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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

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.

Majority of chapters of this book show usefulness of greedy like algorithms for solving various combinatorial optimization problems. The aim of this chapter is to warn the reader that not always a greedy like approach is a good option and, in certain cases, it is a very bad option being sometimes among the worst possible options. Our message is not a discouragement from using greedy like algorithms altogether; we think that for every combinatorial optimization problem of importance, researchers and practitioners should simply investigate the appropriateness of greedy like algorithms and the existence of better alternatives to them (considering both quality of solution and running time). In many cases, especially when the running time must be very short, the conclusion may still be that the most practical of known approaches is a greedy like algorithm. The Traveling Salesman and Multidimensional Assignment Problems are optimization problems for which greedy like approaches are usually not very successful. We demonstrate this by providing both theoretical and experimental results on greedy like algorithms as well as on some other algorithms that produce (in theory and/or in experiments) much better results without spending significantly more time. There are some general theoretical results that indicate that there are, in fact, many combinatorial optimization problems for which greedy like algorithms are not the best option even among fast construction heuristics, see, e.g., [3, 5, 17]. We will not consider these general results in order to avoid most mathematical details that are not necessary for understanding the results of this chapter. For this reason we will not give proofs here apart from two simple proofs: that of Theorem 8 which shows that some instances on which the greedy algorithm fails are not exotic in a sense and that of Theorem 11 since Theorem 11 is a new result. It is not a trivial question whether a certain algorithm is greedy like or not. In the next section we define an independence system and give the classic definition of the greedy algorithm for such a system. We extend this definition to so-called greedy type algorithms that include such well-known algorithms as the Prim’s algorithm for the minimum spanning tree problem and the nearest neighbor algorithm for the traveling salesman problem. We use the term ‘greedy like’ in an informal way and we include in this class simple and fast construction heuristics that seem to us to be of greedy nature. O pe n A cc es s D at ab as e w w w .in te ch w eb .o rg

There has always been a need to solve real-life large-scale problems, suchas efficiently allocating limited resources, and other complex and conflicting situations related to combinatorial optimization genre. A class of combinato- rial optimization problems is NP-hard and, among many well-known, several of them are assignment, planning and rescheduling problems. Assignment problems can deal with optimal assignment of teams of collaborating agents; planning problems can be effects-based planning that search for promising plans to get desired end states with minimal cost; rescheduling problems can be multi-criteria optimization of rescheduling resources that modify existing original schedule. These large scale optimization problems are complex with intractable and highly complex search spaces. Currently, there are no known algorithms with polynomial time complexity, which can solve these problems. Genetic Algorithms have been successfully applied to solve many complex optimization problems but not to the specific problems mentioned above.The aim of the research, presented in this thesis, is to use Genetic Algo- rithms for large scale optimization of assignment, planning and rescheduling problems. More specifically, the contributions of the thesis are to: (i) adapt existing and develop new efficient Genetic Algorithms to solve large scale as- signment problems, and (ii) adapt existing Genetic Algorithms to solve large scale effects-based planning, and multi-objective rescheduling optimization problems.In case of assignment, we solve a team assignment problem and investigate specific regions in a solution space for assignment problems with huge search spaces.For the team assignment, an existing Genetic Algorithm is adapted and applied for optimal assignment of tasks to teams of collaborating agents. The algorithm is scalable, stable, robust and produces a near optimal solution. The results of the team assignment problem show that the existing Genetic Algorithms are not efficient for optimal assignment of tasks to teams of agents. Hence, to solve larger instances of the problem efficiently, new Genetic Algo- rithms are developed with emphasis on the construction of crossover opera- tors. Since teams assignment can be multi-criteria, a multi-objective model is constructed and two widely used multi-objective evolutionary algorithms are applied. Further, for the assignment problems with huge search spaces, an existing Genetic Algorithm is adapted to extract possible combinations of input parameters from a specified solution space region. To solve the large scale effects-based planning, a multi-objective optimization problem is formu- lated for the evaluation of operational plans and a multi-objective Genetic Algorithm is adapted and applied to the problem. The results show that the suggested algorithm is much more efficient than A*. For the rescheduling problem, a multi-objective optimization model for rescheduling of resources is proposed and a multi-objective Genetic Algorithm is adapted and applied to obtain the Pareto-optimal solutions.The research presented in this thesis confirms that Genetic Algorithms can be used for large scale assignment, planning and rescheduling problems since they have shown to be suitable in solving these problems efficiently.

