature can be formulated as special cases of COPIC. The main contributions of this paper are results on the computational complexity and approximability of COPIC for different families of combinatorial structures (e.g. spanning trees, paths, matroids), and special structures of the interaction costs. More specifically, we analyze the complexity if the interaction cost matrix is parameterized by its rank and if it is a diagonal matrix. Also, we determine the structure of the intersection cost matrix, such that COPIC is equivalent to independently solving linear optimization problems for the two given families of combinatorial structures.

Το Shortest Superstring Problem (SSP) είναι ένα πρόβλημα συνδυαστικής βελτιστοποίησης που έχει προσελκύσει το ενδιαφέρων πολλών ερευνητών, λόγω των εφαρμογών του. Μπορεί να χρησιμοποιηθεί σε προβλήματα Υπολογιστικής Μοριακής Βιολογίας όπως η αλληλούχιση του DNA και σε προβλήματα της επιστήμης υπολογιστών όπως η συμπίεση δεδομένων. Το SSP είναι ένα NP-hard πρόβλημα. Ένα άρθρο ανασκόπησης για το SSP παρουσιάζεται στο πρώτο κεφάλαιο της παρούσας διατριβής με έναν περιεκτικό και σαφή τρόπο, καλύπτοντας ολόκληρη τη σχετική βιβλιογραφία, αναδεικνύοντας την κατακτημένη γνώση και βοηθώντας στην μελλοντική έρευνα.Η μέθοδος GRASP (Greedy Randomized Adaptive Search Procedure) είναι μια επαναληπτική ευρετική μέθοδος για συνδυαστική βελτιστοποίηση. Η μέθοδος Path Relinking (PR) αποτελεί έναν τρόπο ενοποίησης των στρατηγικών εντατικοποίησης και διαφοροποίησης στην αναζήτηση για βέλτιστες λύσεις. Η PR στα πλαίσια του GRASP εισήχθη ως μηχανισμός μνήμης για την αξιοποίηση των δεδομένων από καλές λύσεις που έχουν ήδη βρεθεί. Στο δεύτερο κεφάλαιο, παρουσιάζεται η υλοποίηση της μεθόδου GRASP με PR για το SSP. Η νέα μέθοδος λύνει στιγμιότυπα μεγάλης κλίμακας και υπερτερεί του φυσικού άπληστου αλγόριθμου στη συντριπτική πλειοψηφία των στιγμιοτύπων που δοκιμάστηκαν. Η προτεινόμενη μέθοδος είναι ικανή να παράγει πολλαπλές λύσεις κοντά στο βέλτιστο, γεγονός το οποίο είναι σημαντικό για την πρακτική της αλληλούχισης του DNA και επιτρέπει μια φυσική και εύκολη παράλληλη υλοποίηση. Ένα σύνολο αναφοράς στιγμιοτύπων με γνωστή βέλτιστη λύση κατασκευάστηκε χρησιμοποιώντας μια νέα Διατύπωση Ακέραιου Προγραμματισμού (Integer Programming Formulation) για το SSP.Η οικογένεια των γράφων επικάλυψης αποτελεί ένα κατάλληλο είδος δομής δεδομένων για την περίπτωση του SSP. Έχουν εφαρμογές στην αλληλούχιση γονιδιώματος, στην συμπίεση ακολουθιών και στον χρονοπρογραμματισμό μηχανών. Ένας κατευθυνόμενος γράφος με βάρη είναι γράφος επικάλυψης αν υπάρχει ένα σύνολο από ακολουθίες, οι οποίες βρίσκονται σε ένα προς ένα αντιστοιχία με τις κορυφές του γράφου, έτσι ώστε κάθε βάρος του γράφου να ισούται με την επικάλυψη μεταξύ των αντίστοιχων ακολουθιών. Στο τρίτο κεφάλαιο της παρούσας διατριβής, παρουσιάζεται ένα θεώρημα χαρακτηρισμού των γράφων επικάλυψης και ο αντίστοιχος αλγόριθμος αναγνώρισής τους.Το Quadratic Assignment Problem (QAP) είναι ένα από τα δυσκολότερα προβλήματα συνδυαστικής βελτιστοποίησης. Το QAP είναι ένα NP-hard πρόβλημα, ενώ η εύρεση ενός ε-προσεγγιστικού αλγόριθμου για αυτό είναι επίσης δύσκολη. Ο κλασικός άπληστος αλγόριθμος για διακριτά προβλήματα βελτιστοποίησης όπου η βέλτιστη λύση είναι ένα μεγιστοτικό ανεξάρτητο υποσύνολο ενός πεπερασμένου συνόλου βάσης με στοιχεία με βάρη, μπορεί να οριστεί με δύο διαφορετικούς τρόπους που είναι δυϊκοί ο ένας προς το άλλο. Τον άπληστο-εισαγωγής (greedy-in) αλγόριθμο, όπου μια λύση κατασκευάζεται από ένα κενό σύνολο με την εισαγωγή του επόμενου καλύτερου στοιχείου, ενός κάθε φορά, μέχρι να προκύψει μια μη εφικτή λύση και τον άπληστο-εξαγωγής (greedy-out) αλγόριθμο, όπου ξεκινώντας από το σύνολο βάσης, διαγράφεται το επόμενο χειρότερο στοιχείο, ένα κάθε φορά, μέχρι να προκύψει κάποια εφικτή λύση. Έχει αποδειχτεί ότι ενώ ο πρώτος αλγόριθμος παρέχει έναν παράγοντα προσέγγισης για τα προβλήματα μεγιστοποίησης, η απόδοσή του στην χειρότερη περίπτωση δεν είναι φραγμένη για τα προβλήματα ελαχιστοποίησης και το αντίστροφο για τον δεύτερο αλγόριθμο. Στο τέταρτο κεφάλαιο αυτής της διατριβής, παρουσιάζεται ο άπληστος-εξαγωγής αλγόριθμος για το QAP, αφότου αναπτύσσεται ένας συνδυαστικός χαρακτηρισμός των λύσεων του προβλήματος.

