Combinatorial optimization problems The need for this domain stems from the intractability of brute-force search which can take thousands of years to find the best solution to relatively search space. Combinatorial optimization is the process of searching for maxima (or minima) of an objective function F whose domain is a discrete but large configuration space (as opposed to an N-dimensional continuous space). e. Some simple examples of typical combinatorial optimization problems are: Feb 12, 2025 · Combinatorial Optimization Problems (COP) apply to a lot of interesting problems with real-world impacts. Jun 5, 2024 · In recent years, graph neural networks (GNNs) have become increasingly popular for solving NP-hard combinatorial optimization (CO) problems, such as maximum cut and maximum independent set. While less commonly used than linear programming, combinatorial optimization is an essential skill for finding optimal solutions for business problems. Mastering combinatorial optimization helps improve problem-solving skills and prepares students for real-world challenges in optimization. But in efforts to develop efficient methods to solve difficult combinatorial optimization problems, novel forms This paper surveys the recent attempts, both from the machine learning and operations research communities, at leveraging machine learning to solve combinatorial optimization problems. Maintained by members in SJTU-Thinklab: Chang Liu, Runzhong Wang, Jiayi Zhang, Zelin Zhao, Haoyu Geng, Tianzhe Wang The QUBO model (1) captures a wide range of integer and combinatorial optimization (COPT) problems; that is, optimization problems where some or all of the decision variables are restricted to be integers [see, *Department Industrial and Systems Engineering, Lehigh University, USA roq219@lehigh. For example, consider the integer 1 programming. Metaheuristic algorithms have been evolved and modified with respect to the problem Foundation Models for Combinatorial Optimization FM4CO contains interesting research papers (1) using Existing Large Language Models for Combinatorial Optimization, and (2) building Domain Foundation Models for Combinatorial Optimization. (b) Examples of combinatorial optimization problems: the travelling salesman problem (top) and MaxCut (bottom). It is based on numerous courses on combinatorial optimization and specialized topics, mostly at graduate level. This work potentially opens up new avenues for algorithm design in neural combinatorial optimization, bridging the gap Jun 11, 2025 · Numerous practical problems—ranging from machine learning to bioinformatics—can be formulated as combinatorial optimization problems. R. The advantage of this approach is that although large search spaces can be infeasible to search in their entireties, large swaths of each search space can be Combinatorial Optimization Problems (COPs) are a class of optimization problems that are commonly encountered in industrial production and everyday life. Recently, two-stage neural methods with divide-and-conquer strategies have shown efficiency in Since many combinatorial optimization problems, such as the set covering problem, can be explicitly or implicitly formulated on graphs, we believe that our work opens up a new avenue for graph algorithm design and discovery with deep learning. Short of proving P = NP, when one deals with an NP-hard problem one can either hope to design an exact algorithm that runs in polynomial time on “many” instances but has exponential worst-case running Abstract Combinatorial Optimization (CO) encompasses a wide range of problems that arise in many real-world scenarios. DISCO's efficacy is twofold: First, it enhances solution quality by constraining the sampling space to a more meaningful domain guided by solution residues, while preserving the multi Oct 1, 2021 · Conventionally, optimization problems in the discrete space are called combinatorial optimization (CO) problems and, typically, have different types of solutions comparing to the ones in the continuous space. However, as COPs in the real world become more complex, traditional algorithms Many traditional algorithms for solving combinatorial optimization problems involve using hand-crafted heuristics that sequentially construct a solution. May 30, 2024 · Scalable addressing of high-dimensional constrained combinatorial optimization problems is a challenge that arises in several science and engineering disciplines. Traditional computational methods often struggle with their exponential complexity, motivating exploration into alternative paradigms such as quantum computing. Some examples of problems in combinatorial optimization are: Grouping: Find a subset of members within Aug 1, 2022 · This chapter reviews a number of typical combinatorial optimization problems. Apr 10, 2023 · The process of optimizing business problems in these large-scale systems is named combinatorial optimization. This paper surveys the recent attempts, both from the machine learning and operations research communities, at leveraging machine learning to solve combinatorial optimization problems. Here we report a scalable ring-oscillator-based integrated circuit for optimization problem solv-ing. May 25, 2024 · Recently, deep neural networks have demonstrated remarkable performance in addressing combinatorial optimization challenges. Any combinatorial optimization problem can be formulated as a minimization or maximization problem, depending on the objective function. 