We incorporate a non-Markovian feedback mechanism into the simulated bifurcation method for dynamical solvers addressing combinatorial optimization problems.By reinjecting a portion of dissipated kinetic energy into e...We incorporate a non-Markovian feedback mechanism into the simulated bifurcation method for dynamical solvers addressing combinatorial optimization problems.By reinjecting a portion of dissipated kinetic energy into each spin in a history-dependent and trajectory-informed manner,the method effectively suppresses early freezing induced by inelastic boundaries and enhances the system's ability to explore complex energy landscapes.Numerical results on the maximum cut(MAX-CUT)instances of fully connected Sherrington–Kirkpatrick(SK)spin glass models,including the 2000-spin K_(2000)benchmark,demonstrate that the non-Markovian algorithm significantly improves both solution quality and convergence speed.Tests on randomly generated SK instances with 100 to 1000 spins further indicate favorable scalability and substantial gains in computational efficiency.Moreover,the proposed scheme is well suited for massively parallel hardware implementations,such as field-programmable gate arrays,providing a practical and scalable approach for solving large-scale combinatorial optimization problems.展开更多
We introduce Quafu-Qcover,an open-source cloud-based software package developed for solving combinatorial optimization problems using quantum simulators and hardware backends.Quafu-Qcover provides a standardized and c...We introduce Quafu-Qcover,an open-source cloud-based software package developed for solving combinatorial optimization problems using quantum simulators and hardware backends.Quafu-Qcover provides a standardized and comprehensive workflow that utilizes the quantum approximate optimization algorithm(QAOA).It facilitates the automatic conversion of the original problem into a quadratic unconstrained binary optimization(QUBO)model and its corresponding Ising model,which can be subsequently transformed into a weight graph.The core of Qcover relies on a graph decomposition-based classical algorithm,which efficiently derives the optimal parameters for the shallow QAOA circuit.Quafu-Qcover incorporates a dedicated compiler capable of translating QAOA circuits into physical quantum circuits that can be executed on Quafu cloud quantum computers.Compared to a general-purpose compiler,our compiler demonstrates the ability to generate shorter circuit depths,while also exhibiting superior speed performance.Additionally,the Qcover compiler has the capability to dynamically create a library of qubits coupling substructures in real-time,utilizing the most recent calibration data from the superconducting quantum devices.This ensures that computational tasks can be assigned to connected physical qubits with the highest fidelity.The Quafu-Qcover allows us to retrieve quantum computing sampling results using a task ID at any time,enabling asynchronous processing.Moreover,it incorporates modules for results preprocessing and visualization,facilitating an intuitive display of solutions for combinatorial optimization problems.We hope that Quafu-Qcover can serve as an instructive illustration for how to explore application problems on the Quafu cloud quantum computers.展开更多
This paper states a new metaheuristic based on Deterministic Finite Automata （DFA） for the multi - objective optimization of combinatorial problems. First, a new DFA named Multi - Objective Deterministic Finite Auto...This paper states a new metaheuristic based on Deterministic Finite Automata （DFA） for the multi - objective optimization of combinatorial problems. First, a new DFA named Multi - Objective Deterministic Finite Automata （MDFA） is defined. MDFA allows the representation of the feasible solutions space of combinatorial problems. Second, it is defined and implemented a metaheuritic based on MDFA theory. It is named Metaheuristic of Deterministic Swapping （MODS）. MODS is a local search strategy that works using a MDFA. Due to this, MODS never take into account unfeasible solutions. Hence, it is not necessary to verify the problem constraints for a new solution found. Lastly, MODS is tested using well know instances of the Bi-Objective Traveling Salesman Problem （TSP） from TSPLIB. Its results were compared with eight Ant Colony inspired algorithms and two Genetic algorithms taken from the specialized literature. The comparison was made using metrics such as Spacing, Generational Distance, Inverse Generational Distance and No-Dominated Generation Vectors. In every case, the MODS results on the metrics were always better and in some of those cases, the superiority was 100%.展开更多
Many problems in science,engineering and real life are related to the combinatorial optimization.However,many combinatorial optimization problems belong to a class of the NP-hard problems,and their globally optimal so...Many problems in science,engineering and real life are related to the combinatorial optimization.However,many combinatorial optimization problems belong to a class of the NP-hard problems,and their globally optimal solutions are usually difficult to solve.Therefore,great attention has been attracted to the algorithms of searching the globally optimal solution or near-optimal solution for the combinatorial optimization problems.As a typical combinatorial optimization problem,the traveling salesman problem(TSP)often serves as a touchstone for novel approaches.It has been found that natural systems,particularly brain nervous systems,work at the critical region between order and disorder,namely,on the edge of chaos.In this work,an algorithm for the combinatorial optimization problems is proposed based on the neural networks on the edge of chaos(ECNN).The algorithm is then applied to TSPs of 10 cities,21 cities,48 cities and 70 cities.The results show that ECNN algorithm has strong ability to drive the networks away from local minimums.Compared with the transiently chaotic neural network(TCNN),the stochastic chaotic neural network(SCNN)algorithms and other optimization algorithms,much higher rates of globally optimal solutions and near-optimal solutions are obtained with ECNN algorithm.To conclude,our algorithm provides an effective way for solving the combinatorial optimization problems.展开更多
Traveling salesman problem(TSP) is one of the typical NP-hard problems, and it has been used in many engineering applications. However, the previous swarm intelligence(SI) based algorithms for TSP cannot coordinate wi...Traveling salesman problem(TSP) is one of the typical NP-hard problems, and it has been used in many engineering applications. However, the previous swarm intelligence(SI) based algorithms for TSP cannot coordinate with the exploration and exploitation abilities and are easily trapped into local optimum. In order to deal with this situation, a new hybrid optimization algorithm based on wolf pack search and local search(WPS-LS)is proposed for TSP. The new method firstly simulates the predatory process of wolf pack from the broad field to a specific place so that it allows for a search through all possible solution spaces and prevents wolf individuals from getting trapped into local optimum. Then, local search operation is used in the algorithm to improve the speed of solving and the accuracy of solution. The test of benchmarks selected from TSPLIB shows that the results obtained by this algorithm are better and closer to the theoretical optimal values with better robustness than those obtained by other methods.展开更多
