Difficult Combinatorial Optimization Problems Research Articles

MIXED-INTEGER MODEL OF OPTIMIZATION OF PRODUCTION AND TECHNOLOGICAL CAPACITIES OF REGIONAL SERVICE CENTRES FOR THE REPAIR OF COMPONENTS AND ASSEMBLIES OF AVIATION EQUIPMENT

The problem of developing an algorithm for structural-parametric optimization of production and technological capacities of regional service centers for the repair of components and assemblies of aviation equipment is considered. A mixed-integer model has been formed, where integer variables determine the composition of maintainable components (MF) of aviation equipment and repair levels, and continuous variables are indicators of the cost and labor intensity of work with them. It is shown that the problem under consideration relates to difficult combinatorial optimization problems, that is, there is no polynomial algorithm for solving it in a reasonable time. However, taking into account the specifics of the task, in some cases, allows you to build calculation schemes that provide sufficient accuracy of the solution and an acceptable time. The criterion for choosing the optimal solution is proposed - costs for ensuring technical operation, which are necessary to achieve a given operational readiness coefficient. An algorithm for calculating the best variant of the service center structure, which is based on the dependence of the specific weight of the decision to repair each product in the service center, has been developed. The specific weight reflects the difference between the costs of using the service center exclusively as an intermediate warehouse for storing each type of component parts with their repair at the manufacturer, and the costs of creating a repair infrastructure in the service center of the components warehouse for the repair of component parts and performing repairs before the development of the assigned product resource. Options for organizing the maintenance, repair, overhaul (MRO) system of a typical product in the traditional way are consistently considered, where it is assumed to organize a two-level warehouse system and with the deployment of regional service centers by industry for the repair of components and aggregates. The possibility of reducing the costs of MRO for the product life cycle by up to 15 % due to a reduction in the level of stocks of expensive MF, as well as the costs of their repair and transportation is show, the results of calculations are presented.

ANT COLONY OPTIMIZATION: AN ECONOMIC TRANSPOSITION

The paper tackles recent developments in the field of social behaviours of insects and swarm intelligence – “stigmergy”. Specifically, this paper aims to explore the paradigm of ant colony optimization from two main perspectives – economic and biological – so that we can attain a clear view of the genomic bases that allow ants to function as complex biological navigation systems. Such systems translate nowadays into metaheuristic algorithms whose purpose is to solve extremely difficult combinatorial optimization problems. The design of these algorithms draws inspiration from the foraging behaviour of real ants. In the case study, an example of using an ant colony optimization algorithm in order to solve a routing problem shows us how only two iterations and two ants were enough to reveal the shortest path, taking into consideration the amount of pheromones emitted.

Warm-Starting Nested Rollout Policy Adaptation with Optimal Stopping

Nested Rollout Policy Adaptation (NRPA) is an approach using online learning policies in a nested structure. It has achieved a great result in a variety of difficult combinatorial optimization problems. In this paper, we propose Meta-NRPA, which combines optimal stopping theory with NRPA for warm-starting and significantly improves the performance of NRPA. We also present several exploratory techniques for NRPA which enable it to perform better exploration. We establish this for three notoriously difficult problems ranging from telecommunication, transportation and coding theory namely Minimum Congestion Shortest Path Routing, Traveling Salesman Problem with Time Windows and Snake-in-the-Box. We also improve the lower bounds of the Snake-in-the-Box problem for multiple dimensions.

A parallel computing based model for online binary computation offloading in mobile edge computing

Mobile-edge computing (MEC) with wireless power transfer has recently emerged as a viable concept for improving the data processing capacity of limited powered networks like wireless sensor networks (WSN) and the internet of things (IoT). In this research, we explore a wireless MEC network with a binary offloading strategy. Each mobile device’s (MDs) computation task is either performed locally or entirely offloaded to a MEC server. We aim to develop an online system that adapts task offloading decisions and resource allocations to changing wireless channel conditions in real-time. This necessitates solving difficult combinatorial optimization problems quickly within the channel coherence time, which is difficult to achieve with traditional optimization approaches. To address this issue, we offer a parallel computing architecture in which several parallel offloading actors as deep neural networks (DNNs) are used as a scalable method to learn binary offloading decisions from experience. It avoids the need to solve combinatorial optimization issues, reducing computational complexity significantly, especially in large networks. Compared to existing optimization approaches, numerical results demonstrate that the proposed algorithm can achieve optimal performance while reducing computing time by an acceptable margin. For instance, our algorithm achieves a latency rate of 0.033 s in a network of 30 MDs.