组合优化 （英語： Combinatorial optimization）是 数学优化 的一个子领域，在应用数学和理论计算机科学的领域中，组合优化是在一个 有限集合 中找出最优对象的一类问题。 Mar 24, 2025 · This work introduces the free-energy machine (FEM), which combines statistical physics principles and machine learning techniques to solve various combinatorial optimization problems. 2 Combinatorial optimization problems A CO problem is an optimization problem where the number of possible solutions is finite and grows combinatorially with the problem size. Over the last few decades, traditional algorithms, such as exact algorithms, approximate algorithms, and heuristic algorithms, have been proposed to solve COPs. G. This tutorial gives an introduction to this evolving area. Panos M. In some applications, it is required to locate multiple optima. Oct 12, 2025 · IJCOPI aim is to provide a forum for the dissemination and discussion of combinatorial optimization problems and informatics. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons, 1988. , a combinatorial optimization problem. L. It aims to look for the perfect solution from a very huge solution space and allows an excellent usage of limited resources in order to attain a fundamental objective within a running time bounded by a polynomial in the Combinatorial Optimization provides a thorough treatment of linear programming and combinatorial optimization. These problems require finding an optimal solution from a finite but exponentially large set of possibilities and have been extensively studied in the May 16, 2025 · Quantum annealing (QA) has the potential to significantly improve solution quality and reduce time complexity in solving combinatorial optimization problems compared to classical optimization Many complex problems encountered in both production and daily life can be conceptualized as combinatorial optimization problems (COPs). Jun 1, 2024 · Combinatorial optimization problems are discrete optimization problems with a finite number of solutions [1]. Until recently, its methods have focused on solving problem instances in isolation, ignoring that they often stem from related data distributions in practice. The Ising model is of fundamental computational interest because any problem in the complexity class NP can be formulated as an Ising problem with only polynomial overhead, and thus a scalable Ising machine that outperforms existing standard digital computers could have a huge Problems with multiple optimal solutions widely exist in the real world. Combinatorial Optimization Problems (COPs) are fundamental in numerous practical applications, including route planning, circuit design, scheduling, and bioinformatics (Papadimitriou & Steiglitz, 1998; Cook et al. To address Combinatorial optimization problems are an important branch of computer science and operations research, which mainly studying the solution methods of discrete structure optimization problems. The integration of transfer learning (TL) with AutoRL in combinatorial optimization is an area that requires further research. Semide nite relaxations are used to derive lower bounds for the optimum while heuristics are applied to either nd upper The problems which our entropy quantum computing devices aim to solve are known as combinatorial optimization problems. Jan 16, 2022 · Combinatorial optimization problems (COPs) are a class of optimization problems with discrete decision variables and a finite search space, although still too large for an exhaustive search to be a realistic option (Korte, Vygen, Korte, & Vygen, 2012). However, the computational resources required to find an Jun 1, 2019 · With the development of machine learning in various fields, it can also be applied to combinatorial optimization problems, automatically discovering generic and fast heuristic algorithms based on training data, and requires fewer theoretical and empirical knowledge. Such a model would offer substantial advantages in terms of efficiency and Recently, neural combinatorial optimization (NCO) methods have been prevailing for solving multiobjective combinatorial optimization problems (MOCOPs). Edges correspond to assigning an energy J i Dec 30, 2022 · Nature Machine Intelligence - Modern graph neural networks do worse than classical greedy algorithms in solving combinatorial optimization problems like maximum independent set Aug 20, 2024 · Neural combinatorial optimization (NCO) is a promising learning-based approach to solving complex combinatorial optimization problems such as the traveling salesman problem (TSP), the vehicle routing problem (VRP), and the orienteering problem (OP). This success drives the scientific community towards the definition of new