The optimization of polymer structures aims to determine an optimal sequence or topology that achieves a given target property or structural performance.This inverse design problem involves searching within a vast com...The optimization of polymer structures aims to determine an optimal sequence or topology that achieves a given target property or structural performance.This inverse design problem involves searching within a vast combinatorial phase space defined by components,se-quences,and topologies,and is often computationally intractable due to its NP-hard nature.At the core of this challenge lies the need to evalu-ate complex correlations among structural variables,a classical problem in both statistical physics and combinatorial optimization.To address this,we adopt a mean-field approach that decouples direct variable-variable interactions into effective interactions between each variable and an auxiliary field.The simulated bifurcation(SB)algorithm is employed as a mean-field-based optimization framework.It constructs a Hamiltonian dynamical system by introducing generalized momentum fields,enabling efficient decoupling and dynamic evolution of strongly coupled struc-tural variables.Using the sequence optimization of a linear copolymer adsorbing on a solid surface as a case study,we demonstrate the applica-bility of the SB algorithm to high-dimensional,non-differentiable combinatorial optimization problems.Our results show that SB can efficiently discover polymer sequences with excellent adsorption performance within a reasonable computational time.Furthermore,it exhibits robust con-vergence and high parallel scalability across large design spaces.The approach developed in this work offers a new computational pathway for polymer structure optimization.It also lays a theoretical foundation for future extensions to topological design problems,such as optimizing the number and placement of side chains,as well as the co-optimization of sequence and topology.展开更多
With the development of technology,diffusion model-based solvers have shown significant promise in solving Combinatorial Optimization(CO)problems,particularly in tackling Non-deterministic Polynomial-time hard(NP-hard...With the development of technology,diffusion model-based solvers have shown significant promise in solving Combinatorial Optimization(CO)problems,particularly in tackling Non-deterministic Polynomial-time hard(NP-hard)problems such as the Traveling Salesman Problem(TSP).However,existing diffusion model-based solvers typically employ a fixed,uniform noise schedule(e.g.,linear or cosine annealing)across all training instances,failing to fully account for the unique characteristics of each problem instance.To address this challenge,we present GraphGuided Diffusion Solvers(GGDS),an enhanced method for improving graph-based diffusion models.GGDS leverages Graph Neural Networks(GNNs)to capture graph structural information embedded in node coordinates and adjacency matrices,dynamically adjusting the noise levels in the diffusion model.This study investigates the TSP by examining two distinct time-step noise generation strategies:cosine annealing and a Neural Network(NN)-based approach.We evaluate their performance across different problem scales,particularly after integrating graph structural information.Experimental results indicate that GGDS outperforms previous methods with average performance improvements of 18.7%,6.3%,and 88.7%on TSP-500,TSP-100,and TSP-50,respectively.Specifically,GGDS demonstrates superior performance on TSP-500 and TSP-50,while its performance on TSP-100 is either comparable to or slightly better than that of previous methods,depending on the chosen noise schedule and decoding strategy.展开更多
Ising machines offer a paradigm shift from traditional computing methods, tackling complex combinatorialoptimization problems (COPs). Despite the proliferation of various Ising machine implementations, their applicati...Ising machines offer a paradigm shift from traditional computing methods, tackling complex combinatorialoptimization problems (COPs). Despite the proliferation of various Ising machine implementations, their application tosolve real-world COPs has been limited. Here, we introduce a high-performance optoelectronic Ising machine (OEIM),based on optoelectronic parametric oscillators, that represents a significant advancement in this field. With 4096 Isingspins, arbitrary coupling capabilities, and unparalleled long-term stability, our OEIM outperforms traditional computingapproaches in both accuracy and speed. By solving the benchmark maximum cut problem, we demonstrate itssuperior performance. More importantly, we apply the OEIM to a real-world traffic optimization problem, using realtraffic data and a classical traffic model, and achieve results that far surpass those of conventional computers. This worknot only validates the OEIM’s capability to solve complex practical challenges but also heralds a new era in real-timetraffic management, where high-performance optoelectronic Ising machines enable rapid and efficient solutions tocritical societal issues.展开更多
Much of our daily tasks have been computerized by machines and sensors communicating with each other in real-time.There is a reasonable risk that something could go wrong because there are a lot of sensors producing a...Much of our daily tasks have been computerized by machines and sensors communicating with each other in real-time.There is a reasonable risk that something could go wrong because there are a lot of sensors producing a lot of data.Combinatorial testing(CT)can be used in this case to reduce risks and ensure conformance to specifications.Numerous existing metaheuristic-based solutions aim to assist the test suite generation for combinatorial testing,also known as t-way testing(where t indicates the interaction strength),viewed as an optimization problem.Much previous research,while helpful,only investigated a small number of interaction strengths up to t=6.For lightweight applications,research has demonstrated good fault-finding ability.However,the number of interaction strengths considered must be higher in the case of interactions that generate large amounts of data.Due to resource restrictions and the combinatorial explosion challenge,little work has been done to produce high-order interaction strength.In this context,the Whale Optimization Algorithm(WOA)is proposed to generate high-order interaction strength.To ensure that WOA conquers premature convergence and avoids local optima for large search spaces(owing to high-order interaction),three variants of WOA have been developed,namely Structurally Modified Whale Optimization Algorithm(SWOA),Tolerance Whale Optimization Algorithm(TWOA),and Tolerance Structurally Modified Whale Optimization Algorithm(TSWOA).Our experiments show that the third strategy gives the best performance and is comparable to existing state-of-thearts based strategies.展开更多