MIXED-INTEGER MODEL OF INFORMATION RESOURCES SECURITY CONTROL IN THE CORPORATE INFORMATION ENVIRONMENT

The problem of developing a fast algorithm for calculating options for choosing the best composition of information resource protection in a corporate information system is considered. The necessary composition of models has been formed, including: threats, vulnerabilities, means of protection and destruction of the resource. It is assumed that the damage to the resource is determined by the joint action of threats and means of protection, while the sequence of application of individual threats does not matter for assessing the total value. The harmfulness of threats and the effectiveness of protection tools are described within the framework of a probabilistic model. A criterion for choosing the optimal set of protections is proposed. The criteria for forming a decision on reconfiguration and restructuring of the corporate information system are consistently considered. It is shown that the present problem relates to difficult combinatorial optimization problems, that is, there is no general polynomial algorithm for it that allows to obtain a solution in polynomial time. However, taking into account the specifics of the task, in some cases, allows you to build calculation schemes that provide sufficient accuracy of the solution and an acceptable time. The following calculation schemes are considered: the search for alternative options by sorting through threats, by means of protection, as well as a sequential search of a variety of means of protection. The effectiveness of each scheme was evaluated based on the number of iteration options. A fast algorithm has been developed for calculating the best protection option based on an ordered search of subsets of protections, and the minimum set of protections that covers all threats is used as the initial set. The results of calculations are presented and the possibility of a sharp reduction in the time of computational procedures is shown.

On the strength of recursive McCormick relaxations for binary polynomial optimization

Recursive McCormick relaxations are among the most popular convexification techniques for binary polynomial optimization. It is well-understood that both the quality and the size of these relaxations depend on the recursive sequence and finding an optimal sequence amounts to solving a difficult combinatorial optimization problem. We prove that any recursive McCormick relaxation is implied by the extended flower relaxation, a linear programming relaxation, which for binary polynomial optimization problems with fixed degree can be solved in strongly polynomial time.

Correlation-Guided Updating Strategy for Feature Selection in Classification With Surrogate-Assisted Particle Swarm Optimization

Classification data are usually represented by many features, but not all of them are useful. Without domain knowledge, it is challenging to determine which features are useful. Feature selection is an effective preprocessing technique for enhancing the discriminating ability of data, but it is a difficult combinatorial optimization problem because of the challenges of the huge search space and complex interactions between features. Particle swarm optimization (PSO) has been successfully applied to feature selection due to its efficiency and easy implementation. However, most existing PSO-based feature selection methods still face the problem of falling into local optima. To solve this problem, this article proposes a novel PSO-based feature selection approach, which can continuously improve the quality of the population at each iteration. Specifically, a correlation-guided updating strategy based on the characteristic of data is developed, which can effectively use the information of the current population to generate more promising solutions. In addition, a particle selection strategy based on a surrogate technique is presented, which can efficiently select particles with better performance in both convergence and diversity to form a new population. Experimental comparing the proposed approach with a few state-of-the-art feature selection methods on 25 classification problems demonstrate that the proposed approach is able to select a smaller feature subset with higher classification accuracy in most cases.

IPO-MAXSAT: Combining the In-Parameter-Order Strategy for Covering Array Generation with MaxSAT Solving (Extended Abstract)

Covering arrays (CAs) are discrete objects appearing in combinatorial design theory that find practical applications, most prominently in software testing. The generation of optimized CAs is a difficult combinatorial optimization problem being subject to ongoing research. Previous studies have shown that many different algorithmic approaches are best suited for different instances of CAs. In this extended abstract we describe the IPO-MAXSAT algorithm, which adopts the prominent IPO strategy for CA generation and uses MaxSAT solving to optimize the occurring sub-problems.

Trigonometric approximation of the Max-Cut polytope is star-like

The Max-Cut polytope appears in the formulation of many difficult combinatorial optimization problems. These problems can also be formulated as optimization problems over the so-called trigonometric approximation which possesses an algorithmically accessible description but is not convex. Hirschfeld conjectured that this trigonometric approximation is star-like. In this article, we provide a proof of this conjecture.