The facility layout problem (FLP) is a very important class of NP-hard problems in operations research that deals with the optimal assignment of facilities to minimize transportation costs. The quadratic assignment problem (QAP) can model the FLP effectively. One of the FLPs is the hospital facility layout problem that aims to place comprehensive clinics, laboratories, and radiology units within predefined boundaries in a way that minimizes the cost of movement of patients and healthcare personnel. We are going to develop a hybrid method based on discrete differential evolution (DDE) algorithm for solving the QAP. In the existing DDE algorithms, certain issues such as premature convergence, stagnation, and exploitation mechanism have not been properly addressed. In this study, we first aim to discover the issues that make the current problem worse and to identify the best solution to the problem, and then we propose to develop a hybrid algorithm (HDDETS) by combining the DDE and tabu search (TS) algorithms to enhance the exploitation mechanism in the DDE algorithm. Then, the performance of the proposed HDDETS algorithm is evaluated by implementing on the benchmark instances from the QAPLIB website and by comparing with DDE and TS algorithms on the benchmark instances. It is found that the HDDETS algorithm has better performance than both the DDE and TS algorithms where the HDDETS has obtained 42 optimal and best-known solutions from 56 instances, while the DDE and TS algorithms have obtained 15 and 18 optimal and best-known solutions out of 56 instances, respectively. Finally, we propose to apply the proposed algorithm to find the optimal distributions of the advisory clinics inside the Azadi Hospital in Iraq that minimizes the total travel distance for patients when they move among these clinics. Our application shows that the proposed algorithm could find the best distribution of the hospital’s rooms, which are modeled as a QAP, with reduced total distance traveled by the patients.