The minimum vertex cover problem(MVCP)is a well-known combinatorial optimization problem of graph theory.The MVCP is an NP(nondeterministic polynomial)complete problem and it has an exponential growing complexity with...The minimum vertex cover problem(MVCP)is a well-known combinatorial optimization problem of graph theory.The MVCP is an NP(nondeterministic polynomial)complete problem and it has an exponential growing complexity with respect to the size of a graph.No algorithm exits till date that can exactly solve the problem in a deterministic polynomial time scale.However,several algorithms are proposed that solve the problem approximately in a short polynomial time scale.Such algorithms are useful for large size graphs,for which exact solution of MVCP is impossible with current computational resources.The MVCP has a wide range of applications in the fields like bioinformatics,biochemistry,circuit design,electrical engineering,data aggregation,networking,internet traffic monitoring,pattern recognition,marketing and franchising etc.This work aims to solve the MVCP approximately by a novel graph decomposition approach.The decomposition of the graph yields a subgraph that contains edges shared by triangular edge structures.A subgraph is covered to yield a subgraph that forms one or more Hamiltonian cycles or paths.In order to reduce complexity of the algorithm a new strategy is also proposed.The reduction strategy can be used for any algorithm solving MVCP.Based on the graph decomposition and the reduction strategy,two algorithms are formulated to approximately solve the MVCP.These algorithms are tested using well known standard benchmark graphs.The key feature of the results is a good approximate error ratio and improvement in optimum vertex cover values for few graphs.展开更多
The All-pairs shortest path problem(ALL-SPP)aims to find the shortest path joining all the vertices in a given graph.This study proposed a new optimal method,Dhouib-matrix-ALL-SPP(DM-ALL-SPP)to solve the ALL-SPP based...The All-pairs shortest path problem(ALL-SPP)aims to find the shortest path joining all the vertices in a given graph.This study proposed a new optimal method,Dhouib-matrix-ALL-SPP(DM-ALL-SPP)to solve the ALL-SPP based on column-row navigation through the adjacency matrix.DM-ALL-SPP is designed to generate in a single execution the shortest path with details among all-pairs of vertices for a graph with positive and negative weighted edges.Even for graphs with a negative cycle,DM-ALL-SPP reported a negative cycle.In addition,DM-ALL-SPP continues to work for directed,undirected and mixed graphs.Furthermore,it is characterized by two phases:the first phase consists of adding by column repeated(n)iterations(where n is the number of vertices),and the second phase resides in adding by row executed in the worst case(n∗log(n))iterations.The first phase,focused on improving the elements of each column by adding their values to each row and modifying them with the smallest value.The second phase is emphasized by rows only for the elements modified in the first phase.Different instances from the literature were used to test the performance of the proposed DM-ALL-SPP method,which was developed using the Python programming language and the results were compared to those obtained by the Floyd-Warshall algorithm.展开更多
Traditional operational research methods have been the primary means of solving combinatorial optimization problems(COPs)for the past few decades.However,with the rapid increase in the scale of problems in real-world ...Traditional operational research methods have been the primary means of solving combinatorial optimization problems(COPs)for the past few decades.However,with the rapid increase in the scale of problems in real-world scenarios and the demand for online optimization,these methods face persistent challenges including computational complexity and optimality.In recent years,combinatorial optimization methods based on deep learning have rapidly evolved,progressing from tackling solely small-scale problems(e.g.,the traveling salesman problem(TSP)with fewer than 100 cities)to swiftly delivering high-quality solutions for graphs containing up to a million nodes.Particularly,in the last two years,a multitude of studies has surfaced,demonstrating the ability to generalize learned models to large-scale problems with diverse distributions.This capability empowers deep learning-based methods to demonstrate robust competitiveness,even when challenged by professional solvers.Consequently,this review summarizes the methods employed in recent years for solving COPs through deep learning(including prompt learning),scrutinizes the strengths and weaknesses of these methods,and concludes by highlighting potential directions for mitigating these weaknesses.展开更多
The Vehicle Routing Problem with Time Windows(VRPTW)presents a significant challenge in combinatorial optimization,especially under real-world uncertainties such as variable travel times,service durations,and dynamic ...The Vehicle Routing Problem with Time Windows(VRPTW)presents a significant challenge in combinatorial optimization,especially under real-world uncertainties such as variable travel times,service durations,and dynamic customer demands.These uncertainties make traditional deterministic models inadequate,often leading to suboptimal or infeasible solutions.To address these challenges,this work proposes an adaptive hybrid metaheuristic that integrates Genetic Algorithms(GA)with Local Search(LS),while incorporating stochastic uncertainty modeling through probabilistic travel times.The proposed algorithm dynamically adjusts parameters—such as mutation rate and local search probability—based on real-time search performance.This adaptivity enhances the algorithm’s ability to balance exploration and exploitation during the optimization process.Travel time uncertainties are modeled using Gaussian noise,and solution robustness is evaluated through scenario-based simulations.We test our method on a set of benchmark problems from Solomon’s instance suite,comparing its performance under deterministic and stochastic conditions.Results show that the proposed hybrid approach achieves up to a 9%reduction in expected total travel time and a 40% reduction in time window violations compared to baseline methods,including classical GA and non-adaptive hybrids.Additionally,the algorithm demonstrates strong robustness,with lower solution variance across uncertainty scenarios,and converges faster than competing approaches.These findings highlight the method’s suitability for practical logistics applications such as last-mile delivery and real-time transportation planning,where uncertainty and service-level constraints are critical.The flexibility and effectiveness of the proposed framework make it a promising candidate for deployment in dynamic,uncertainty-aware supply chain environments.展开更多