This paper aims to highlight the motivations for investigating genetic algorithms to solve the DNA Fragments Assembly problem (DNA_FA). DNA_FA is an optimization problem that attempts to reconstruct the original DNA sequence by finding the shortest DNA sequence from a given set of fragments. We showed that the DNA_FA optimization problem is a special case of the two well-known optimization problems: The Traveling Salesman Problem (TSP) and the Quadratic Assignment Problem (QAP). TSP and QAP are important problems in the field of combinatorial optimization and for which there exists an abundant literature. Genetic Algorithms (GA) applied to these problems have led to very satisfactory results in practice. In the perspective of designing efficient genetic algorithms to solve DNA_FA we showed the existence of a polynomial-time reduction of DNA-FA into TSP and QAP enabling us to point out some technical similarities in terms of solutions and search space complexity. We then conceptually designed a genetic algorithm platform for solving the DNA-FA problem inspired from the existing efficient genetic algorithms in the literature solving TSP and QAP problems. This platform offers several ingredients enabling us to create several variants of GA solvers for the DNA assembly optimization problems.

This paper studies a specific class of multiobjective combinatorial optimization problems (MOCOPs), namely the permutation-based MOCOPs. Many commonly seen MOCOPs, e.g., multiobjective traveling salesman problem (MOTSP), multiobjective project scheduling problem (MOPSP), belong to this problem class and they can be very different. However, as the permutation-based MOCOPs share the inherent similarity that the structure of their search space is usually in the shape of a permutation tree, this paper proposes a generic multiobjective set-based particle swarm optimization methodology based on decomposition, termed MS-PSO/D. In order to coordinate with the property of permutation-based MOCOPs, MS-PSO/D utilizes an element-based representation and a constructive approach. Through this, feasible solutions under constraints can be generated step by step following the permutation-tree-shaped structure. And problem-related heuristic information is introduced in the constructive approach for efficiency. In order to address the multiobjective optimization issues, the decomposition strategy is employed, in which the problem is converted into multiple single-objective subproblems according to a set of weight vectors. Besides, a flexible mechanism for diversity control is provided in MS-PSO/D. Extensive experiments have been conducted to study MS-PSO/D on two permutation-based MOCOPs, namely the MOTSP and the MOPSP. Experimental results validate that the proposed methodology is promising.

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 bat algorithm is one of the recent nature-inspired algorithms, which has been emerged as a powerful search method for solving continuous as well as discrete problems. The quadratic assignment problem is a well-known NP-hard problem in combinatorial optimization. The goal of this problem is to assign n facilities to n locations in such a way as to minimize the assignment cost. For that purpose, this paper introduces a novel discrete variant of bat algorithm to deal with this combinatorial optimization problem. The proposed algorithm was evaluated on a set of benchmark instances from the QAPLIB library and the performance was compared to other algorithms. The empirical results of exhaustive experiments were promising and illustrated the efficacy of the suggested approach.

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.

Bi-level optimization problems (BOPs) are a class of challenging problems with two levels of optimization tasks. The main goal is to optimize the upper level problem which has another optimization problem as a constraint. In these problems, the optimal solutions to the lower level problem become possible feasible candidates to the upper level one. Such a requirement makes the optimization problem difficult to solve, and has kept the researchers busy towards devising methodologies, which can efficiently handle the problem. Recently, a new research field, called EBO (Evolutionary Bi-Level Optimization) has appeared thanks to the promising results obtained by the use of EAs (Evolutionary Algorithms) to solve such kind of problems. However, most of these promising results are restricted to the continuous case. The number of existing EBO works for the discrete (combinatorial case) bi-level problems is relatively small when compared to the field of evolutionary continuous BOP. Motivated by this observation, we have recently proposed a Co-evolutionary Decomposition-Based Algorithm (CODBA) to solve combinatorial bi-level problems. The recently proposed approach applies a Genetic Algorithm to handle BOPs. Besides, a new recently proposed meta-heuristic called CRO has been successfully applied to several practical NP-hard problems. To this end, we propose in this work a CODBA-CRO (CODBA with Chemical Reaction Optimization) to solve BOP. The experimental comparisons against other works within this research area on a variety of benchmark problems involving various difficulties show the effectiveness and the efficiency of our proposal.

