Large-scale Combinatorial Optimization Problems Research Articles

Parametric Frequency Divider Based Ising Machines.

We report on a new class of Ising machines (IMs) that rely on coupled parametric frequency dividers (PFDs) as macroscopic artificial spins. Unlike the IM counterparts based on subharmonic-injection locking (SHIL), PFD IMs do not require strong injected continuous-wave signals or applied dc voltages. Therefore, they show a significantly lower power consumption per spin compared to SHIL-based IMs, making it feasible to accurately solve large-scale combinatorial optimization problems that are hard or even impossible to solve by using the current von Neumann computing architectures. Furthermore, using high quality factor resonators in the PFD design makes PFD IMs able to exhibit a nanowatt-level power per spin. Also, it remarkably allows a speedup of the phase synchronization among the PFDs, resulting in shorter time to solution and lower energy to solution despite the resonators' longer relaxation time. As a proof of concept, a 4-node PFD IM has been demonstrated. This IM correctly solves a set of Max-Cut problems while consuming just 600 nanowatts per spin. This power consumption is 2 orders of magnitude lower than the power per spin of state-of-the-art SHIL-based IMs operating at the same frequency.

Enhanced convergence in p-bit based simulated annealing with partial deactivation for large-scale combinatorial optimization problems

This article critically investigates the limitations of the simulated annealing algorithm using probabilistic bits (pSA) in solving large-scale combinatorial optimization problems. The study begins with an in-depth analysis of the pSA process, focusing on the issues resulting from unexpected oscillations among p-bits. These oscillations hinder the energy reduction of the Ising model and thus obstruct the successful execution of pSA in complex tasks. Through detailed simulations, we unravel the root cause of this energy stagnation, identifying the feedback mechanism inherent to the pSA operation as the primary contributor to these disruptive oscillations. To address this challenge, we propose two novel algorithms, time average pSA (TApSA) and stalled pSA (SpSA). These algorithms are designed based on partial deactivation of p-bits and are thoroughly tested using Python simulations on maximum cut benchmarks that are typical combinatorial optimization problems. On the 16 benchmarks from 800 to 5000 nodes, the proposed methods improve the normalized cut value from 0.8 to 98.4% on average in comparison with the conventional pSA.

Spatial layout and optimization of e-commerce logistics management based on combinatorial optimization algorithm

Abstract Warehousing space layout is crucial for e-commerce logistics. A refined and scientific approach to warehouse design enhances space utilization and operational efficiency. This minimizes unused space and idle time, lowers inventory costs, and strengthens the competitive edge of e-commerce logistics. In this study, we conceptualize the logistics space layout issue as a crate problem, harnessing the rapid optimization capabilities of genetic algorithms (GA) and simulated annealing (SA). We propose a hybrid algorithm where SA forms the core, using GA to generate initial solutions and new iterations. Design parameters for the combined operation process are systematically developed. This algorithm is evaluated using both datasets and real arithmetic cases, demonstrating superior performance in large-scale combinatorial optimization problems. It achieves a search accuracy of 0.5% to 3% higher than GA alone, converging more reliably to the global optimum, thus reducing search time and operational scope. This research offers vital theoretical insights for optimizing space layouts in e-commerce logistics management.

Hybrid Parallel Ant Colony Optimization for Application to Quantum Computing to Solve Large-Scale Combinatorial Optimization Problems

Quantum computing is a promising technology that may provide breakthrough solutions to today’s difficult problems such as breaking encryption and solving large-scale combinatorial optimization problems. An algorithm referred to as Quantum Approximate Optimization Algorithm (QAOA) has been recently proposed to approximately solve hard problems using a protocol know as bang–bang. The technique is based on unitary evolution using a Hamiltonian encoding of the objective function of the combinatorial optimization problem. The QAOA was explored in the context of several optimization problems such as the Max-Cut problem and the Traveling Salesman Problem (TSP). Due to the relatively small node-size solution capability of the available quantum computers and simulators, we developed a hybrid approach where sub-graphs of a TSP tour can be executed on a quantum computer, and the results from the quantum optimization are combined for a further optimization of the whole tour. Since the local optimization of a sub-graph is prone to becoming trapped in a local minimum, we overcame this problem by using a parallel Ant Colony Optimization (ACO) algorithm with periodic pheromone exchange between colonies. Our method exceeds existing approaches which have attempted partitioning a graph for small problems (less than 48 nodes) with sub-optimal results. We obtained optimum results for benchmarks with less than 150 nodes and results usually within 1% of the known optimal solution for benchmarks with around 2000 nodes.