and better-performing heuristics and results in an increased interest in this research field. In this work, we develop a new framework to solve any combinatorial optimization problem over graphs that can be formulated as a single player game defined by states, actions, and rewards, including minimum spanning Combinatorial optimization algorithms for graph problems are usually designed afresh for each new problem with careful attention by an expert to the problem structure. Due to above, we do not make a clear cut to exclude other parts of dis-crete optimization. These situations in-clude disaster response planning, economic policy decision-making, and investment portfolio optimization. As a successful transformation to tackle complex dimensional problems, metaheuristic algorithms have been implemented in a wide area of combinatorial optimization problems. edu Abstract Combinatorial optimization is a well-established area in operations research and computer science. Dec 29, 2023 · Discover a Comprehensive Guide to some common combinatorial optimization problems in ai: Your go-to resource for understanding the intricate language of artificial intelligence. Recent work introduced novel Apr 16, 2021 · We advocate for pushing further the integration of machine learning and combinatorial optimization and detail a methodology to do so. The full name of COP is C ombinatorial O ptimization P roblems. The core idea behind these methods is to represent a CO problem as a graph and then use GNNs to learn the node/graph embedding with combinatorial information. Conventional techniques to solve discrete optimization problems that use simple continuous relaxation of the objective function followed by gradient-descent minimization are inherently unable Oct 1, 2021 · Many traditional algorithms for solving combinatorial optimization problems involve using hand-crafted heuristics that sequentially construct a solution. This paper deals with several combinatorial optimization problems. The main objective of this paper is characterizing the agent–environment interactions, namely, the state space Hence, the name combinatorial optimization. A range of quantum-, optical- and spintronic-based approaches have been explored for solving such combinatorial optimization problems, but they remain com-plicated to build and to scale. When the size of the feasible solution set is large, the complexity of the problem increases, and it is not easy to solve in a reasonable time with the current classical computer technology. Oct 19, 2021 · Combinatorial optimization problems are often considered NP-hard problems in the field of decision science and the industrial revolution. (a) An example of an 8 spin Ising model. Abstract. However, these methods still suffer from insufficient proximity and Sep 19, 2024 · Combinatorial optimization problems pose significant computational challenges across various fields, from logistics to cryptography. This comprehensive textbook on combinatorial optimization places special emphasis on theoretical results and algorithms with provably good performance, in contrast to heuristics. Such heuristics are designed by domain experts and may often be suboptimal due to the hard nature of the problems. This lesson will explain what those are and why they are valuable to be solved. On each node is a two valued spin (arrows). Despite significant progress, existing RL approaches face challenges such as diminish-ing reward signals and ineficient exploration in vast combinatorial action spaces, leading to inef-ficiency. Most NCO methods are based on the “Learning to Construct” (L2C) paradigm, where the trained model(s) can directly generate a set of approximate Pareto optimal solutions. Jan 1, 2018 · This paper examines various ways of utilizing machine learning to enhance or develop heuristic and metaheuristic algorithms for combinatorial optimization problems (COPs). In recent years the unconstrained quadratic binary program (UQP) has emerged as a unified framework for modeling and solving a wide variety of combi-natorial optimization problems. In Combinatorial optimization is a subfield of the optimization field of mathematics. The Ising model, combinatorial problems, and its energy landscape. edu Some examples of combinatorial optimization problems include the vehicle routing problem, scheduling problem, joint relay assignment and power allocation problem in green cooperative CRNs, and the Traveling Salesman Problem. 