Combinatorial optimization problems are found in many application fields such as computer science, engineering and economy. In this paper, a new efficient meta-heuristic, Intersection-Based Scaling (IBS for abbreviati...Combinatorial optimization problems are found in many application fields such as computer science, engineering and economy. In this paper, a new efficient meta-heuristic, Intersection-Based Scaling (IBS for abbreviation), is proposed and it can be applied to the combinatorial optimization problems. The main idea of IBS is to scale the size of the instance based on the intersection of some local optima, and to simplify the search space by extracting the intersection from the instance, which makes the search more efficient. The combination of IBS with some local search heuristics of different combinatorial optimization problems such as Traveling Salesman Problem (TSP) and Graph Partitioning Problem (GPP) is studied, and comparisons are made with some of the best heuristic algorithms and meta-heuristic algorithms. It is found that it has significantly improved the performance of existing local search heuristics and significantly outperforms the known best algorithms. Keywords combinatorial optimization - TSP (Traveling Salesman Problem) - GPP (Graph Partitioning Problem) - IBS (Intersection-Based Scaling) - meta heuristic Regular PaperThis work is supported by the National Basic Research 973 Program of China (Grant No.TG1998030401).Peng Zou was born in 1979. He received the B.S. degree in computer software from University of Science and Technology of China (USTC) in 2000. Now he is a Ph.D. candidate in computer science of USTC. His current research interests include algorithms for NP-hard problems and parallel computing.Zhi Zhou was born in 1976. He received the B.S. degree in computer software from USTC in 1995. Now he is a Ph.D. candidate in computer science of USTC. His current research interests include combinatorial problem and parallel computing.Ying-Yu Wan was born in 1976. He received the B.S. degree in computer software from USTC in 1997, and the Ph.D. degree from USTC in 2002. His current research interests include parallel computing and combinatorial problem.Guo-Liang Chen was born in 1938. Now he is an Academician of CAS and Ph.D. supervisor in Department of Computer Science at USTC, director of the National High Performance Computing Center at Hefei. His current research interests include parallel computing, computer architecture and combinatorial optimization.Jun Gu was born in 1956. He received the B.S. degree in electronic engineering from USTC in 1982, and the Ph.D. degree in computer science from University of Utah. Now he is a professor and Ph.D. supervisor in computer science at USTC and Hong Kong University of Science and Technology. His main research interrests include algorithms for NP-hard problems.展开更多
Combinatorial Optimization Problems(COPs)are a class of optimization problems that are commonly encountered in industrial production and everyday life.Over the last few decades,traditional algorithms,such as exact alg...Combinatorial Optimization Problems(COPs)are a class of optimization problems that are commonly encountered in industrial production and everyday life.Over the last few decades,traditional algorithms,such as exact algorithms,approximate algorithms,and heuristic algorithms,have been proposed to solve COPs.However,as COPs in the real world become more complex,traditional algorithms struggle to generate optimal solutions in a limited amount of time.Since Deep Neural Networks(DNNs)are not heavily dependent on expert knowledge and are adequately flexible for generalization to various COPs,several DNN-based algorithms have been proposed in the last ten years for solving COPs.Herein,we categorize these algorithms into four classes and provide a brief overview of their applications in real-world problems.展开更多
Combinatorial Optimization(CO)problems have been intensively studied for decades with a wide range of applications.For some classic CO problems,e.g.,the Traveling Salesman Problem(TSP),both traditional planning algori...Combinatorial Optimization(CO)problems have been intensively studied for decades with a wide range of applications.For some classic CO problems,e.g.,the Traveling Salesman Problem(TSP),both traditional planning algorithms and the emerging reinforcement learning have made solid progress in recent years.However,for CO problems with nested sub-tasks,neither end-to-end reinforcement learning algorithms nor traditional evolutionary methods can obtain satisfactory strategies within a limited time and computational resources.In this paper,we propose an algorithmic framework for solving CO problems with nested sub-tasks,in which learning and planning algorithms can be combined in a modular way.We validate our framework in the Job-Shop Scheduling Problem(JSSP),and the experimental results show that our algorithm has good performance in both solution qualities and model generalizations.展开更多
基金supported by the National Key Research and Development Program of China(Grant No.2024YFA1408500)the National Natural Science Foundation of China(Grant Nos.12174028 and 12574115)the Open Fund of the State Key Laboratory of Spintronics Devices and Technologies(Grant No.SPL-2408)。
文摘We incorporate a non-Markovian feedback mechanism into the simulated bifurcation method for dynamical solvers addressing combinatorial optimization problems.By reinjecting a portion of dissipated kinetic energy into each spin in a history-dependent and trajectory-informed manner,the method effectively suppresses early freezing induced by inelastic boundaries and enhances the system's ability to explore complex energy landscapes.Numerical results on the maximum cut(MAX-CUT)instances of fully connected Sherrington–Kirkpatrick(SK)spin glass models,including the 2000-spin K_(2000)benchmark,demonstrate that the non-Markovian algorithm significantly improves both solution quality and convergence speed.Tests on randomly generated SK instances with 100 to 1000 spins further indicate favorable scalability and substantial gains in computational efficiency.Moreover,the proposed scheme is well suited for massively parallel hardware implementations,such as field-programmable gate arrays,providing a practical and scalable approach for solving large-scale combinatorial optimization problems.