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.

This paper presents GHOST , a combinatorial optimization framework that a real-time strategy (RTS) AI developer can use to model and solve any problem encoded as a constraint satisfaction/optimization problem (CSP/COP). We show a way to model three different problems as a CSP/COP, using instances from the RTS game StarCraft as test beds. Each problem belongs to a specific level of abstraction (the target selection as reactive control problem, the wall-in as a tactics problem, and the build order planning as a strategy problem). In our experiments, GHOST shows good results computed within some tens of milliseconds. We also show that GHOST outperforms state-of-the-art constraint solvers, matching them on the resources allocation problem, a common combinatorial optimization problem.

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).

This paper proposes a method for visualizing combinatorial search spaces named low-dimensional Euclidean embedding (LDEE). The proposed method transforms the original search space, such as a set of permutations or binary vectors, to $\pmb {\mathbb {R}}^{k}$ (with ${k} {=} {2}$ or 3 in practice) while aiming to preserve spatial relationships existing in the original space. The LDEE method uses the t-distributed stochastic neighbor embedding (t-SNE) to transform solutions from the original search space to the Euclidean one. In this paper, it is mathematically shown that the assumptions underlying the t-SNE method are valid in the case of permutation spaces with the Mallows distribution. The same is true for other metric spaces provided that the distribution of points is assumed to be normal with respect to the adopted metric. The embedding obtained using t-SNE is further refined to ensure visual separation of individual solutions. The visualization obtained using the LDEE method can be used for analyzing the behavior of the population in a population-based metaheuristic, the working of the genetic operators, etc. Examples of visualizations obtained using this method for the four peaks problem, the firefighter problem, the knapsack problem, the quadratic assignment problem, and the traveling salesman problem are presented in this paper.

Leonhard Euler introduced the concept of Graph Theory in his paper about the seven bridges of Konigsberg published in 1736. It is the study of pair-wise relationships between objects. Each object is represented using a vertex, and in case of a relationship between a pair of vertices, they will be connected using an edge. In this dissertation, graph theory is used to study several important combinatorial optimization problems. In chapter 2, we study the multi-dimensional assignment problem using its underlying hypergraphs. It will be shown how the MAP can be represented by a k-partite graph and how any solution to MAP is associated to a k-clique in the respective k-partite graph. Two heuristics are proposed to solve the MAP and computational studies are performed to compare the performance of the proposed methods with exact solutions. On the heels of chapter 2, a new branch-and-bound method is proposed to solve the problem of finding all k-cliques in a k-partite graph in chapter 3. The new method utilizes bitsets as the datastructure to represent graph data. A new pruning method is introduced in BitCLQ, and CPU instructions are used to improve the performance of the branch-and-bound method. BitCLQ gains up to 300% speed up over existing methods. In chapter 4, two new heuristics to solve decomposable cost MAP’s are proposed. The proposed heuristic are based on the partitioning of the underlying graph representing the MAP. In the first heuristic method, MAP’s are partitioned into several smaller MAP’s with the same dimensial

The idea of k-adaptability in two-stage robust optimization is to calculate a fixed number k of second-stage policies here-and-now. After the actual scenario is revealed, the best of these policies is selected. This idea leads to a min---max---min problem. In this paper, we consider the case where no first stage variables exist and propose to use this approach to solve combinatorial optimization problems with uncertainty in the objective function. We investigate the complexity of this special case for convex uncertainty sets. We first show that the min---max---min problem is as easy as the underlying certain problem if k is greater than the number of variables and if we can optimize a linear function over the uncertainty set in polynomial time. We also provide an exact and practical oracle-based algorithm to solve the latter problem for any underlying combinatorial problem. On the other hand, we prove that the min---max---min problem is NP-hard for every fixed number k, even when the uncertainty set is a polyhedron, given by an inner description. For the case that k is smaller or equal to the number of variables, we finally propose a fast heuristic algorithm and evaluate its performance.