1 a, are prevalent in diverse fields, including logistics, resource allocation, communication network design, finance, drug discovery Feb 14, 2025 · This survey study provides a comprehensive investigation of the theories and recent advancements in applying RL to solve hard CO problems, such as vehicle routing, bin packing, assignment, scheduling, and planning problems, and, in addition, summarizes the applications of neural combinatorial optimization with reinforcement learning (NCO-RL). Since the introduction of the Quantum Approximate Optimization Algorithm (QAOA), numerous variants have been proposed to enhance its performance. However, most studies are dedicated to the continuous multi-solution optimization, while few works contribute to the discrete multi-solution optimization. Pardalos, University of Florida Given an optimization problem and a feasible solution to it, the corresponding inverse optimization problem is to find a minimal adjustment of the cost vector under some norm such that the given solution becomes optimum. However, these methods still suffer from insufficient proximity and Combinatorial Optimization (CO) drives many of the decision-support tools that we use today, across industries like logistics, finance, and healthcare. It always belongs to discrete optimization. stanford. Aug 16, 2023 · Many problem-specific heuristic frameworks have been developed to solve combinatorial optimization problems, but these frameworks do not generalize we… Sep 18, 2024 · Reinforcement Learning (RL) has emerged as a powerful tool for neural combinatorial optimization, enabling models to learn heuristics that solve complex problems without requiring optimal A modern platform for learningMathematics \ Combinatorics \ Combinatorial Optimization Description: Combinatorial optimization is a subfield of combinatorics and optimization theory that focuses on optimizing a specific objective function over a finite set of feasible solutions. These topics include the shortest path problem and network ows, which can be extended to solve more complex problems. Sep 19, 2024 · Combinatorial optimization problems pose significant computational challenges across various fields, from logistics to cryptography. QAOA was later extended to the Quantum Alternating Operator Ansatz (QAOA+), which generalizes the initial state a powerful tool for neural combinatorial opti-mization, enabling models to learn heuristics that solve complex problems without requiring expert knowledge. Although these methods have achieved promising Feb 5, 2022 · This is apparently because discrete/combinatorial optimization problems involve "treating the problem as a continuous problem" to first come up with a solution, and then determine if the solution lies within the feasible region, thus effectively solving two optimization problems in one. Many various important applied and theoretical problems of different degree of complexity can be presented as problems in graph theory. edu May 1, 2012 · Discovering the conditions under which an optimization algorithm or search heuristic will succeed or fail is critical for understanding the strengths and weaknesses of different algorithms, and for automated algorithm selection. The expressive power of graph neural networks combined with Reinforcement Learning (RL) enabled learning heuristics that rival or even surpass Mar 29, 2024 · Solving combinatorial optimization problems using quantum or quantum-inspired machine learning models would benefit from strategies able to work with arbitrary objective functions. However, many popular GNNs struggle to generalize to heterophilic scenarios where adjacent nodes tend to be with different labels or dissimilar features, such as graph coloring problem. Therefore, to solve the problem, we need only to study partitions of this special type, i. These problems include the travelling salesman problem, maximum independent set, maximum cut, and various other related problems. One prominent collective state computing paradigm embodied in the so-called Ising machines has recently attracted considerable research attention due to its ability to optimize Recently, neural combinatorial optimization (NCO) methods have been prevailing for solving multiobjective combinatorial optimization problems (MOCOPs). Vygen, Combinatorial Optimization: Theory and Algorithms, Algorithms and Combinatorics 21 Springer, Berlin Heidelberg New York, 2012. There are usually many ways to do so; ideally, one tries to get the “tightest” description (in which the feasible set of the linear program is the convex hull of integer solutions); but this is often too com-plicated to determine, and we have to Oct 1, 2025 · Permutation-based combinatorial optimization problems (PCOPs) [1] are a widely studied class of combinatorial optimization problems (COPs) in which solutions are represented as permutation vectors. Reinforcement learning (RL) proposes a good alternative to automate the search of these heuristics by training an agent in a Abstract Ising machines have generated much excitement in recent years due to their promise for solving the Ising problem, a graph-based hard combinatorial optimization problem. Oct 25, 2023 · The problem is conventionally solved with digital hardware and heuristic algorithms. In this work, we develop a new framework to solve any combinatorial optimization problem over graphs that can be formulated as a single player game defined by states, actions, and rewards, including minimum spanning tree E. Combinatorial optimization problems have numerous real-life applications, and adding explanation techniques for them can have a far-reaching impact on improving the understanding and acceptance of Jul 13, 2021 · In the past few decades, metaheuristics have demonstrated their suitability in addressing complex problems over different domains. 