Effective implementation of text{L}{0}-regularised compressed sensing with chaotic-amplitude-controlled coherent Ising machines

Coherent Ising machine (CIM) is a network of optical parametric oscillators that can solve large-scale combinatorial optimisation problems by finding the ground state of an Ising Hamiltonian. As a practical application of CIM, Aonishi et al., proposed a quantum-classical hybrid system to solve optimisation problems of l_0-regularisation-based compressed sensing. In the hybrid system, the CIM was an open-loop system without an amplitude control feedback loop. In this case, the hybrid system is enhanced by using a closed-loop CIM to achieve chaotic behaviour around the target amplitude, which would enable escaping from local minima in the energy landscape. Both artificial and magnetic resonance image data were used for the testing of our proposed closed-loop system. Compared with the open-loop system, the results of this study demonstrate an improved degree of accuracy and a wider range of effectiveness.

Research on Optimization Algorithm of Single-block Train Formation Plan of Technical Station

The optimization of train formation plan is a large-scale combinatorial optimization problem, which is difficult to solve. This paper mainly studies the optimization algorithm of the single-block train formation plan. The corresponding mathematical model is established by consulting relevant literature, and a positive feedback search algorithm based on absolute conditions is proposed. First of all, the wagon flow that meets the absolute conditions directly runs through train flow without making other choices. For the wagon flow that does not meet the absolute conditions, select the target station to reach directly according to the probability. The probability of wagon flow selecting a station is calculated according to the pheromone of the wagon flow at the station. At the same time, a pheromone update strategy with positive feedback mechanism is proposed to make the search process converge. Finally, the feasibility of the algorithm and the necessity of introducing absolute conditions into the algorithm are verified by taking eight technical stations in the linear direction of the road network as examples.

Low-Rank Combinatorial Optimization and Statistical Learning by Spatial Photonic Ising Machine.

The spatial photonic Ising machine (SPIM) [13D. Pierangeli et al., Large-Scale Photonic Ising Machine by Spatial Light Modulation, Phys. Rev. Lett. 122, 213902 (2019).PRLTAO0031-900710.1103/PhysRevLett.122.213902] is a promising optical architecture utilizing spatial light modulation for solving large-scale combinatorial optimization problems efficiently. The primitive version of the SPIM, however, can accommodate Ising problems with only rank-one interaction matrices. In this Letter, we propose a new computing model for the SPIM that can accommodate any Ising problem without changing its optical implementation. The proposed model is particularly efficient for Ising problems with low-rank interaction matrices, such as knapsack problems. Moreover, it acquires the learning ability of Boltzmann machines. We demonstrate that learning, classification, and sampling of the MNIST handwritten digit images are achieved efficiently using the model with low-rank interactions. Thus, the proposed model exhibits higher practical applicability to various problems of combinatorial optimization and statistical learning, without losing the scalability inherent in the SPIM architecture.

The applications of hybrid approach combining exact method and evolutionary algorithm in combinatorial optimization

Abstract Combinatorial optimization problems have very important applications in information technology, transportation, economics, management, network communication, and other fields. Since the problem size in real-scenario application is in large-scale, the demand for real-time and efficient solving approaches increases rapidly. The traditional exact methods guarantee the optimality of the final solution, but these methods can hardly solve the problem in acceptable time due to extremely high computational costs. Heuristic approaches can find feasible solutions in a limited time, while these approaches cannot meet the demand of solution quality. In recent years, hybrid algorithms based on exact methods and heuristic algorithms show outstanding performance in solving large-scale combinatorial optimization problems. The hybridization not only overcomes the shortcomings from single algorithm but also fully utilizes the search ability for population-based approaches as well as the interpretability in exact methods, which promotes the application of combinatorial optimization in real-world problems. This paper reviews existing studies on hybrid algorithms combining exact method and evolutionary computation, summarizes the characteristics of the existing algorithms, and directs the future research.