In the classical min-max approach to robust combinatorial optimization, a single feasible solution is computed that optimizes the worst case over a given set of considered scenarios. As is well known, this approach is very conservative, leading to solutions that in the average case are far from being optimal. In this paper, we present a different approach: the objective is to compute k feasible solutions such that the best of these solutions for each given scenario is worst-case optimal, i.e., we model the problem as a min-max-min problem. In practice, these k solutions can be computed once in a preprocessing phase, while choosing the best of the k solutions can be done in real time for each scenario occuring. Using a polynomial-time oracle algorithm, we show that the problem of choosing k min-max-min optimal solutions is as easy as the underlying combinatorial problem if k ≥ n + 1 and if the uncertainty set is a polytope or an ellipsoid. On contrary, in the discrete scenario case, many tractable problems such as the shortest path problem or the minimum spanning tree problem turn NP-hard in the new approach.

Hyper-heuristics are optimization techniques for solving hard combinatorial problems. Their main feature is that their design involves an important decoupling of the search components from the problem domain ones. This allows them to extend their applicability to different problem domains without major redesign, unlike traditional methods such as metaheuristics. In this work, a hyper-heuristic is evaluated for a transportation problem. The implemented hyper-heuristic uses a greedy operator, and it implements an adapter layer that would allow it to be used in other similar problems. Experimental results shows balanced solution quality and CPU time performance, regarding other metaheuristics in literature.

Linear unconstrained problem of combinatorial optimization on arrangements under stochastic uncertainty is being solved. The minimum is defined as the result of sequential comparison of numerical characteristics of random variables. The properties of the solution of the optimization problem under study are obtained. These properties use the properties of special constructed deterministic problems. The authors also propose the reduction method to solve linear unconstrained problem of combinatorial stochastic optimization, which is based on obtained solution's properties.

A generic scheme is proposed for designing and classifying simple probabilistic population-based optimization (SPPBO) algorithms that use principles from population-based ant colony optimization (PACO) and simplified swarm optimization (SSO) for solving combinatorial optimization problems. The scheme, called SPPBO, identifies different types of populations (or archives) and their influence on the construction of new solutions. The scheme is used to show how SSO can be adapted for solving combinatorial optimization problems and how it is related to PACO. Moreover, several new variants and combinations of these two metaheuristics are generated with the proposed scheme. An experimental study is done to evaluate and compare the influence of different population types on the optimization behavior of SPPBO algorithms, when applied to the traveling salesperson problem and the quadratic assignment problem.

The real world problems in the supply-chain domain are generally constrained and combinatorial in nature. Several nature-/bio-/socio-inspired metaheuristic methods have been proposed so far solving such problems. An emerging metaheuristic methodology referred to as Cohort Intelligence (CI) in the socio-inspired optimization domain is applied in order to solve three selected combinatorial optimization problems. The problems considered include a new variant of the assignment problem which has applications in healthcare and inventory management, a sea-cargo mix problem and a cross-border shipper selection problem. In each case, we use two benchmarks for evaluating the effectiveness of the CI method in identifying optimal solutions. To assess the quality of solutions obtained by using CI, we do comparative testing of its performance against solutions generated by using CPLEX. Furthermore, we also compare the performance of the CI method to that of specialized multi-random-start local search optimization methods that can be used to find solutions to these problems. The results are robust with a reasonable computational time and accuracy.

The paper establishes the necessary condition for the solution of a linear unconditional problem of combinatorial optimization on arrangements where coefficients of objective function are positive. These results are used to establish the properties of the solution of linear unconditional optimization problem on arrangements for the case where probabilistic uncertainty takes place in the definition of the feasible domain and the minimum is defined according to the linear order introduced on the set of discrete random variables: we formulate and prove the condition that can underlie the search for solution and the ways of constructing the solution in some special cases.

Quadratic Assignment Problems (QAPs) are the hardest of combinatorial optimization problems, with some problems of sizes of the order of 30 still remaining unsolved optimally. Solving these problems with exact optimization methods is cumbersome and hence the use of non-traditional optimization methods is advisable. The discrete version of Particle Swarm Optimization (DPSO) is a widely accepted non-conventional optimization algorithm for combinatorial optimization problems. This paper uses Taguchi's robust design approach to determine the best combination of parameters for discrete particle swarm optimization algorithm for solving quadratic assignment problems.