The Reformulation Linearization Technique (RLT) of Sherali and Adams (Manag Sci 32(10):1274–1290, 1986; SIAM J Discrete Math 3(3):411–430, 1990), when applied to a pure 0–1 quadratic optimization problem with linear constraints (P), constructs a hierarchy of LP (i.e., continuous and linear) models of increasing sizes. These provide monotonically improving continuous bounds on the optimal value of (P) as the level, i.e., the stage in the process, increases. When the level reaches the dimension of the original solution space, the last model provides an LP bound equal to the IP optimum. In practice, unfortunately, the problem size increases so rapidly that for large instances, even computing bounds for RLT models of level k (called RLTk) for small k may be challenging. Their size and their complexity increase drastically with k. To our knowledge, only results for bounds of levels 1, 2, and 3 have been reported in the literature. We are proposing, for certain quadratic problem types, a way of producing stronger bounds than continuous RLT1 bounds in a fraction of the time it would take to compute continuous RLT2 bounds. The approach consists in applying a specific decomposable Lagrangean relaxation to a specially constructed RLT1-type linear 0–1 model. If the overall Lagrangean problem does not have the integrality property, and if it can be solved as a 0–1 rather than a continuous problem, one may be able to obtain 0–1 RLT1 bounds of roughly the same quality as standard continuous RLT2 bounds, but in a fraction of the time and with much smaller storage requirements. If one actually decomposes the Lagrangean relaxation model, this two-step procedure, reformulation plus decomposed Lagrangean relaxation, will produce linear 0–1 Lagrangean subproblems with a dimension no larger than that of the original model. We first present numerical results for the Crossdock Door Assignment Problem, a special case of the Generalized Quadratic Assignment Problem. These show that just solving one Lagrangean relaxation problem in 0–1 variables produces a bound close to or better than the standard continuous RLT2 bound (when available) but much faster, especially for larger instances, even if one does not actually decompose the Lagrangean problem. We then present numerical results for the 0–1 quadratic knapsack problem, for which no RLT2 bounds are available to our knowledge, but we show that solving an initial Lagrangean relaxation of a specific 0–1 RLT1 decomposable model drastically improves the quality of the bounds. In both cases, solving the fully decomposed rather than the decomposable Lagrangean problem to optimality will make it feasible to compute such bounds for instances much too large for computing the standard continuous RLT2 bounds.

Theoretical Computer Science, 745, ISSN:0304-3975

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.

We consider new polynomially solvable cases of the well-known Quadratic Assignment Problem involving coefficient matrices with a special diagonal structure. By combining the new special cases with polynomially solvable special cases known in the literature we obtain a new and larger class of polynomially solvable special cases of the QAP where one of the two coefficient matrices involved is a Robinson matrix with an additional structural property: this matrix can be represented as a conic combination of cut matrices in a certain normal form. The other matrix is a conic combination of a monotone anti-Monge matrix and a down-benevolent Toeplitz matrix. We consider the recognition problem for the special class of Robinson matrices mentioned above and show that it can be solved in polynomial time.

The Quadratic Assignment Problem (QAP) is a well-known NP-hard problem that is equivalent to optimizing a linear objective function over the QAP polytope. The QAP polytope with parameter n – \(\mathrm {QAP}_{n}\) – is defined as the convex hull of rank-1 matrices \(xx^T\) with x as the vectorized \(n\times n\) permutation matrices.

In this paper, a novel bi-level grouping optimization (BIGO) model is proposed for solving the storage location assignment problem with grouping constraint (SLAP-GC). A major challenge in this problem is the grouping constraint which restricts the number of groups each product can have and the locations of items in the same group. In SLAP-GC, the problem consists of two subproblems, one is how to group the items, and the other one is how to assign the groups to locations. It is an arduous task to solve the two subproblems simultaneously. To overcome this difficulty, we propose a BIGO. BIGO optimizes item grouping in the upper level, and uses the lower-level optimization to evaluate each item grouping. Sophisticated fitness evaluation and search operators are designed for both upper and lower level optimization so that the feasibility of solutions can be guaranteed, and the search can focus on promising areas in the search space. Based on the BIGO model, a multistart random search method and a tabu search algorithm are proposed. The experimental results on the real-world dataset validate the efficacy of the BIGO model and the advantage of the tabu search method over the random search method. © 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.