1 Introduction Hundreds of interesting and important combinatorial optimization problems are NP-hard, and so it is unlikely that any of them can be solved by an efficient exact algorithm. It also carries a prestigious selection of top quality, refereed articles with the emphasis on new problems, methods, techniques and algorithms to solve the combinatorial optimization problems. Apr 1, 2022 · The Ising model, combinatorial problems, and its energy landscape. The framework consists of an agent-based generalized solution algorithm (GSA) to Abstract Combinatorial Optimization (CO) problems are fundamentally important in numerous real-world applications across diverse industries, characterized by entailing enormous solution space and demanding time-sensitive response. Objective function f(x): F ®R. They have roots in combinatorics, operations research, and theoretical computer science. B. Here, the Oct 14, 2024 · The demand for efficient solvers of complicated combinatorial optimization problems, especially those classified as NP-complete or NP-hard, has recently led to increased exploration of novel computing architectures. In combinatorial optimization, a problem has a finite set of possible solutions. This paper employs both AutoRL and TL to effectively tackle combinatorial optimization challenges, specifically the asymmetric traveling salesman problem (ATSP) and the sequential ordering problem (SOP). Combinatorial Optimization ¶ Introduction ¶ Combinatorial optimization gathers different types of optimization problems where the decision maker needs to select an alternative from a finite set of possible alternatives with different outcomes, given a set of requirements (i. The combinatorial optimization is a proper sub eld of discrete optimiza-tion. This Review surveys Ising machines — special A combinatorial optimization problem (COP) is the problem of finding the optimal solution in a finite set. In this manuscript, the authors benchmark several quantum-annealing-inspired Apr 17, 2025 · Solving combinatorial optimization problems using variational quantum algorithms (VQAs) has emerged as a promising research direction. Brisset, J. FEM Jun 28, 2024 · We propose DISCO, an efficient DIffusion Solver for large-scale Combinatorial Optimization problems that excels in both solution quality and inference speed. Over the years, many classic algorithms have been proposed to solve CO problems on graphs Abstract Combinatorial optimization algorithms for graph problems are usually designed afresh for each new problem with careful attention by an expert to the problem structure. Inverse optimization problems have been applied in diverse areas, ranging from geophysical sciences, traffic networks, communication Jun 3, 2025 · Learn what combinatorial optimization is in AI. It illustrates the tenuous border that sometimes exists between an easy problem, for which effective algorithms are known, and an intractable one that differs merely by a small detail that Abstract. Mar 25, 2020 · As known, most of the combinatorial optimization problems are NP-hard in terms of complexity, and they are solved as part of one of the three predefined classifications: solution construction, solution improvement (or trajectory algorithms), and population-based Jul 19, 2024 · One of the most considered applications for quantum-inspired algorithms is solving combinatorial optimization problems. Introduction Combinatorial Optimization Problems (COPs) indicate a class of optimization problems. In many cases, combinatorial optimization seeks to find the best solution to problems that have a discrete set of possible solutions. Rein-forcement learning (RL) proposes a good alternative to automate the search of these heuristics by training an agent in a 1. The challenge is to find the optimal solution within that finite set. , 1994; Korte et al. Quantum annealing (QA) is a method that replaces classical simulated annealing (SA) methods that The QUBO model (1) captures a wide range of integer and combinatorial optimization (COPT) problems; that is, optimization problems where some or all of the decision variables are restricted to be integers [see, *Department Industrial and Systems Engineering, Lehigh University, USA roq219@lehigh. The methodology is illustrated by several examples and substantial computa-tional experience demonstrating the viability and robustness of the The only book offering solved exercises for integer and combinatorial optimization, this book contains 102 problems of varying scope and difficulty chosen from a plethora of topics and applications. In this paper, we propose Preference Summary. Combinatorial optimization is an emerging field at the forefront of combinatorics and theoretical computer science that aims to use combinatorial techniques to solve discrete optimization problems. This problem class includes well-studied problems such as