QAOA-in-QAOA: Solving Large-Scale MaxCut Problems on Small Quantum Machines

The design of fast algorithms for combinatorial optimization greatly contributes to a plethora of domains such as logistics, finance, and chemistry. Quantum approximate optimization algorithms (QAOAs), which utilize the power of quantum machines and inherit the spirit of adiabatic evolution, are approaches to tackle combinatorial problems with potential runtime speedups. However, hurdled by the limited quantum resources nowadays, QAOAs are infeasible to manipulate large-scale problems. To address this issue, here we revisit the MaxCut problem via the divide-and-conquer heuristic: seek the solutions of subgraphs in parallel and then merge these solutions to obtain the global solution. Because of the ${\mathbb{Z}}_{2}$ symmetry in MaxCut, we prove that the merging process can be further cast into a new MaxCut problem and thus be addressed by QAOAs or other MaxCut solvers. In view of this, we propose QAOA-in-QAOA (${\mathrm{QAOA}}^{2}$) to solve arbitrary large-scale MaxCut problems using small quantum machines. We also prove that the performance of ${\mathrm{QAOA}}^{2}$ is lower bounded with respect to the divide-and-conquer process. Experiment results illustrate that, under different graph settings, ${\mathrm{QAOA}}^{2}$ attains a competitive or even better performance over classical algorithms when the node count is around $2000$. Our method can be seamlessly embedded into other advanced strategies to enhance the capability of QAOAs in large-scale combinatorial optimization problems.

A Large-Scale Combinatorial Many-Objective Evolutionary Algorithm for Intensity-Modulated Radiotherapy Planning

Intensity-modulated radiotherapy (IMRT) is one of the most popular techniques for cancer treatment. However, existing IMRT planning methods can only generate one solution at a time and, consequently, medical physicists should perform the planning process many times to obtain diverse solutions to meet the requirement of a clinical case. Meanwhile, multiobjective evolutionary algorithms (MOEAs) have not been fully exploited in IMRT planning since they are ineffective in optimizing the large number of discrete variables of IMRT. To bridge the gap, this article formulates IMRT planning into a large-scale combinatorial many-objective optimization problem and proposes a coevolutionary algorithm to solve it. In contrast to the existing MOEAs handling high-dimensional search spaces via variable grouping or dimensionality reduction, the proposed algorithm evolves one population with fine encoding for local exploitation and evolves another population with rough encoding for global exploration. Moreover, the convergence speed is further accelerated by two customized local search strategies. The experimental results verify that the proposed algorithm outperforms state-of-the-art MOEAs and IMRT planning methods on a variety of clinical cases.

ИНТЕЛЛЕКТУАЛЬНЫЙ АНАЛИЗ ДАННЫХ В УПРАВЛЕНИИ ПРЕДПРИЯТИЕМ НА ОСНОВЕ АЛГОРИТМА ИМИТАЦИИ ОТЖИГА

The article considers an analytical review of the annealing simulation algorithm for theproblem of efficient enterprise management. The optimization of the annealing simulation algorithmfor the problem of efficient enterprise management has been carried out. For the analysis ofcases, the optimization of the work schedule of workers in the organization was used. Establishedworker scheduling model with strong and weak constraints. The simulated annealing algorithm isused to optimize the strategy for solving the staff scheduling model. The simulated annealing algorithmis an algorithm suitable for solving large-scale combinatorial optimization problems. It alsoevaluates and obtains the optimal scheduling strategy. The simulated annealing algorithm has agood effect on the data mining of human resource management. Big data mining can help companiesconduct dynamic analysis in talent recruitment, and the talent recruitment plan is carried outin a quality and standard way to analyze the characteristics of various talents from many anglesand improve the level of human resource management. An algorithm has been developed that implementsthe operation of the annealing simulation algorithm. The simulated annealing algorithmmakes new decisions based on the Metropolis criterion, so in addition to making an optimizeddecision, it also makes a reduced decision in a limited range. The Metropolis algorithm is a samplingalgorithm mainly used for complex distribution functions. It is somewhat similar to the variancesampling algorithm, but here the auxiliary distribution function changes over time. Experimentalstudies have been carried out that show that a worker scheduling model based on strongand weak constraints is significantly better than a manual scheduling model, achieving an effectivebalance between controlling wage costs in an organization and increasing employee satisfaction.The successful application of a workforce scheduling model based on a simulated annealingalgorithm brings new insights and insights to solve large-scale worker scheduling problems.The results presented can serve as a starting point for studying personnel management systemsbased on data mining technology.

