Graph theory has been a powerful tool in solving difficult and complex problems arising in all disciplines. In particular, graph matching is a classical problem in pattern analysis with enormous applications. Many graph problems have been formulated as a mathematical program and then solved using exact, heuristic, and/or approximated-guaranteed procedures. On the other hand, graph theory has been a powerful tool in visualizing and understanding complex mathematical programming problems, especially integer programs. Formulating a graph problem as a natural integer program (IP) is often a challenging task. However, an IP formulation of the problem has many advantages. Several researchers have noted the need for natural IP formulation of graph theoretic problems. The present study aims to provide a unified framework for IP formulation of graph-matching problems. Although there are many surveys on graph matching problems, none is concerned with IP formulation. This paper is the first to provide a comprehensive IP formulation for such problems. The framework includes a variety of graph optimization problems in the literature. While these problems have been studied by different research communities, however, the framework presented here helps to bring efforts from different disciplines to tackle such diverse and complex problems. We hope the present study can significantly help to simplify some of the difficult problems arising in practice, especially in pattern analysis.

翻译：图论已成为解决各学科领域中涌现的复杂难题的有力工具。其中，图匹配是模式分析中的一个经典问题，具有广泛的应用价值。许多图论问题常被建模为数学规划问题，并采用精确算法、启发式方法或近似保证算法进行求解。另一方面，图论在可视化与理解复杂数学规划问题（尤其是整数规划）方面同样发挥着重要作用。将图论问题自然地表述为整数规划模型通常具有挑战性，但该建模方式具有诸多优势。多位学者已指出对图论问题建立自然整数规划模型的需求。本研究旨在为图匹配问题构建统一的整数规划建模框架。尽管现有大量关于图匹配问题的综述研究，但尚未有文献系统探讨其整数规划建模。本文首次为此类问题提供了完整的整数规划建模体系。该框架涵盖了文献中多种图优化问题，虽然这些问题由不同研究群体分别探索，但本框架有助于整合跨学科力量以应对此类复杂多样的问题。我们期望本研究能显著简化实践（尤其是模式分析领域）中出现的若干难题。