The Travelling Salesman Problem (TSP) is an NP-hard problem with high number of possible solutions. The complexity increases with the factorial of n nodes in each specific problem. Meta-heuristic algorithms are an optimization algorithm that able to solve TSP problem towards a satisfactory solution. To date, there are many meta-heuristic algorithms introduced in literatures which consist of different philosophies of intensification and diversification. This paper focuses on 6 heuristic algorithms: Nearest Neighbor, Genetic Algorithm, Simulated Annealing, Tabu Search, Ant Colony Optimization and Tree Physiology Optimization. The study in this paper includes comparison of computation, accuracy and convergence.

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.

We study the q -adic assignment problem. We first give an O ( n (q−1)/2) )-approximation algorithm for the Koopmans--Beckman version of the problem, improving upon the result of Barvinok. Then, we introduce a new family of instances satisfying “tensor triangle inequalities” and give a constant factor approximation algorithm for them. We show that many classical optimization problems can be modeled by q -adic assignment problems from this family. Finally, we give several integrality gap examples for the natural LP relaxations of the problem.

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.

Mixed integer quadratic programs with quadratic constraints (MIQQP) occur frequently in various areas of engineering practice and management science, but most solution methods for this kind of problems are often designed for its special cases. In this paper, we present a simple global optimization algorithm for solving problem (MIQQP). We first convert problem (MIQQP) into an equivalent generalized bilinear programming problem with integer variables (EIQQP). We next show that replacing the quadratic objective and constraint functions with their convex envelopes is dominated by an alternative methodology based on convexifying the range of the bilinear terms on the feasible region. Finally, by incorporating the reduction-correction techniques and sampling strategies into the branch and bound scheme, the proposed algorithm is developed for solving (MIQQP). Convergence and optimality of the algorithm are presented and numerical examples taken from some recent literature and MINLPLib2 are carried out to validate the performance of the proposed algorithm.

In this paper we study the complexity and domination analysis in the context of the \emph{bipartite quadratic assignment problem}. Two variants of the problem, denoted by BQAP1 and BQAP2, are investigated. A formula for calculating the average objective function value $\mathcal{A}$ of all solutions is presented whereas computing the median objective function value is shown to be NP-hard. We show that any heuristic algorithm that produces a solution with objective function value at most $\mathcal{A}$ has the domination ratio at least $\frac{1}{mn}$. Analogous results for the standard \emph{quadratic assignment problem} is an open question. We show that computing a solution whose objective function value is no worse than that of $n^mm^n-{\lceil\frac{n}{\alpha}\rceil}^{\lceil\frac{m}{\alpha}\rceil}{\lceil\frac{m}{\alpha}\rceil}^{\lceil\frac{n}{\alpha}\rceil}$ solutions of BQAP1 or $m^mn^n-{\lceil\frac{m}{\alpha}\rceil}^{\lceil\frac{m}{\alpha}\rceil}{\lceil\frac{n}{\alpha}\rceil}^{\lceil\frac{n}{\alpha}\rceil}$ solutions of BQAP2, is NP-hard for any fixed natural numbers $a$ and $b$ such that $\alpha=\frac{a}{b}>1$. However, a solution with the domination number $\Omega(m^{n-1}n^{m-1}+m^{n+1}n+mn^{m+1})$ for BQAP1 and $\Omega(m^{m-1}n^{n-1}+m^2n^{n}+m^mn^2)$ for BQAP2, can be found in $O(m^3n^3)$ time.

A new location problem is formulated and solved. It is the continuous version of the grey pattern problem which is a special case of the Quadratic Assignment Problem. The problem is a minimization of a convex function subject to non-convex constraints and has infinitely many optimal solutions. We propose several mathematical programming formulations that are suitable for a multi-start heuristic algorithm. In addition to solving these formulations by the Solver in Excel and Mathematica, a special Nelder–Mead algorithm is proposed. This special algorithm provided the best results. One suggested modification may improve the performance of the Nelder–Mead algorithm for other optimization problems as well.

LibCoopt is an open-source matlab code library which provides a general and convenient tool to approximately solve the combinatorial optimization problems on the set of partial permutation matrices, which are frequently encountered in computer vision, bioinformatics, social analysis, etc. To use the library, the user needs only to give the objective function and its gradient function associated with the problem. Two typical problems, the subgraph matching problem and the quadratic assignment problem, are employed to illustrate how to use the library and also its flexibility on different types of problems.