SDP-Based Bounds for the Quadratic Cycle Cover Problem via Cutting-Plane Augmented Lagrangian Methods and Reinforcement Learning

We study the quadratic cycle cover problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the reduction and the structure of the graph, which is exploited in an efficient algorithm that constructs this matrix for any instance of the problem. To solve our relaxations, we propose an algorithm that incorporates an augmented Lagrangian method into a cutting-plane framework by utilizing Dykstra’s projection algorithm. Our algorithm is suitable for solving SDP relaxations with a large number of cutting-planes. Computational results show that our SDP bounds and efficient cutting-plane algorithm outperform other QCCP bounding approaches from the literature. Finally, we provide several SDP-based upper bounding techniques, among which is a sequential Q-learning method that exploits a solution of our SDP relaxation within a reinforcement learning environment. Summary of Contribution: The quadratic cycle cover problem (QCCP) is the problem of finding a set of node-disjoint cycles covering all the nodes in a graph such that the total interaction cost between successive arcs is minimized. The QCCP has applications in many fields, among which are robotics, transportation, energy distribution networks, and automatic inspection. Besides this, the problem has a high theoretical relevance because of its close connection to the quadratic traveling salesman problem (QTSP). The QTSP has several applications, for example, in bioinformatics, and is considered to be among the most difficult combinatorial optimization problems nowadays. After removing the subtour elimination constraints, the QTSP boils down to the QCCP. Hence, an in-depth study of the QCCP also contributes to the construction of strong bounds for the QTSP. In this paper, we study the application of semidefinite programming (SDP) to obtain strong bounds for the QCCP. Our strongest SDP relaxation is very hard to solve by any SDP solver because of the large number of involved cutting-planes. Because of that, we propose a new approach in which an augmented Lagrangian method is incorporated into a cutting-plane framework by utilizing Dykstra’s projection algorithm. We emphasize an efficient implementation of the method and perform an extensive computational study. This study shows that our method is able to handle a large number of cuts and that the resulting bounds are currently the best QCCP bounds in the literature. We also introduce several upper bounding techniques, among which is a distributed reinforcement learning algorithm that exploits our SDP relaxations.

Scaling Up Search with Partial Initial States in Optimization Crosswords

Heuristic search remains a leading approach to difficult combinatorial optimization problems. Search algorithms can utilize pruning based on comparing a target score with an admissible (optimistic) estimate of the best score that can be achieved from a given state. If the former is larger they prune the state. However, when the target score is too high the search can fail by exhausting the space without finding a solution. In this paper we show that such failed searches can still be valuable. Specifically, best partial solutions encountered in such failed searches can often bear a high similarity to the corresponding part of a full high-quality or even optimal solution. Thus, a new search for a full solution, with a lower target score, can start with a best known partial solution, rather than starting from scratch. We demonstrate our ideas in a constraint optimization problem modelled on the Romanian Crosswords Competition, a challenging problem where humans perform much better than computers. Utilizing partial solutions produced by a failed search cuts down the running time of an existing state-of-the-art solver by orders of magnitude on competition-level crossword puzzle instances and allows to solve more instances.

Optimal Screening of Populations with Heterogeneous Risk Profiles Under the Availability of Multiple Tests

We study the design of large-scale group testing schemes under a heterogeneous population (i.e., subjects with potentially different risk) and with the availability of multiple tests. The objective is to classify the population as positive or negative for a given binary characteristic (e.g., the presence of an infectious disease) as efficiently and accurately as possible. Our approach examines components often neglected in the literature, such as the dependence of testing cost on the group size and the possibility of no testing, which are especially relevant within a heterogeneous setting. By developing key structural properties of the resulting optimization problem, we are able to reduce it to a network flow problem under a specific, yet not too restrictive, objective function. We then provide results that facilitate the construction of the resulting graph and finally provide a polynomial time algorithm. Our case study, on the screening of HIV in the United States, demonstrates the substantial benefits of the proposed approach over conventional screening methods. Summary of Contribution: This paper studies the problem of testing heterogeneous populations in groups in order to reduce costs and hence allow for the use of more efficient tests for high-risk groups. The resulting problem is a difficult combinatorial optimization problem that is NP-complete under a general objective. Using structural properties specific to our objective function, we show that the problem can be cast as a network flow problem and provide a polynomial time algorithm.