traveling salesman problems (TSPs) [2] and quadratic assignment problems (QAPs) [3]. Short of proving P = NP, when one deals with an NP-hard problem one can either hope to design an exact algorithm that runs in polynomial time on “many” instances but has exponential worst-case running This paper explores advanced algorithms and techniques in optimization and machine learning, focusing on exact methods for solving complex problems. In this paper, we investigate the application of photonic quantum computing to solve combinatorial May 19, 2018 · Combinatorial optimization is a subset of mathematical optimization for identifying how to optimize their finite set of resources to optimize production or profits. In this work, we take a step towards this goal by designing an unified end-to-end deep reinforcement learning framework named Jan 10, 2025 · These methods are essential for solving a range of combinatorial problems and are frequently encountered in assignments, where students apply these techniques to find optimal or near-optimal solutions. Considered one of the essential areas within applied mathematics, it deals with problems where the objective is to Therefore, this is a combinatorial optimization problem. In the minimization version the problem consists in Finding x*Î F, such that f(x*) £ f(x) " x Î F. See full list on theory. Thus Why optimization? In some sense, all engineering design is optimization: choosing design parameters to improve some objective Apr 30, 2025 · In recent years, combinatorial optimization methods based on deep learning have rapidly evolved, progressing from tackling solely small-scale problems (e. A general agent-based framework is introduced, which enables learning both constructive and perturbative heuristics for a wide range of COPs. Many ad-hoc deep learning methods have been proposed to solve these problems, but there still lacks an effective unified framework. Large scale experimental studies – studying the performance of a variety of optimization algorithms across a large collection of diverse problem instances DIMES: A Differentiable Meta Solver for Combinatorial Optimization Problems A meta-learner for large-scale combinatorial optimization problems including the traveling salesman problem (TSP) and the maximum independent set problem (MIS). Lawler, Combinatorial Optimization: Networks and Matroids, Holt, Rinehart and Winston, 1976. It involves finding an optimal object from a finite set of objects. Molwitz and T. 2 Combinatorial Optimization Problem (COP) Formulation torial optimization problem, abbreviated as C is a formulation of the following typ Input: A R and a collection F of \fea Goal: minfP j2S cj : S 2 Fg. The repository is devoted to sharing advanced and lasted papers that solve challenging Combinatorial Optimization Problems. We will also brie y cover some basics of graph theory and solving linear programming problems to give context to the reader. The only book offering solved exercises for integer and combinatorial optimization, this book contains 102 problems of varying scope and difficulty chosen from a plethora of topics and applications. This book reviews the fundamentals, covers the classical topics (paths, flows, matching, matroids, NP CHR and (real world) optimization CHR proved to be an effective tool for related problems A notable application: optimal placement of local telecommunication transmitter stations [P. Topics include network flow, matching theory, matroid optimization, and approximation algorithms for NP-hard problems. To promote the multi-solution research in the discrete area, we design a benchmark test suite for multi Jan 24, 2025 · This paper presents a systematic review on reinforcement learning approaches for combinatorial optimization problems based on real-world industrial applications. Sep 20, 2023 · This research paper provides a comprehensive overview of combinatorial optimization problems, focusing on their significance, algorithmic approaches, real-world applications, and emerging research directions. We examine the theoretical foundations of combinatorial optimization and explore some of the most widely used algorithms. In the travelling salesman problem, the aim is to find the Combinatorial optimization problems are of great significance in many fields, and this paper explores the application of quantum annealing-based QUBO (Quadratic Unconstrained Binary Optimization) model to combinatorial optimization problems, including the 0/1 knapsack problem, the traveler's problem, the maximum cut problem, and the graph coloring problem. Is this correct? Combinatorial optimization problems are often too complex to be solved within reasonable time limits by exact methods, in spite of the theoretical guarantee that such methods will ultimately obtain an optimal solution. While significant progress has been made in developing learning-based methods for specialized CO problems, a unified model with a single architecture and parameter set for diverse CO problems remains elusive. Over the years, many classic algorithms have been proposed to solve CO problems on graphs Attractors Figure 1. In particular, Oscillator-based Ising Machines (OIMs), which consist of resistively-coupled nonlinear oscillators, are a promising on-chip iteration of Ising Machines. 