This paper presents an artificial neural network to solve the quadratic zero-one programming problems under linear constraints. In this paper, by using the connection between integer and nonlinear programming, the quadratic zero-one programming problem is transformed into the quadratic programming problem with nonlinear constraints. Then, by using the nonlinear complementarity problem (NCP) function and penalty method this problem is transformed into an unconstrained optimization problem. It is shown that the Hessian matrix of the associated function in the unconstrained optimization problem is positive definite in the optimal point. To solve the unconstrained optimization problem an artificial neural network is used. The proposed neural network has a simple structure and a low complexity of implementation. It is shown here that the proposed artificial neural network is stable in the sense of Lyapunov. Finally, some numerical examples are given to show that the proposed model finds the optimal solution of this problem in the low convergence time.

A new repeated iterated tabu search for the quadratic assignment problem is developed. A comparative study of this algorithm and the best existing algorithms for solving this problem has shown its competitiveness with respect to both its performance and quality of solutions.

In this article, we will examine whether the Problem Aware Local Search, an efficient method initially proposed for the DNA Fragment Assembly Problem, can also be used in other application domains and with other optimization problems. The main idea is to maintain the key features of PALS and apply it to different permutation-based combinatorial problems. In order to carry out a comprehensive analysis, we use a wide benchmark of well-known problems with different kinds of variation operators and fitness functions, such as the Quadratic Assignment Problem, the Flow Shop Scheduling Problem, and the Multiple Knapsack Problem. We also discuss the main design alternatives for building an efficient and accurate version of PALS for these problems in a competitive manner. In general, the results show that PALS can achieve high-quality solutions for these problems and do it efficiently.

Definition of the equivalence between assignments and edit paths.Graph edit distance formulation as a quadratic assignment problem.New quadratic cost function for computing graph edit distance.Improvement of the accuracy of the approximation of graph edit distance.Approximation computable in reasonable time. The Graph Edit Distance (GED) is a flexible measure of dissimilarity between graphs which arises in error-correcting graph matching. It is defined from an optimal sequence of edit operations (edit path) transforming one graph into another. Unfortunately, the exact computation of this measure is NP-hard. In the last decade, several approaches were proposed to approximate the GED in polynomial time, mainly by solving linear programming problems. Among them, the bipartite GED received much attention. It is deduced from a linear sum assignment of the nodes of the two graphs, which can be efficiently computed by Hungarian-type algorithms. However, edit operations on nodes and edges are not handled simultaneously, which limits the accuracy of the approximation. To overcome this limitation, we propose to extend the linear assignment model to a quadratic one. This is achieved through the definition of a family of edit paths induced by assignments between nodes. We formally show that the GED, restricted to the paths in this family, is equivalent to a quadratic assignment problem. Since this problem is NP-hard, we propose to compute an approximate solution by adapting two algorithms: Integer Projected Fixed Point method and Graduated Non Convexity and Concavity Procedure. Experiments show that the proposed approach is generally able to reach a more accurate approximation of the exact GED than the bipartite GED, with a computational cost that is still affordable for graphs of non trivial sizes.

In this paper we study the effect of a decision maker׳s risk attitude on the median and center problems, two well-known location problems, with uncertain demand in the mean–variance framework. We provide a mathematical programming formulation for both problems in the form of quadratic programming and develop solution procedures. In particular, we consider the vertex and absolute median problems separately, and identify a dominant set for the center problem. Glover׳s linearization method is applied to solve the vertex median problem. We also develop a branch and bound algorithm and a heuristic as the linearization technique takes too long for the vertex median problem on large networks. A computational experiment is conducted to compare the performance of the algorithms. We demonstrate the importance of taking into account the volatility and correlation structure when a location decision is made. The closest assignment property is also discussed for these location problems under the mean–variance objective.