基金supported by the National Natural Science Foundation of China(Grant No.92365206)the support of the China Postdoctoral Science Foundation(Certificate Number:2023M740272)+1 种基金supported by the National Natural Science Foundation of China(Grant No.12247168)China Postdoctoral Science Foundation(Certificate Number:2022TQ0036)。
文摘We introduce Quafu-Qcover,an open-source cloud-based software package developed for solving combinatorial optimization problems using quantum simulators and hardware backends.Quafu-Qcover provides a standardized and comprehensive workflow that utilizes the quantum approximate optimization algorithm(QAOA).It facilitates the automatic conversion of the original problem into a quadratic unconstrained binary optimization(QUBO)model and its corresponding Ising model,which can be subsequently transformed into a weight graph.The core of Qcover relies on a graph decomposition-based classical algorithm,which efficiently derives the optimal parameters for the shallow QAOA circuit.Quafu-Qcover incorporates a dedicated compiler capable of translating QAOA circuits into physical quantum circuits that can be executed on Quafu cloud quantum computers.Compared to a general-purpose compiler,our compiler demonstrates the ability to generate shorter circuit depths,while also exhibiting superior speed performance.Additionally,the Qcover compiler has the capability to dynamically create a library of qubits coupling substructures in real-time,utilizing the most recent calibration data from the superconducting quantum devices.This ensures that computational tasks can be assigned to connected physical qubits with the highest fidelity.The Quafu-Qcover allows us to retrieve quantum computing sampling results using a task ID at any time,enabling asynchronous processing.Moreover,it incorporates modules for results preprocessing and visualization,facilitating an intuitive display of solutions for combinatorial optimization problems.We hope that Quafu-Qcover can serve as an instructive illustration for how to explore application problems on the Quafu cloud quantum computers.
文摘This paper states a new metaheuristic based on Deterministic Finite Automata （DFA） for the multi - objective optimization of combinatorial problems. First, a new DFA named Multi - Objective Deterministic Finite Automata （MDFA） is defined. MDFA allows the representation of the feasible solutions space of combinatorial problems. Second, it is defined and implemented a metaheuritic based on MDFA theory. It is named Metaheuristic of Deterministic Swapping （MODS）. MODS is a local search strategy that works using a MDFA. Due to this, MODS never take into account unfeasible solutions. Hence, it is not necessary to verify the problem constraints for a new solution found. Lastly, MODS is tested using well know instances of the Bi-Objective Traveling Salesman Problem （TSP） from TSPLIB. Its results were compared with eight Ant Colony inspired algorithms and two Genetic algorithms taken from the specialized literature. The comparison was made using metrics such as Spacing, Generational Distance, Inverse Generational Distance and No-Dominated Generation Vectors. In every case, the MODS results on the metrics were always better and in some of those cases, the superiority was 100%.
基金supported by the National Natural Science Foundation of China(Grant No.12074335)the National Science and Technology Major Project of the Ministry of Science and Technology of China(Grant No.2016YFA0300402).
文摘Many problems in science,engineering and real life are related to the combinatorial optimization.However,many combinatorial optimization problems belong to a class of the NP-hard problems,and their globally optimal solutions are usually difficult to solve.Therefore,great attention has been attracted to the algorithms of searching the globally optimal solution or near-optimal solution for the combinatorial optimization problems.As a typical combinatorial optimization problem,the traveling salesman problem(TSP)often serves as a touchstone for novel approaches.It has been found that natural systems,particularly brain nervous systems,work at the critical region between order and disorder,namely,on the edge of chaos.In this work,an algorithm for the combinatorial optimization problems is proposed based on the neural networks on the edge of chaos(ECNN).The algorithm is then applied to TSPs of 10 cities,21 cities,48 cities and 70 cities.The results show that ECNN algorithm has strong ability to drive the networks away from local minimums.Compared with the transiently chaotic neural network(TCNN),the stochastic chaotic neural network(SCNN)algorithms and other optimization algorithms,much higher rates of globally optimal solutions and near-optimal solutions are obtained with ECNN algorithm.To conclude,our algorithm provides an effective way for solving the combinatorial optimization problems.
