scholarly journals Constrained Nets for Graph Matching and Other Quadratic Assignment Problems

1991 ◽  
Vol 3 (2) ◽  
pp. 268-281 ◽  
Author(s):  
Petar D. Simić
Keyword(s):  
Graph Matching ◽  
Optimization Problems ◽  
Quadratic Assignment Problem ◽  
Semiclassical Approximation ◽  
Geometric Optimization ◽  
Elastic Net ◽  
Physical Models ◽  
Quadratic Assignment ◽  
Matching Problem ◽  
Special Case

Some time ago Durbin and Willshaw proposed an interesting parallel algorithm (the “elastic net”) for approximately solving some geometric optimization problems, such as the Traveling Salesman Problem. Recently it has been shown that their algorithm is related to neural networks of Hopfield and Tank, and that they both can be understood as the semiclassical approximation to statistical mechanics of related physical models. The main point of the elastic net algorithm is seen to be in the way one deals with the constraints when evaluating the effective cost function (free energy in the thermodynamic analogy), and not in its geometric foundation emphasized originally by Durbin and Willshaw. As a consequence, the elastic net algorithm is a special case of the more general physically based computations and can be generalized to a large class of nongeometric problems. In this paper we further elaborate on this observation, and generalize the elastic net to the quadratic assignment problem. We work out in detail its special case, the graph matching problem, because it is an important problem with many applications in computational vision and neural modeling. Simulation results on random graphs, and on structured (hand-designed) graphs of moderate size (20-100 nodes) are discussed.

scholarly journals A NOVEL APPROACH FOR PART BASED OBJECT MATCHING USING DISTANCE METRIC LEARNING WITH GRAPH CONVOLUTIONAL NETWORKS

2021 ◽  
Vol XLIV-2/W1-2021 ◽  
pp. 149-154
Author(s):  
V. Kozlov ◽  
A. Maysuradze
Keyword(s):  
Object Representation ◽  
Graph Matching ◽  
Quadratic Assignment Problem ◽  
Metric Learning ◽  
Quadratic Assignment ◽  
Distance Metric ◽  
Matching Problem ◽  
Matching Problems ◽  
Convolutional Networks ◽  
Novel Approach

Abstract. Part-based object representation and part matching problem often appear in various areas of data analysis. A special case of particular interest is when parts are not fully separated, but in relations with each other. The natural way to model such objects are graphs, and part matching problem becomes graph matching problem. Over the years, many methods to solve graph matching problems have been proposed, but it remains relevant due to its complexity. We propose a novel approach to solving graph matching problem based on learning distance metric on graph vertices. We empirically demonstrate that our method outperforms traditional methods based on solving quadratic assignment problem. We also provide an theoretical estimation of computational complexity of proposed method.

scholarly journals Garden optimization problems for benchmarking quantum annealers

2021 ◽  
Vol 20 (9) ◽  
Author(s):  
Carlos D. Gonzalez Calaza ◽  
Dennis Willsch ◽  
Kristel Michielsen
Keyword(s):  
Real World ◽  
Assignment Problem ◽  
Optimization Problems ◽  
Quadratic Assignment Problem ◽  
Quadratic Assignment ◽  
New Class ◽  
Qubit System ◽  
Hybrid Solver ◽  
Qubo Formulation ◽  
D Wave

AbstractWe benchmark the 5000+ qubit system coupled with the Hybrid Solver Service 2 released by D-Wave Systems Inc. in September 2020 by using a new class of optimization problems called garden optimization problems known in companion planting. These problems are scalable to an arbitrarily large number of variables and intuitively find application in real-world scenarios. We derive their QUBO formulation and illustrate their relation to the quadratic assignment problem. We demonstrate that the system and the new hybrid solver can solve larger problems in less time than their predecessors. However, we also show that the solvers based on the 2000+ qubit system sometimes produce more favourable results if they can solve the problems.

scholarly journals A Methodology to Find the Elementary Landscape Decomposition of Combinatorial Optimization Problems

2011 ◽  
Vol 19 (4) ◽  
pp. 597-637 ◽  
Author(s):  
Francisco Chicano ◽  
L. Darrell Whitley ◽  
Enrique Alba
Keyword(s):  
Combinatorial Optimization ◽  
Objective Function ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Quadratic Assignment Problem ◽  
Combinatorial Optimization Problem ◽  
Quadratic Assignment ◽  
Neighborhood Structure ◽  
Combinatorial Optimization Problems ◽  
Elementary Landscape

A small number of combinatorial optimization problems have search spaces that correspond to elementary landscapes, where the objective function f is an eigenfunction of the Laplacian that describes the neighborhood structure of the search space. Many problems are not elementary; however, the objective function of a combinatorial optimization problem can always be expressed as a superposition of multiple elementary landscapes if the underlying neighborhood used is symmetric. This paper presents theoretical results that provide the foundation for algebraic methods that can be used to decompose the objective function of an arbitrary combinatorial optimization problem into a sum of subfunctions, where each subfunction is an elementary landscape. Many steps of this process can be automated, and indeed a software tool could be developed that assists the researcher in finding a landscape decomposition. This methodology is then used to show that the subset sum problem is a superposition of two elementary landscapes, and to show that the quadratic assignment problem is a superposition of three elementary landscapes.

scholarly journals Solving the Quadratic Assignment Problem using the Swallow Swarm Optimization Problem

2019 ◽  
Vol 8 (6) ◽  
pp. 3116-3120
Keyword(s):  
Assignment Problem ◽  
Optimization Problems ◽  
Quadratic Assignment Problem ◽  
Combinatorial Problems ◽  
Computational Method ◽  
Quadratic Assignment ◽  
Swarm Optimization ◽  
Natural Behavior ◽  
Combinatorial Optimization Problems ◽  
Benchmark Instances