When Hashing Met Matching: Efficient Spatio-Temporal Search for Ridesharing

Shared on-demand mobility holds immense potential for urban transportation. However, finding ride matches in real-time at urban scale is a very difficult combinatorial optimization problem and mostly heuristic approaches are applied. In this work, we introduce a principled approach to this combinatorial problem. Our approach proceeds by constructing suitable representations for rides and driver routes capturing their essential spatio-temporal aspects in an appropriate vector space, and defining a similarity metric in this space that expresses matching utility. This then lets us mathematically model the problem of finding ride matches as that of Near Neighbor Search (NNS). Exploiting this modeling, we devise a novel spatio-temporal search algorithm for finding ride matches based on the theory of Locality Sensitive Hashing (LSH). Apart from being highly efficient, our algorithm enjoys several practically useful properties and extension possibilities. Experiments with large real-world datasets show that our algorithm consistently outperforms state-of-the-art heuristic methods thereby proving its practical applicability.

IMPROVED ANTLION OPTIMIZATION ALGORITHM FOR QUADRATIC ASSIGNMENT PROBLEM

The Antlion Optimization (ALO) algorithm is a meta-heuristic optimization algorithm based on the hunting of ants by antlions. The basic inadequacy of this algorithm is that it has long run time because of the random walk model used for the ant's movement. We improved some mechanisms in ALO algorithm, such as ants' random walking, reproduction, sliding ants towards antlion, elitism, and selection procedure. This proposed algorithm is called Improved Antlion Optimization (IALO) algorithm. To show the performance of the proposed IALO algorithm, we used different measurement metrics, such as mean best, standard deviation, optimality, accuracy, CPU time, and number of function evaluations (NFE). The proposed IALO algorithm was tested for different benchmark test functions taken from the literature. There are no studies regarding time analysis of ALO algorithm found in the literature. This study firstly aims to demonstrate the success of the proposed IALO algorithm especially in runtime analysis. Secondly, the IALO algorithm was also applied to the Quadratic Assignment Problem (QAP) known as a difficult combinatorial optimization problem. In QAP tests, the performance of the IALO algorithm was compared with the performances of the classic ALO algorithm and 14 well-known and recent meta-heuristic algorithms. The results of the benchmark test functions show that IALO algorithm is able to provide very competitive results in terms of mean best/standard deviation, optimality, accuracy, CPU time and NFE metrics. The CPU time results prove that IALO algorithm is faster than the classic ALO algorithm. As a result of the QAP tests, the proposed IALO algorithm has the best performance according to the mean cost, worst cost and standard deviation. The source codes of QAP with the proposed IALO algorithm are publicly available at https://github.com/uguryuzgec/QAP-with-IALO.

Hybrid Genetic Simulated Annealing Algorithm for Job Shop Scheduling Problem

The job shop scheduling problem (JSSP) is a well-known difficult combinatorial optimization problem, as it is classified as NP-hard problem and therefore no deterministic algorithms can solve them in a reasonable amount of time. The main objective of solving this problem is to find suitable job sequences on machines to optimize the performance criteria. In this paper, a meta-heuristic approach for solving the job-shop scheduling problem (JSSP) is presented. This approach uses a hybrid genetic algorithm that is suggested by previous stuody, to generate the best solutions and then a simulated annealing algorithm to improve the quality and performance of the best solutions to produce the optimal/near-optimal solution. Ten benchmark problems adopted from the previous study are used to evaluate the performance of the proposed algorithm. The computational results validate the quality of the proposed algorithm; this is done by calculating the completion time Cmax

Frequent Pattern-Based Search: A Case Study on the Quadratic Assignment Problem

We present frequent pattern-based search (FPBS) that combines data mining and optimization. FPBS is a general-purpose method that unifies data mining and optimization within the population-based search framework. The method emphasizes the relevance of a modular- and component-based approach, making it applicable to optimization problems by instantiating the underlying components. To illustrate its potential for solving difficult combinatorial optimization problems, we apply the method to the well-known and challenging quadratic assignment problem. We show the computational results and comparisons on the hardest QAPLIB benchmark instances. This work reinforces the recent trend toward closer cooperations between the optimization methods and machine learning or data mining techniques.

