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<p>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.</p>

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Combinatorial optimization: From deep learning to large language models

  • 期刊论文

  • 发表日期：

    2025年4月30日

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  • Peng Tao,
  • Luonan Chen

摘要

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.