1 Introduction Combinatorial optimization is the science that studies nding the optimal solution from a nite set of discrete possibilities. In such situations, it is often necessary to solve a “combinatorial optimization problem,” which involves evaluating different combinations of various factors and selecting the optimum combination. Graph neural networks for combinatorial optimization problems Soledad Villar based on work with Afonso Bandeira, Joan Bruna, Zhengdao Chen, Lei Chen, Alex Nowak, Weichi Yao Center for Data Science Courant Institute of Mathematical Sciences A fundamental method in combinatorial optimization is to write a combinatorial op-timization problem as a linear program with integer variables. One can formulate a CO problem as follows. Instead, heuristic methods, which do not offer a convergence guarantee, but which have greater flexibility to take advantage of special properties of the search space, are . Jan 1, 2023 · Combinatorial optimization (CO) [9] problems on graphs are a class of integer-constrained optimization problems and NP-hard problems, such as the representative traveling salesman problem (TSP) [31] and vehicle routing problem (VRP) [29], [33], [45], which are difficult to solve in polynomial time. Despite significant progress, existing RL approaches face challenges such as diminishing reward signals and inefficient exploration in vast combinatorial action spaces, leading to inefficiency. Jan 24, 2025 · Combinatorial optimization problems over graphs, such as the traveling salesman problem, longest path problem, and maximum independent set problem, are well-known for being computationally costly, some even NP-hard problems. Moreover, most existing methods are typically optimized for specific instances and lack generalizability. The most challenging such problem is the quadratic assign-ment problem. HypOp is also able to solve certain combinatorial problems that can’t be solved as effectively by prior methods. Single-stage neural combinatorial optimization solvers have achieved near-optimal results on various small-scale combinatorial optimization (CO) problems without needing expert knowledge. 5 days ago · Combinatorial optimization problems (COPs) encompass a class of problems that are aimed at finding optimal or near-optimal solutions within a finite solution space and that are prevalent in both Jan 1, 2014 · We consider combinatorial optimization problems (COP), i. May 4, 2022 · Minimizing the energy of the Ising model is a prototypical combinatorial optimization problem, ubiquitous in our increasingly automated world. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. Korte and J. A main point of the paper is seeing generic optimization problems as data points and inquiring what is the relevant distribution of problems to use for learning on a given task. One prominent collective state computing paradigm embodied in the so-called Ising machines has recently attracted considerable research attention due to its ability to optimize May 7, 2025 · Combinatorial Optimization (CO) encompasses a wide range of problems that arise in many real-world scenarios. In this paper, we propose a general quantum algorithm framework searching for approximate solutions to combinatorial optimization problems with linear objective functions Optimization problems with nonlinear cost functions and combinatorial constraints appear in many real-world applications but remain challenging to solve efficiently compared to their linear counterparts. g. Nemhauser and L. The research of combinatorial optimization problem involves the fields of information technology, communication network, computer vision and so on. , the traveling salesman problem (TSP) with fewer than 100 cities) to swiftly delivering high-quality solutions for graphs containing up to a million nodes. Subset Sums Combinatorial optimization problems are a very common type of Dec 7, 2017 · Abstract Original approach to determination of the concepts “combinatorial object” and “fuzzy combinatorial object” is offered, which allows to strictly formalize both the known and new classes of problems of combinatorial optimization. Discover how it solves complex problems by finding the best solution among many possible combinations. Historically, combinatorial problems have been approached with exact algorithms, which guarantee the optimality of the solution. We mark work contributed by Thinklab with ⭐. However, these solvers exhibit significant performance degradation when applied to large-scale CO problems. While significant progress has been made in developing learning-based meth-ods for specialized CO problems, a unified model with a single architecture and parameter set for di-verse CO problems remains elusive. COPs seek the values of a set of discrete variables that cause the objective function to reach