In this paper, we investigate the energy-efficient resource allocation problem in a single-cell orthogonal frequency division multiple access (OFDMA) system to achieve the energy efficiency (EE) tradeoff among users. Rather than overall system EE, our objective is to maximize the EE for each individual user. Therefore, a multiple-objective optimization problem is formulated, which in general has many Pareto optimal solutions and is hard to solve. To find its solution, we first convert it into two different single-objective optimization problems using the weighted-sum approach and the max-min approach, respectively. The single-objective optimization problems are non-convex due to the combinatorial channel allocation variables. Therefore, for both problems, we first provide an upper bound algorithm by relaxing the combinatorial variables and then develop a suboptimal heuristic algorithm. The sum-of-ratios optimization and the generalized fractional programming are utilized for the weighted-sum problem and the max-min problem, respectively. Numerical results demonstrate that both the weighted-sum and the max-min approaches can effectively solve the EE maximization problem, and the suboptimal heuristic algorithms can achieve a close performance to the corresponding upper bound algorithm.

Optimal flight gate assignment is a highly relevant optimization problem from airport management. Among others, an important goal is the minimization of the total transit time of the passengers. The corresponding objective function is quadratic in the binary decision variables encoding the flight-to-gate assignment. Hence, it is a quadratic assignment problem being hard to solve in general. In this work we investigate the solvability of this problem with a D-Wave quantum annealer. These machines are optimizers for quadratic unconstrained optimization problems (QUBO). Therefore the flight gate assignment problem seems to be well suited for these machines. We use real world data from a mid-sized German airport as well as simulation based data to extract typical instances small enough to be amenable to the D-Wave machine. In order to mitigate precision problems, we employ bin packing on the passenger numbers to reduce the precision requirements of the extracted instances. We find that, for the instances we investigated, the bin packing has little effect on the solution quality. Hence, we were able to solve small problem instances extracted from real data with the D-Wave 2000Q quantum annealer., Comment: Updated figure 2

The cooperative multi-targets assignment for multiple unmanned aerial vehicles (UAV) is a complex combinatorial optimization problem. Multi-UAVs cooperation increases the scale of problems which cause a noticeable increase in task planning time. Moreover, it is difficult to build a unified assignment model because different tasks often require different numbers of UAVs and targets. Besides, the cooperative constraints of multi-UAVs in a three-dimensional environments are more complex than that in a two-dimensional environments, which makes it difficult to obtain an optimal solution. To solve these problems, we present a unified gene coding strategy to handle various models in a consistent framework. Then, a cooperative target assignment algorithm in a three-dimensional environments based on discrete mapping differential evolution is given. First, we use flight path cost to indicate the assignment relationship between the UAV and the target, which turns the optimization problem from discrete space to continuous space, and so the solving process can be simplified. Secondly, in order to obtain reasonable offspring for differential evolution, we map the solution back to the assignment relationship space according to inverse mapping rules. Finally, to avoid falling into a local optimal, a balance between exploration and exploitation is achieved by combining the dynamic crossover rate with the hybrid evolution strategy. The simulation results show that the proposed discrete mapping differential evolution algorithm with the unified gene coding strategy not only effectively solves the cooperative multi-targets assignment problem, but also improves the accuracy of the multi-targets assignment. It is also suitable for solving the large scale problem of assignment.

We consider a general approach to the construction of necessary, sufficient, and necessary and sufficient conditions that allow to ‘adapt’ a known optimal solution of an abstract combinatorial problem with a certain structure to a change in the initial data set for a fixed cost function ‘easily’ from the combinatorial point of view. We call this approach adaptive stability. Apparently, it is the first time that the approach is described for an abstract problem in a rigorous mathematical formalization.