基金the National Natural Science Foundation of China(No.61502198)the Science&Technology Development Project of Jilin Province(Nos.20180101334JC and 20190302117GX)the"3th-Five Year" Science and Technology Research Project of Education Department of Jilin Province(No.JJKH20170574KJ)
文摘Traveling salesman problem(TSP) is one of the typical NP-hard problems, and it has been used in many engineering applications. However, the previous swarm intelligence(SI) based algorithms for TSP cannot coordinate with the exploration and exploitation abilities and are easily trapped into local optimum. In order to deal with this situation, a new hybrid optimization algorithm based on wolf pack search and local search(WPS-LS)is proposed for TSP. The new method firstly simulates the predatory process of wolf pack from the broad field to a specific place so that it allows for a search through all possible solution spaces and prevents wolf individuals from getting trapped into local optimum. Then, local search operation is used in the algorithm to improve the speed of solving and the accuracy of solution. The test of benchmarks selected from TSPLIB shows that the results obtained by this algorithm are better and closer to the theoretical optimal values with better robustness than those obtained by other methods.
基金supported by the Fundamental Research Funds for the Central Universities(No.2024JBZX029)Shijiazhuang High Level Science and Technology Innovation and Entrepreneurship Talent Project(No.08202307)the National Natural Science Foundation of China(NSFC)(No.22173004).
文摘The optimization of polymer structures aims to determine an optimal sequence or topology that achieves a given target property or structural performance.This inverse design problem involves searching within a vast combinatorial phase space defined by components,se-quences,and topologies,and is often computationally intractable due to its NP-hard nature.At the core of this challenge lies the need to evalu-ate complex correlations among structural variables,a classical problem in both statistical physics and combinatorial optimization.To address this,we adopt a mean-field approach that decouples direct variable-variable interactions into effective interactions between each variable and an auxiliary field.The simulated bifurcation(SB)algorithm is employed as a mean-field-based optimization framework.It constructs a Hamiltonian dynamical system by introducing generalized momentum fields,enabling efficient decoupling and dynamic evolution of strongly coupled struc-tural variables.Using the sequence optimization of a linear copolymer adsorbing on a solid surface as a case study,we demonstrate the applica-bility of the SB algorithm to high-dimensional,non-differentiable combinatorial optimization problems.Our results show that SB can efficiently discover polymer sequences with excellent adsorption performance within a reasonable computational time.Furthermore,it exhibits robust con-vergence and high parallel scalability across large design spaces.The approach developed in this work offers a new computational pathway for polymer structure optimization.It also lays a theoretical foundation for future extensions to topological design problems,such as optimizing the number and placement of side chains,as well as the co-optimization of sequence and topology.
基金supported by the National Science and Technology Council,Taiwan,under grant no.NSTC 114-2221-E-197-005-MY3.
文摘With the development of technology,diffusion model-based solvers have shown significant promise in solving Combinatorial Optimization(CO)problems,particularly in tackling Non-deterministic Polynomial-time hard(NP-hard)problems such as the Traveling Salesman Problem(TSP).However,existing diffusion model-based solvers typically employ a fixed,uniform noise schedule(e.g.,linear or cosine annealing)across all training instances,failing to fully account for the unique characteristics of each problem instance.To address this challenge,we present GraphGuided Diffusion Solvers(GGDS),an enhanced method for improving graph-based diffusion models.GGDS leverages Graph Neural Networks(GNNs)to capture graph structural information embedded in node coordinates and adjacency matrices,dynamically adjusting the noise levels in the diffusion model.This study investigates the TSP by examining two distinct time-step noise generation strategies:cosine annealing and a Neural Network(NN)-based approach.We evaluate their performance across different problem scales,particularly after integrating graph structural information.Experimental results indicate that GGDS outperforms previous methods with average performance improvements of 18.7%,6.3%,and 88.7%on TSP-500,TSP-100,and TSP-50,respectively.Specifically,GGDS demonstrates superior performance on TSP-500 and TSP-50,while its performance on TSP-100 is either comparable to or slightly better than that of previous methods,depending on the chosen noise schedule and decoding strategy.
基金National Key Research and Development Program of China grant 2022YFF1500900(M.L.)Key Research Program of the Chinese Academy of Sciences grant ZDRW-XX-2023-3(M.L.)+1 种基金National Natural Science Foundation of China grant 62135014(M.L.)International Partnership Program of the Chinese Academy of Sciences grant 100GJHZ2022028GC(M.L.).
文摘Ising machines offer a paradigm shift from traditional computing methods, tackling complex combinatorialoptimization problems (COPs). Despite the proliferation of various Ising machine implementations, their application tosolve real-world COPs has been limited. Here, we introduce a high-performance optoelectronic Ising machine (OEIM),based on optoelectronic parametric oscillators, that represents a significant advancement in this field. With 4096 Isingspins, arbitrary coupling capabilities, and unparalleled long-term stability, our OEIM outperforms traditional computingapproaches in both accuracy and speed. By solving the benchmark maximum cut problem, we demonstrate itssuperior performance. More importantly, we apply the OEIM to a real-world traffic optimization problem, using realtraffic data and a classical traffic model, and achieve results that far surpass those of conventional computers. This worknot only validates the OEIM’s capability to solve complex practical challenges but also heralds a new era in real-timetraffic management, where high-performance optoelectronic Ising machines enable rapid and efficient solutions tocritical societal issues.