Heuristic algorithms based on deep reinforcement learning for quadratic unconstrained binary optimization

The unconstrained binary quadratic programming (UBQP) problem is a difficult combinatorial optimization problem that has been intensively studied in the past decades. Due to its NP-hardness, many heuristic algorithms have been developed for the solution of the UBQP. These algorithms are usually problem-tailored, which lack generality and scalability. To address these issues, a heuristic algorithm based on deep reinforcement learning (DRLH) is proposed in this paper. It features in inputting specific features and using a neural network model called NN to guild the selection of variable at each solution construction step. Also, to improve the algorithm speed and efficiency, two algorithm variants named simplified DRLH (DRLS) and DRLS with hill climbing (DRLS-HC) are developed as well. These three algorithms are examined through extensive experiments in comparison with famous heuristic algorithms from the literature. Experimental results show that the DRLH, DRLS, and DRLS-HC outperform their competitors in terms of both solution quality and computational efficiency. Precisely, the DRLH achieves the best-quality results, while DRLS offers a high-quality solution in a very short time. By adding a hill-climbing procedure to DRLS, the resulting DRLS-HC algorithm is able to obtain almost the same quality result as DRLH with however 5 times less computing time on average. We conducted additional experiments on large-scale instances and various data distributions to verify the generality and scalability of the proposed algorithms, and the results on benchmark instances indicate the ability of the algorithms to be applied to practical problems.

A quantum annealer with fully programmable all-to-all coupling via Floquet engineering

Quantum annealing is a promising approach to heuristically solving difficult combinatorial optimization problems. However, the connectivity limitations in current devices lead to an exponential degradation of performance on general problems. We propose an architecture for a quantum annealer that achieves full connectivity and full programmability while using a number of physical resources only linear in the number of spins. We do so by application of carefully engineered periodic modulations of oscillator-based qubits, resulting in a Floquet Hamiltonian in which all the interactions are tunable. This flexibility comes at the cost of the coupling strengths between qubits being smaller than they would be compared with direct coupling, which increases the demand on coherence times with increasing problem size. We analyze a specific hardware proposal of our architecture based on Josephson parametric oscillators. Our results show how the minimum-coherence-time requirements imposed by our scheme scale, and we find that the requirements are not prohibitive for fully connected problems with up to at least 1000 spins. Our approach could also have impact beyond quantum annealing, since it readily extends to bosonic quantum simulators, and would allow the study of models with arbitrary connectivity between lattice sites.

Using ZDDs in the Mapping of Quantum Circuits

A critical step in quantum compilation is the transformation of a technology-independent quantum circuit into a technology-dependent form for a targeted device. In addition to mapping quantum gates into the supported gate set, it is necessary to map pseudo qubits in the technology-independent circuit into physical qubits of the technology-dependent circuit such that coupling constraints among qubits acting in multiple-qubit gates are satisfied. It is usually not possible to find such a mapping without adding SWAP gates into the circuit. To cope with the technical limitations of NISQ-era quantum devices, it is advantageous to find a mapping that requires as few additional gates as possible. The large search space of possible mappings makes this task a difficult combinatorial optimization problem. In this work, we demonstrate how zero-suppressed decision diagrams (ZDDs) can be used for typical implementation tasks in quantum mapping algorithms. We show how to maximally partition a quantum circuit into blocks of adjacent gates, and if adjacent gates within a circuit do not share common mapping permutations, we attempt to combine them using parallelized SWAP operations represented in a ZDD. Boundaries for the partitions are formed where adjacent gates are unable to be combined. Within each partition block, ZDDs represent all possible mappings of pseudo qubits to physical qubits.

Biogeography Based Optimization for Mobile Station Reporting Cell System Design

Fast service access involves keeping track of the location of mobile users, while they are moving around the mobile network for a satisfactory level of QoS (Quality of Service) in a cost-effective manner. The location databases are used to keep track of Mobile Terminals (MT) so that incoming calls can be directed to requested mobile terminals at all times. MT reporting cell system used in location management is to designate each cell in the network as a reporting cell or a non-reporting cell. Determination of an optimal number of reporting cells (or reporting cell configuration) for a given network is reporting cell planning (RCP) problem. This is a difficult combinatorial optimization problem which has an exponential complexity. We can see that a cell in a network is either a reporting cell or a non-reporting cell. Hence, for a given network with N cells, the number of possible solutions is  . We propose a biogeography based optimization (BBO) for design of mobile station location management system in wireless communication network. The number and locations of reporting cells should be determined to balance the registration for location update and paging operations for search the mobile stations to minimize the cost of system. Experimental results show that our proposed BBO is a fairly effective and competitive approach with respect to solution quality for optimally designing location management system because BBO is suitable for combinatorial optimization and multi-functional problems.