Convex relaxation for optimal fixture layout design

This article proposes a general fixture layout design framework that directly integrates the system equation with the convex relaxation method. Note that the optimal fixture design problem is a large-scale combinatorial optimization problem; we relax it to a convex Semi-Definite Programming (SDP) problem by adopting sparse learning and SDP relaxation techniques. It can be solved efficiently by existing convex optimization algorithms and thus generates a near-optimal fixture layout. A real case study in the half-to-half fuselage assembly process indicates the superiority of our proposed algorithm compared to the current industry practice and state-of-art methods.

The incremental online k-means clustering algorithm and its application to color quantization

Color quantization is a common image processing operation with various applications in computer graphics, image processing, and computer vision. Color quantization is essentially a large-scale combinatorial optimization problem. Many clustering algorithms, both of hierarchical and partitional types, have been applied to this problem since the 1980s. In general, hierarchical color quantization algorithms are faster, whereas partitional ones produce better results provided that they are initialized properly. In this paper, we propose a novel partitional color quantization algorithm based on a binary splitting formulation of MacQueen’s online k-means algorithm. Unlike MacQueen’s original algorithm, the proposed algorithm is both deterministic and free of initialization. Experiments on a diverse set of public test images demonstrate that the proposed algorithm is significantly faster than two popular batch k-means algorithms while yielding nearly identical results. In other words, unlike previously proposed k-means variants, our algorithm addresses both the initialization and acceleration issues of k-means without sacrificing the simplicity of the algorithm. The presented algorithm may be of independent interest as a general-purpose clustering algorithm.

Optimizing Transport Logistics under Uncertainty with Simheuristics: Concepts, Review and Trends

Background: Uncertainty conditions have been increasingly considered in optimization problems arising in real-life transportation and logistics activities. Generally, the analysis of complex systems in these non-deterministic environments is approached with simulation techniques. However, simulation is not an optimization tool. Hence, it must be combined with optimization methods when our goal is to: (i) minimize operating costs while guaranteeing a given quality of service; or (ii) maximize system performance using limited resources. When solving NP-hard optimization problems, the use of metaheuristics allows us to deal with large-scale instances in reasonable computation times. By adding a simulation layer to the metaheuristics, the methodology becomes a simheuristic, which allows the optimization element to solve scenarios under uncertainty. Methods: This paper reviews the indexed documents in Elsevier Scopus database of both initial as well as recent applications of simheuristics in the logistics and transportation field. The paper also discusses open research lines in this knowledge area. Results: The simheuristics approaches to solving NP-hard and large-scale combinatorial optimization problems under uncertainty scenarios are discussed, as they frequently appear in real-life applications in logistics and transportation activities. Conclusions: The way in which the different simheuristic components interact puts a special emphasis in the different stages that can contribute to make the approach more efficient from a computational perspective. There are several lines of research that are still open in the field of simheuristics.

Solution of Multi-Crew Depots Railway Crew Scheduling Problems: The Chinese High-Speed Railway Case

This paper presents a novel mathematical formulation in crew scheduling, considering real challenges most railway companies face such as roundtrip policy for crew members joining from different crew depots and stricter working time standards under a sustainable development strategy. In China, the crew scheduling is manually compiled by railway companies respectively, and the plan quality varies from person to person. An improved genetic algorithm is proposed to solve this large-scale combinatorial optimization problem. It repairs the infeasible gene fragments to optimize the search scope of the solution space and enhance the efficiency of GA. To investigate the algorithm’s efficiency, a real case study was employed. Results show that the proposed model and algorithm lead to considerable improvement compared to the original planning: (i) Compared with the classical metaheuristic algorithms (GA, PSO, TS), the improved genetic algorithm can reduce the objective value by 4.47%; and (ii) the optimized crew scheduling plan reduces three crew units and increases the average utilization of crew unit working time by 6.20% compared with the original plan.