基金This work was supported by the Ministry of Education,Malaysia(FRGS/1/2019/ICT02/UKM/01/1)the Universiti Kebangsaan Malaysia(DIP-2016-024).
文摘Much of our daily tasks have been computerized by machines and sensors communicating with each other in real-time.There is a reasonable risk that something could go wrong because there are a lot of sensors producing a lot of data.Combinatorial testing(CT)can be used in this case to reduce risks and ensure conformance to specifications.Numerous existing metaheuristic-based solutions aim to assist the test suite generation for combinatorial testing,also known as t-way testing(where t indicates the interaction strength),viewed as an optimization problem.Much previous research,while helpful,only investigated a small number of interaction strengths up to t=6.For lightweight applications,research has demonstrated good fault-finding ability.However,the number of interaction strengths considered must be higher in the case of interactions that generate large amounts of data.Due to resource restrictions and the combinatorial explosion challenge,little work has been done to produce high-order interaction strength.In this context,the Whale Optimization Algorithm(WOA)is proposed to generate high-order interaction strength.To ensure that WOA conquers premature convergence and avoids local optima for large search spaces(owing to high-order interaction),three variants of WOA have been developed,namely Structurally Modified Whale Optimization Algorithm(SWOA),Tolerance Whale Optimization Algorithm(TWOA),and Tolerance Structurally Modified Whale Optimization Algorithm(TSWOA).Our experiments show that the third strategy gives the best performance and is comparable to existing state-of-thearts based strategies.
文摘The minimum vertex cover problem(MVCP)is a well-known combinatorial optimization problem of graph theory.The MVCP is an NP(nondeterministic polynomial)complete problem and it has an exponential growing complexity with respect to the size of a graph.No algorithm exits till date that can exactly solve the problem in a deterministic polynomial time scale.However,several algorithms are proposed that solve the problem approximately in a short polynomial time scale.Such algorithms are useful for large size graphs,for which exact solution of MVCP is impossible with current computational resources.The MVCP has a wide range of applications in the fields like bioinformatics,biochemistry,circuit design,electrical engineering,data aggregation,networking,internet traffic monitoring,pattern recognition,marketing and franchising etc.This work aims to solve the MVCP approximately by a novel graph decomposition approach.The decomposition of the graph yields a subgraph that contains edges shared by triangular edge structures.A subgraph is covered to yield a subgraph that forms one or more Hamiltonian cycles or paths.In order to reduce complexity of the algorithm a new strategy is also proposed.The reduction strategy can be used for any algorithm solving MVCP.Based on the graph decomposition and the reduction strategy,two algorithms are formulated to approximately solve the MVCP.These algorithms are tested using well known standard benchmark graphs.The key feature of the results is a good approximate error ratio and improvement in optimum vertex cover values for few graphs.
文摘The All-pairs shortest path problem(ALL-SPP)aims to find the shortest path joining all the vertices in a given graph.This study proposed a new optimal method,Dhouib-matrix-ALL-SPP(DM-ALL-SPP)to solve the ALL-SPP based on column-row navigation through the adjacency matrix.DM-ALL-SPP is designed to generate in a single execution the shortest path with details among all-pairs of vertices for a graph with positive and negative weighted edges.Even for graphs with a negative cycle,DM-ALL-SPP reported a negative cycle.In addition,DM-ALL-SPP continues to work for directed,undirected and mixed graphs.Furthermore,it is characterized by two phases:the first phase consists of adding by column repeated(n)iterations(where n is the number of vertices),and the second phase resides in adding by row executed in the worst case(n∗log(n))iterations.The first phase,focused on improving the elements of each column by adding their values to each row and modifying them with the smallest value.The second phase is emphasized by rows only for the elements modified in the first phase.Different instances from the literature were used to test the performance of the proposed DM-ALL-SPP method,which was developed using the Python programming language and the results were compared to those obtained by the Floyd-Warshall algorithm.
基金supported by National Natural Science Foundation of China(Grant Nos.T2350003,31930022,12131020 and T2341007)the National Basic Research Program of China(Grant No.2022YFA1004800)Strategic Priority Research Program of the Chinese Academy of Sciences(Grant No.XDB38040400)。
文摘Traditional operational research methods have been the primary means of solving combinatorial optimization problems(COPs)for the past few decades.However,with the rapid increase in the scale of problems in real-world scenarios and the demand for online optimization,these methods face persistent challenges including computational complexity and optimality.In recent years,combinatorial optimization methods based on deep learning have rapidly evolved,progressing from tackling solely small-scale problems(e.g.,the traveling salesman problem(TSP)with fewer than 100 cities)to swiftly delivering high-quality solutions for graphs containing up to a million nodes.Particularly,in the last two years,a multitude of studies has surfaced,demonstrating the ability to generalize learned models to large-scale problems with diverse distributions.This capability empowers deep learning-based methods to demonstrate robust competitiveness,even when challenged by professional solvers.Consequently,this review summarizes the methods employed in recent years for solving COPs through deep learning(including prompt learning),scrutinizes the strengths and weaknesses of these methods,and concludes by highlighting potential directions for mitigating these weaknesses.