In recent years, there is a growing interest in swarm intelligent algorithms inspired by the observation of the natural behavior of swarm to define a computational method, which may resolve the hardest combinatorial optimization problems. The Quadratic Assignment Problem is one of the well-known combinatorial problems, which simulate with the assignment problem in several domains such as the industrial domain. This paper proposes an adaptation of a recent algorithm called the swallow swarm optimization to solve the Quadratic Assignment Problem; this algorithm is characterized by a hierarchy of search who allow it to search in a totality of research space. The obtained results in solving some benchmark instances from QAPLIB are compared with those obtained from other know metaheuristics in other to evaluate the performance of the proposed adaptation.

scholarly journals ITERATED TABU SEARCH: AN IMPROVEMENT TO STANDARD TABU SEARCH

2006 ◽  
Vol 35 (3) ◽  
Author(s):  
Alfonsas Misevicius ◽  
Antanas Lenkevicius ◽  
Dalius Rubliauskas
Keyword(s):  
Combinatorial Optimization ◽  
Tabu Search ◽  
High Efficiency ◽  
Optimization Problems ◽  
Quadratic Assignment Problem ◽  
Special Kind ◽  
Quadratic Assignment ◽  
Combinatorial Optimization Problems ◽  
Diversification Mechanism ◽  
Iterated Tabu Search

The goal of this paper is to discuss the tabu search (TS) meta-heuristic and its enhancement for combinatorial optimization problems. Firstly, the issues related to the principles and specific features of the standard TS are concerned. Further, a promising extension to the classical tabu search scheme is introduced. The most important component of this extension is a special kind of diversification mechanism. We give the paradigm of this new improved TS strategy, which is called an iterated tabu search (ITS). ITS was applied to the difficult combinatorial optimization problems, the traveling salesman problem (TSP) and the quadratic assignment problem (QAP). The results of the experiments with the TSP and QAP show the high efficiency of the ITS strategy. The outstanding performance of ITS is also demonstrated by the fact that the new record-breaking solutions were found for the hard QAP instances - tai80a and tai100a.

scholarly journals Algorithms for exact solutions to combinatorial optimization problems

2013 ◽  
Author(s):  
Θεόδωρος Γκεβεζές
Keyword(s):  
Integer Programming ◽  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Quadratic Assignment Problem ◽  
Quadratic Assignment ◽  
Search Procedure ◽  
Np Hard ◽  
Adaptive Search ◽  
Programming Formulation ◽  
Combinatorial Optimization Problems

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.

scholarly journals Heuristic Approaches to Stochastic Quadratic Assignment Problem: VaR and CVar Cases

MENDEL ◽  
2017 ◽  
Vol 23 (1) ◽  
pp. 73-78 ◽  
Author(s):  
Radomil Matousek ◽  
Pavel Popela ◽  
Jakub Kudela
Keyword(s):  
At Risk ◽  
Assignment Problem ◽  
Value At Risk ◽  
Optimization Problems ◽  
Quadratic Assignment Problem ◽  
Risk Measures ◽  
Conditional Value At Risk ◽  
Quadratic Assignment ◽  
Heuristic Approaches ◽  
Combinatorial Optimization Problems

The goal of this paper is to continue our investigation of the heuristic approaches of solving thestochastic quadratic assignment problem (StoQAP) and provide additional insight into the behavior of di erentformulations that arise through the stochastic nature of the problem. The deterministic Quadratic AssignmentProblem (QAP) belongs to a class of well-known hard combinatorial optimization problems. Working with severalreal-world applications we have found that their QAP parameters can (and should) be considered as stochasticones. Thus, we review the StoQAP as a stochastic program and discuss its suitable deterministic reformulations.The two formulations we are going to investigate include two of the most used risk measures - Value at Risk(VaR) and Conditional Value at Risk (CVaR). The focus is on VaR and CVaR formulations and results of testcomputations for various instances of StoQAP solved by a genetic algorithm, which are presented and discussed.

scholarly journals A Hybrid Ant Colony Algorithm for Quadratic Assignment Problem

2013 ◽  
Vol 7 (1) ◽  
pp. 51-54 ◽  
Author(s):  
Guo Hong
Keyword(s):  
Ant Colony Optimization ◽  
Assignment Problem ◽  
Large Scale ◽  
Ant Colony Algorithm ◽  
Optimization Problems ◽  
Quadratic Assignment Problem ◽  
Ant Colony ◽  
Quadratic Assignment ◽  
Optimal Solutions ◽  
Combinatorial Optimization Problems

Quadratic assignment problem (QAP) is one of fundamental combinatorial optimization problems in many fields. Many real world applications such as backboard wiring, typewriter keyboard design and scheduling can be formulated as QAPs. Ant colony algorithm is a multi-agent system inspired by behaviors of real ant colonies to solve optimization problems. Ant colony optimization (ACO) is one of new bionic optimization algorithms and it has some characteristics such as parallel, positive feedback and better performances. ACO has achieved in solving quadratic assignment problems. However, its solution quality and its computation performance need be improved for a large scale QAP. In this paper, a hybrid ant colony optimization (HACO) has been proposed based on ACO and particle swarm optimization (PSO) for a large scale QAP. PSO algorithm is combined with ACO algorithm to improve the quality of optimal solutions. Simulation experiments on QAP standard test data show that optimal solutions of HACO are better than those of ACO for QAP.

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