[论文解读] New Techniques for Graph Edit Distance Computation

David B. Blumenthal|arXiv (Cornell University)|Aug 1, 2019

Graph Theory and Algorithms参考文献 83被引用 6

一句话总结

本论文提出了一套新颖的图编辑距离（GED）计算技术，这是模式识别中用于有标签图的关键相似性度量。论文提出了新的LSAPE到LSAP的归约方法，基于精确MIP的算法，八种新的近似启发式算法，以及一个可扩展的C++库，显著提升了在图像分析和分子分析等多样化应用中GED计算的效率与准确性。

ABSTRACT

Due to their capacity to encode rich structural information, labeled graphs are often used for modeling various kinds of objects such as images, molecules, and chemical compounds. If pattern recognition problems such as clustering and classification are to be solved on these domains, a (dis-)similarity measure for labeled graphs has to be defined. A widely used measure is the graph edit distance (GED), which, intuitively, is defined as the minimum amount of distortion that has to be applied to a source graph in order to transform it into a target graph. The main advantage of GED is its flexibility and sensitivity to small differences between the input graphs. Its main drawback is that it is hard to compute. In this thesis, new results and techniques for several aspects of computing GED are presented. Firstly, theoretical aspects are discussed: competing definitions of GED are harmonized, the problem of computing GED is characterized in terms of complexity, and several reductions from GED to the quadratic assignment problem (QAP) are presented. Secondly, solvers for the linear sum assignment problem with error-correction (LSAPE) are discussed. LSAPE is a generalization of the well-known linear sum assignment problem (LSAP), and has to be solved as a subproblem by many GED algorithms. In particular, a new solver is presented that efficiently reduces LSAPE to LSAP. Thirdly, exact algorithms for computing GED are presented in a systematic way, and improvements of existing algorithms as well as a new mixed integer programming (MIP) based approach are introduced. Fourthly, a detailed overview of heuristic algorithms that approximate GED via upper and lower bounds is provided, and eight new heuristics are described. Finally, a new easily extensible C++ library for exactly or approximately computing GED is presented.

研究动机与目标

• 为解决图编辑距离（GED）计算的困难性，开发高效的确切算法与近似算法。
• 统一并澄清关于GED的相互竞争的定义，并确立其计算复杂度。
• 改进线性和分配问题带误差校正（LSAPE）的求解效率，这是GED计算中的核心子问题。
• 提供一个系统化的框架，用于精确与启发式GED计算，包括一种新的混合整数规划（MIP）公式。
• 提供一个模块化、可扩展的C++库，支持精确与近似GED计算。

提出的方法

• 提出一种新型求解器，将LSAPE问题归约为标准LSAP问题，从而可利用高度优化的LSAP求解器。
• 提出一种新的混合整数规划（MIP）公式，用于精确GED计算，提升了求解质量与性能。
• 开发了八种新颖的启发式算法，通过先进的剪枝与搜索策略，计算GED的上下界。
• 提出从GED到二次分配问题（QAP）的多种归约，将GED与一个著名的NP难问题联系起来。
• 设计并实现了模块化的C++库，支持精确与近似GED计算，并提供可扩展的接口。
• 采用理论分析，统一GED的定义并确立其复杂度类。

实验结果

研究问题

• RQ1 如何通过归约为LSAP，高效求解GED计算中的LSAPE子问题？
• RQ2 哪些是最有效的启发式策略，可用于在保持紧密上下界的情况下近似GED？
• RQ3 如何通过新颖的MIP公式与算法优化，改进精确GED计算？
• RQ4 GED与二次分配问题（QAP）之间存在怎样的理论关系？
• RQ5 如何设计一个通用的、可扩展的软件库，以支持多样化的GED计算需求？

主要发现

• 所提出的LSAPE到LSAP归约方法通过利用成熟的LSAP求解器，实现了更快、更可扩展的解决方案。
• 基于新MIP的精确算法在基准数据集上的求解质量与收敛速度方面优于现有方法。
• 八种新启发式算法显著降低了计算时间，同时在近似GED时保持了高精度。
• 从GED到QAP的理论归约，深化了对问题复杂性的理解，并开辟了新的算法路径。
• C++库已成功实现，并证明可轻松扩展，适用于各种研究与应用流程的集成。
• 对GED定义的统一消除了先前文献中的不一致性，强化了理论基础。