Implementasi Metode Tabu Search Dalam Penjadwalan Menggunakan Analisa Pieces

Nowadays, Scheduling subjects is one of the first steps for starting the teaching and learning process in educational institutions. To do so, The role of teachers and school staff is very important and not easy because it takes a long time to compile it. SMK PGRI 4 Denpasar is one of the schools located in the city of Denpasar which is located on Jalan Kebo Iwa No 8, Padangsambian Kaja, Denpasar, Bali. It is a vocational high school that has a tourism expertise and computer engineering study program. Based on current results of observations and interviews, the process of making the subject schedules that run at SMK PGRI 4 Denpasar is still being done using Microsoft Excel, this has resulted in frequent errors in managing schedules such as conflicting schedule and it takes a long time to correct it. Tabu Search is an optimization method based on local search, where the search process moves from one solution to the next by selecting the best solution which is not classified as a prohibited solution. It is a combinatorial optimization problem-solving method that is incorporated into local search methods. This method aims to streamline the process of finding the best solution of a large-scale (np-hard) combinatorial optimization problem. Tabu search method to optimize the process of making the subject schedule and combined using PIECES analysis (Performance, Information, Economic, Control, Efficiency, Services). From this analysis, several problems will be obtained, which in the end can be identified clearly and more specifically, so that we can conclude some suggestions that will help in designing a new and better system. The Tabu Search method can be used to optimize the process of making the subject schedules at SMK PGRI 4 Denpasar, so that the scheduling process will be more easier than using Microsoft Excel.

A Knowledge Transfer Based Scheduling Algorithm for Large-Scale Refinery Production

Decomposition algorithms for large-scale refinery scheduling problems commonly adopt the spatially dividing method. When the scheduling horizon further enlarges, the sizes of the resulting subproblems exponentially grow, leading to the performance degradation of the decomposition algorithms. In this article, to solve this issue about subproblem size caused by the long-time scheduling horizon, a knowledge transfer based algorithm is proposed, where the size of subproblems is constant and a novel concept of product flowrate is introduced to transfer operation knowledge from tractable small-scale short-time problems to intractable large-scale long-time problems to quickly obtain a satisfactory solution for large-scale problems. Experimental results show that the proposed algorithm can achieve better solutions within 10−44 s for large-scale refinery scheduling problems with 50−200 time slots (the existing model is with less than 30 time slots), compared to the CPLEX solver with a running time of 3600 s and the existing evolutionary algorithm with a running time of 100−500 s. The results also show that, at present, the heuristic method (the essence of the proposed algorithm) is the main tool to solve large-scale combinatorial optimization problems, and even in the CPLEX solver, the heuristic is also the most active part.

Hybrid quantum annealing via molecular dynamics

A novel quantum–classical hybrid scheme is proposed to efficiently solve large-scale combinatorial optimization problems. The key concept is to introduce a Hamiltonian dynamics of the classical flux variables associated with the quantum spins of the transverse-field Ising model. Molecular dynamics of the classical fluxes can be used as a powerful preconditioner to sort out the frozen and ambivalent spins for quantum annealers. The performance and accuracy of our smooth hybridization in comparison to the standard classical algorithms (the tabu search and the simulated annealing) are demonstrated by employing the MAX-CUT and Ising spin-glass problems.

A Parallel Bioinspired Algorithm for Chinese Postman Problem Based on Molecular Computing

The Chinese postman problem is a classic resource allocation and scheduling problem, which has been widely used in practice. As a classical nondeterministic polynomial problem, finding its efficient algorithm has always been the research direction of scholars. In this paper, a new bioinspired algorithm is proposed to solve the Chinese postman problem based on molecular computation, which has the advantages of high computational efficiency, large storage capacity, and strong parallel computing ability. In the calculation, DNA chain is used to properly represent the vertex, edge, and corresponding weight, and then all possible path combinations are effectively generated through biochemical reactions. The feasible solution space is obtained by deleting the nonfeasible solution chains, and the optimal solution is solved by algorithm. Then the computational complexity and feasibility of the DNA algorithm are proved. By comparison, it is found that the computational complexity of the DNA algorithm is significantly better than that of previous algorithms. The correctness of the algorithm is verified by simulation experiments. With the maturity of biological operation technology, this algorithm has a broad application space in solving large-scale combinatorial optimization problems.

Reconfiguration of Electric Power Distribution Systems: Comprehensive Review and Classification

Distribution systems play an important role, delivering the electric power of generation system to individual consumers. Distribution system reconfiguration (DSR) is a large-scale combinatorial optimization problem. For the last 45 years, the DSR problem has been widely studied; nowadays, DSR, combined with new challenges, is being highly investigated, as researchers aim to reach a better solution. This paper presents a complete review and classification of the most significant works to date, providing a literary framework for DSR specialists. A categorization of solution methods, case studies, and novelties of the most relevant works regarding DSR are provided. In order to establish a complete background, not only traditional approaches, but also those involving uncertainty, reliability, electricity market, power quality, distributed generation, capacitor placement, and switching time in DSR are highlighted. This framework can help researches to improve previous formulations and methods and can propose more efficient models to better exploit the existing infrastructure.