We consider the boolean quadratic programming problem with generalized upper bound constraints (BQP-GUB) which subsumes the well-known quadratic semi-assignment problem. BQP-GUB has applications in engineering, production planning and biology. We present various complexity results on the problem along with different metaheuristic algorithms. Results of extensive experimental analysis are presented demonstrating the efficacy of our proposed algorithms.

A new version of BLS algorithm is introduced.Levenshtein Distance (LD) metric is applied to the BLS heuristic.The QAP is optimized with several processors using OpenMP.Significant improvements are obtained.The BLS-OpenMP can be reported as one of the best heuristics for the QAP. The Quadratic Assignment Problem (QAP) is one of the most challenging NP-Hard combinatorial optimization problems. Circuit-layout design, transportation/traffic engineering, and assigning gates to airplanes are some of the interesting applications of the QAP. In this study, we introduce an enhanced version of a recent local search heuristic, Breakout Local Search Algorithm (BLS), by using the Levenshtein Distance metric for checking the similarity of the new starting points to previously explored QAP permutations. The similarity-checking process prevents the local search algorithm, BLS, from getting stuck in already-explored areas. In addition, the proposed BLS Algorithm (BLS-OpenMP) incorporates multi-threaded computation using OpenMP. The stagnation-aware search for the optimal solutions of the QAP is executed concurrently on several cores with diversified trajectories while considering their similarity to already-discovered local optima. The exploration of the search space is improved by selecting the starting points intelligently and speeding up the fitness evaluations linearly with number of processors/threads. BLS-OpenMP has been tested on the hardest 59 problem instances of the QAPLIB, and it obtained 57 of the best known results. The overall deviation of the achieved solutions from the best known results is 0.019% on average, which is a significant improvement compared with state-of-the-art algorithms.

In this paper, we address continuous, integer and combinatorial k-sum optimization problems. We analyze different formulations of this problem that allow to solve it through the minimization of a relatively small number of minisum optimization problems. This approach provides a general tool for solving a variety of k-sum optimization problems and at the same time, improves the complexity bounds of many ad-hoc algorithms previously reported in the literature for particular versions of this problem. Moreover, the results developed for k-sum optimization have been extended to the more general case of the convex ordered median problem, improving upon existing solution approaches.

Combinatorially complex problems are often optimised with heuristic solvers which generally provide acceptable results but no indication as to how the quality achieved compares to the best possible. In previous work we have introduced Predictive Diagnostic Optimisation (PDO), a heuristic based on local search that provides information about the search space structure through a set of indicators whilst searching for the optimal solution. PDO can collect useful information about the search process, such as the variation in the number of steps needed to locally optimise a random solution and the error between the expected and actual qualities of the local optimum, known as the prediction error. Given previous experimental results on the quadratic assignment problem, it appears that a high prediction error coincides with lower search quality and vice versa. This work confirms this assumption with the help of two additional problems but also shows that the reliability of the prediction error is challenged by structural properties that lead to a homogeneity of the optima basins. Conversely, a high variation in the number of steps that lead to the local optima increases the reliability of the prediction error as an indicator of search quality.

The generalized quadratic assignment problem (GQAP) is the task of assigning a set of facilities to a set of locations such that the sum of the assignment and transportation costs is minimized. The facilities may have different space requirements, and the locations may have varying space capacities. Also, multiple facilities may be assigned to each location such that space capacity is not exceeded. In this paper, an application of the GQAP is presented for assigning a set of machines to a set of locations on the plant floor. Construction algorithms and a simple tabu search heuristic are developed for the GQAP. A set of test problems available in the literature was used to evaluate the performances of the TS heuristic using different construction algorithms. The results show that the simple TS heuristic is effective for solving the GQAP.

In this paper we propose a new lower bounding procedure for the Quadratic Assignment Problem based on a generalization of the well-known Gilomore-Lawler procedure for a higher order reformulation. Computational results on some benchmark instances show the strength of the new approach compared with other lower bounds.

We review some complexity results and present a viable heuristic approach based on the Variable Neighborhood Search (VNS) framework for multimode extension of combinatorial optimization problems, such as the the Set Covering Problem (SCP) and the Covering Location Problem (CLP).