文摘The Vehicle Routing Problem with Time Windows(VRPTW)presents a significant challenge in combinatorial optimization,especially under real-world uncertainties such as variable travel times,service durations,and dynamic customer demands.These uncertainties make traditional deterministic models inadequate,often leading to suboptimal or infeasible solutions.To address these challenges,this work proposes an adaptive hybrid metaheuristic that integrates Genetic Algorithms(GA)with Local Search(LS),while incorporating stochastic uncertainty modeling through probabilistic travel times.The proposed algorithm dynamically adjusts parameters—such as mutation rate and local search probability—based on real-time search performance.This adaptivity enhances the algorithm’s ability to balance exploration and exploitation during the optimization process.Travel time uncertainties are modeled using Gaussian noise,and solution robustness is evaluated through scenario-based simulations.We test our method on a set of benchmark problems from Solomon’s instance suite,comparing its performance under deterministic and stochastic conditions.Results show that the proposed hybrid approach achieves up to a 9%reduction in expected total travel time and a 40% reduction in time window violations compared to baseline methods,including classical GA and non-adaptive hybrids.Additionally,the algorithm demonstrates strong robustness,with lower solution variance across uncertainty scenarios,and converges faster than competing approaches.These findings highlight the method’s suitability for practical logistics applications such as last-mile delivery and real-time transportation planning,where uncertainty and service-level constraints are critical.The flexibility and effectiveness of the proposed framework make it a promising candidate for deployment in dynamic,uncertainty-aware supply chain environments.
文摘Combinatorial optimization problems are found in many application fields such as computer science, engineering and economy. In this paper, a new efficient meta-heuristic, Intersection-Based Scaling (IBS for abbreviation), is proposed and it can be applied to the combinatorial optimization problems. The main idea of IBS is to scale the size of the instance based on the intersection of some local optima, and to simplify the search space by extracting the intersection from the instance, which makes the search more efficient. The combination of IBS with some local search heuristics of different combinatorial optimization problems such as Traveling Salesman Problem (TSP) and Graph Partitioning Problem (GPP) is studied, and comparisons are made with some of the best heuristic algorithms and meta-heuristic algorithms. It is found that it has significantly improved the performance of existing local search heuristics and significantly outperforms the known best algorithms. Keywords combinatorial optimization - TSP (Traveling Salesman Problem) - GPP (Graph Partitioning Problem) - IBS (Intersection-Based Scaling) - meta heuristic Regular PaperThis work is supported by the National Basic Research 973 Program of China (Grant No.TG1998030401).Peng Zou was born in 1979. He received the B.S. degree in computer software from University of Science and Technology of China (USTC) in 2000. Now he is a Ph.D. candidate in computer science of USTC. His current research interests include algorithms for NP-hard problems and parallel computing.Zhi Zhou was born in 1976. He received the B.S. degree in computer software from USTC in 1995. Now he is a Ph.D. candidate in computer science of USTC. His current research interests include combinatorial problem and parallel computing.Ying-Yu Wan was born in 1976. He received the B.S. degree in computer software from USTC in 1997, and the Ph.D. degree from USTC in 2002. His current research interests include parallel computing and combinatorial problem.Guo-Liang Chen was born in 1938. Now he is an Academician of CAS and Ph.D. supervisor in Department of Computer Science at USTC, director of the National High Performance Computing Center at Hefei. His current research interests include parallel computing, computer architecture and combinatorial optimization.Jun Gu was born in 1956. He received the B.S. degree in electronic engineering from USTC in 1982, and the Ph.D. degree in computer science from University of Utah. Now he is a professor and Ph.D. supervisor in computer science at USTC and Hong Kong University of Science and Technology. His main research interrests include algorithms for NP-hard problems.
基金supported by the National Natural Science Foundation of China(Nos.62173258 and 61773296).
文摘Combinatorial Optimization Problems(COPs)are a class of optimization problems that are commonly encountered in industrial production and everyday life.Over the last few decades,traditional algorithms,such as exact algorithms,approximate algorithms,and heuristic algorithms,have been proposed to solve COPs.However,as COPs in the real world become more complex,traditional algorithms struggle to generate optimal solutions in a limited amount of time.Since Deep Neural Networks(DNNs)are not heavily dependent on expert knowledge and are adequately flexible for generalization to various COPs,several DNN-based algorithms have been proposed in the last ten years for solving COPs.Herein,we categorize these algorithms into four classes and provide a brief overview of their applications in real-world problems.
基金supported by the National Key Research and Development Program of China(No.2020AAA0106302)National Natural Science Foundation of China(Nos.62061136001,92248303,62106123,and 61972224)Tsinghua Institute for Guo Qiang,and the High Performance Computing Center,Tsinghua University.
文摘Combinatorial Optimization(CO)problems have been intensively studied for decades with a wide range of applications.For some classic CO problems,e.g.,the Traveling Salesman Problem(TSP),both traditional planning algorithms and the emerging reinforcement learning have made solid progress in recent years.However,for CO problems with nested sub-tasks,neither end-to-end reinforcement learning algorithms nor traditional evolutionary methods can obtain satisfactory strategies within a limited time and computational resources.In this paper,we propose an algorithmic framework for solving CO problems with nested sub-tasks,in which learning and planning algorithms can be combined in a modular way.We validate our framework in the Job-Shop Scheduling Problem(JSSP),and the experimental results show that our algorithm has good performance in both solution qualities and model generalizations.