Mar 18, 2024 · Combinatorial optimization problems (COPs), as shown in Fig. , finding extrema of an objective function on a combinatorial space. Learning to Solve Combinatorial Optimization Problems on Real-World Graphs in Linear Time Example Problems Over Graphs Polynomial Minimum Spanning Tree (MST) Single-Source Shortest Paths (SSP) NP-hard Traveling Salesman Problem (TSP) 1 Introduction Hundreds of interesting and important combinatorial optimization problems are NP-hard, and so it is unlikely that any of them can be solved by an efficient exact algorithm. Discrete Optimization Jan 1, 2023 · Combinatorial optimization (CO) [9] problems on graphs are a class of integer-constrained optimization problems and NP-hard problems, such as the representative traveling salesman problem (TSP) [31] and vehicle routing problem (VRP) [29], [33], [45], which are difficult to solve in polynomial time. Sep 1, 2025 · Graph Neural Networks (GNNs) are widely used to address combinatorial optimization problems. At its core, combinatorial optimization is about making the best possible decisions from a finite set of choices while respecting constraints. In fact, there exists problem in discrete optimization, which does not belong to combinatorial optimization. However, recent years have seen a surge of interest in using machine learning, especially graph neural networks (GNNs), as a key Feb 25, 2025 · Combinatorial optimization problems (COPs) stand out for the ir intrinsic complexity and widespread use in real-world applications such as schedu ling, routing, and resource allocation. In this paper, we investigate the application of photonic quantum computing to solve combinatorial Mar 29, 2024 · Solving combinatorial optimization problems using quantum or quantum-inspired machine learning models would benefit from strategies able to work with arbitrary objective functions. Modern deep learning tools are poised to solve these problems at unprecedented scales, but a unifying framework that 加權 平面圖 的 最小生成樹。尋找最小生成樹是涉及組合最佳化的常見問題。 組合最佳化 （英語： Combinatorial optimization）是 數學最佳化 的一個子領域，在應用數學和理論電腦科學的領域中，組合最佳化是在一個 有限集合 中找出最佳對象的一類問題。 [1] 在很多組合最佳化的問題中，窮舉搜尋 Ising machines are hardware solvers that aim to find the absolute or approximate ground states of the Ising model. The "best" solution is May 13, 2025 · Reinforcement Learning (RL) has emerged as a powerful tool for neural combinatorial optimization, enabling models to learn heuristics that solve complex problems without requiring expert knowledge. While this topic is increasing in popularity, explicit implementation details are not always available in the literature. Nevertheless, new studies have been focused on developing new algorithms without providing Understanding Combinatorial Optimization Combinatorial optimization is a topic that lies at the intersection of computer science, mathematics, and operations research. constraints). The quantum annealing algorithm can We present a method to approximately solve general instances of combinatorial optimization problems using the physical dynamics of three-dimensional (3D) rotors obeying Landau-Lifshitz-Gilbert dynamics. Apr 21, 2022 · Combinatorial optimization problems are pervasive across science and industry. , 2011). Fruehwirth] P r eface Com binatorial optimization is a liv ely eld of applied mathematics com bining tec hniques from com binatorics linear programming and the theory of algo rithms to solv e optimization problems o v er discrete structures There are a n um b er of classic texts in this eld but w e felt that there is a place for a new treatmen t of the sub We would like to maintain a list of resources that utilize machine learning technologies to solve combinatorial optimization problems. Edges correspond to assigning an energy Jij = 1. In this tutorial, we’ll learn about major problems and their solutions. It is considered in both two dimensions (QAP) and in three dimensions (Q3AP) and in the context of communication engi-neering. Oct 22, 2024 · This paper proposes a framework that formulates a wide range of graph combinatorial optimization problems using permutation-based representations. Given the hard nature of these problems, state-of-the-art algorithms rely on hand-crafted heuristics for making decisions that are otherwise too expensive to compute or mathematically not well defined. For each combinatorial optimization problem, two variants are distinguished: the search variant, which finds a solution with minimal or maximal objective function value, and the evaluation variant, which finds Combinatorial Optimization Combinatorial Optimization problem Given a set of elements E = {1, 2, , n} Set of feasible solutions F Each element of F is a subset of E. Jun 10, 2024 · The framework is able to solve combinatorial optimization problems significantly faster than existing methods. This paper will cover some topics of combinatorial optimization, the study of nding the best possible arrangement of a set of discrete objects. wsjxs olaj zegx twbpy xayk ousxy aukj xypq ekjrl vkebpjy svdlx dwkeolxf ukkhm ezwxr bgcag