In this paper, we introduce a new concept of hierarchical structure into solution space of a combinatorial optimization problem, and develop a novel combinatorial optimization method based on the concept. The introduction of the above new hierarchical structure concept "basin of attraction," which is a set binding solution by utilizing properties of local optimal solutions, enables us to construe solution space as not only set of solutions but also set of basins of attraction hierarchically. It is well known that the appropriate balance of two policies, intensification and diversification, is essential in the search of metaheuristics. The proposed method clarifies the search policy by relating the hierarchical structure in solution space with intensification and diversification. With regard to diversification, we incorporate a movement strategy that has longer term or more macroscopic viewpoint than before in the algorithm by utilizing the concept basin of attraction. The performance of the proposed combinatorial optimization method is inspected by numerical experiments using some typical benchmark problems of a traveling salesman problem, a knapsack problem, a flow-shop scheduling problem, and a quadratic assignment problem.

We present a hyper-heuristic algorithm for solving combinatorial black-box optimization problems. The algorithm named CMA-VNS stands for a hybrid of variants of Covariance Matrix Adaptation Evolution Strategy (CMA-ES) and Variable Neighborhood Search (VNS). The framework design and the design profiles of variants of CMA-VNS are introduced to enhance the intensification of searching for conventional CMA-ES solvers. We explain the parameter configuration details, the heuristic profile selection, and the rationale of incorporating machine learning methods during the study. Experimental tests and the results of the first and the second Combinatorial Black-Box Optimization Competitions (CB-BOC 2015, 2016) confirmed that CMA-VNS is a competitive hyper-heuristic algorithm.

There are many nature-inspired algorithms being proposed and researched to solve combinatorial optimization problems. The theory of Five-elements in Chinese traditional culture implies a possible approach to solve present-day problems of science and engineering. In this paper, the Five-elements Cycle Model (FECM) is developed based on the mechanism of generation and restriction among five elements, and Five-elements Cycle Optimization (FECO) algorithm is proposed for finding the optimal solution of travelling salesman problems. The performance and parameter comparison of FECO is given by experiments, the comparison with 7 optimization algorithms based on various mechanisms for some TSP instances from TSPLIB are also given, which indicate the availability of FECO.

The development of optimization algorithms for combinatorial problems is a complicated process, both guided and validated by the computational experiments over the different scenarios. Since the number of experiments can be very large and each experiment can take substantial execution time, distributing the load over the cloud speeds up the whole process significantly. In this paper we present the system used for experimental validation and comparison of stochastic combinatorial optimization algorithms, applied in the specific case of project scheduling problems.

Traveling salesman problem (TSP) is a classic combinatorial optimization problem. It is a simplified form of many complex problems. In the process of study and research, it is understood that the parameters that affect the performance of genetic algorithm mainly include the quality of initial population, the population size, and crossover probability and mutation probability values. As a result, an improved genetic algorithm for solving TSP problems is put forward. The population is graded according to individual similarity, and different operations are performed to different levels of individuals. In addition, elitist retention strategy is adopted at each level, and the crossover operator and mutation operator are improved. Several experiments are designed to verify the feasibility of the algorithm. Through the experimental results analysis, it is proved that the improved algorithm can improve the accuracy and efficiency of the solution.

K-coverage optimization is widely used in healthcare environments, which minimizing the number of directional receivers that guarantee a given region is covered k times. As K-coverage optimization is NP-hard, a commonly used approach for finding the optimal solution is integer linear programming. However, it can be slow for many practical instances. In this article, we propose an exact dynamic programming algorithm based on tree-decomposition. A probability-distribution density function is introduced to describe the relative importance of various areas and compute the minimal optimized sub-structures. We also show that this algorithm can be easily extended to provide exact solutions for different coverage ratio requirements. When compared with the ILP approach and greedy algorithm, our algorithm provides accurate solutions without scarifying too much on efficiency.

This paper introduces a new self-tuning mechanism to the local search heuristic for solving of combinatorial optimization problems. Parameter tuning of heuristics makes them difficult to apply, as parameter tuning itself is an optimization problem. For this purpose, a modified local search algorithm free from parameter tuning, called Self-Adaptive Local Search (SALS), is proposed for obtaining qualified solutions to combinatorial problems within reasonable amount of computer times. SALS is applied to several combinatorial optimization problems, namely, classical vehicle routing, permutation flow-shop scheduling, quadratic assignment, and topological design of networks. It is observed that self-adaptive structure of SALS provides implementation simplicity and flexibility to the considered combinatorial optimization problems. Detailed computational studies confirm the performance of SALS on the suit of test problems for each considered problem type especially in terms of solution quality.

In a multi-objective combinatorial optimization (MOCO) problem, multiple objectives must be optimized simultaneously. In past years, several constraint-based algorithms have been proposed for finding Pareto-optimal solutions to MOCO problems that rely on repeated calls to a constraint solver. Understanding the properties of these algorithms and analyzing their performance is an important problem. Previous work has focused on empirical evaluations on benchmark instances. Such evaluations, while important, have their limitations. Our paper adopts a different, purely theoretical approach, which is based on characterizing the search space into subspaces and analyzing the worst-case performance of a MOCO algorithm in terms of the expected number of calls to the underlying constraint solver. We apply the approach to two important constraint-based MOCO algorithms. Our analysis reveals a deep connection between the search mechanism of a constraint solver and the exploration of the search space of a MOCO problem.

Robust combinatorial optimization problems with cardinality constrained uncertainty may be solved by a finite number of nominal problems. In this paper, we show that the number of nominal problems to be solved can be reduced significantly.

Combined with quantum computing and genetic algorithm, quantum genetic algorithm (QGA) shows considerable ability of parallelism. Experiments have shown that QGA performs quite well on TSP, job shop problem and some other typical combinatorial optimization problems. The other problems like nutritional diet which can be transformed into specific combinational optimization problem also can be solved through QGA smoothly. This paper sums up the main points of QGA for general combinatorial optimization problems. These points such as modeling of the problem, qubit decoding and rotation strategy are useful to enhance the convergence speed of QGA and avoid premature at the same time.

Hybrid techniques of combinatorial optimization are a growing research area, designed to solve complex optimization problems. In the first part of this paper, we focus on the methodological background of hybrid techniques of combinatorial optimization, paying special attention to the important concepts in the field of combinatorial optimization and computational complexity theory, as well as to hybridization strategies that are important in the development of hybrid techniques of combinatorial optimization. According to the presented relations among the techniques of combinatorial optimization, the strategies of combining them and the concepts for solving combinatorial optimization problems, this paper presents an example of the hybrid technique for feature selection and classification in credit risk assessment. This study emphasizes the importance of hybridization as a concept of cooperation among metaheuristics and other optimization techniques. The importance of such cooperation is confirmed by the results that are presented in the experimental part of the paper, which were obtained on a German credit dataset using the hybrid technique of combinatorial optimization based on a low-level relay strategy. The experimental results show that the proposed method outperforms, on the same dataset, the methods presented in the literature in terms of the average prediction accuracy.

We introduce the quadratic balanced optimization problem (QBOP) which can be used to model equitable distribution of resources with pairwise interaction. QBOP is strongly NP-hard even if the family of feasible solutions has a very simple structure. Several general purpose exact and heuristic algorithms are presented. Results of extensive computational experiments are reported using randomly generated quadratic knapsack problems as the test bed. These results illustrate the efficacy of our exact and heuristic algorithms. We also show that when the cost matrix is specially structured, QBOP can be solved as a sequence of linear balanced optimization problems. As a consequence, we have several polynomially solvable cases of QBOP.

A novel swarm intelligence approach for combinatorial optimization is proposed, which we call probability increment based swarm optimization (PIBSO). The population evolution mechanism of PIBSO is depicted. Each state in search space has a probability to be chosen. The rule of increasing the probabilities of states is established. Incremental factor is proposed to update probability of a state, and its value is determined by the fitness of the state. It lets the states with better fitness have higher probabilities. Usual roulette wheel selection is employed to select states. Population evolution is impelled by roulette wheel selection and state probability updating. The most distinctive feature of PIBSO is because roulette wheel selection and probability updating produce a trade-off between global and local search; when PIBSO is applied to solve the printed circuit board assembly optimization problem (PCBAOP), it performs superiorly over existing genetic algorithm and adaptive particle swarm optimization on length of tour and CPU running time, respectively. The reason for having such advantages is analyzed in detail. The success of PCBAOP application verifies the effectiveness and efficiency of PIBSO and shows that it is a good method for combinatorial optimization in engineering.