[论文解读] A Quadratic Assignment Formulation of the Graph Edit Distance

Sébastien Bougleux, Luc Brun+4 more|arXiv (Cornell University)|Dec 23, 2015

Graph Theory and Algorithms参考文献 32被引用 10

一句话总结

本文通过引入一个ϵ-分配矩阵来将图编辑距离（GED）表述为一个二次分配问题（QAP），该矩阵编码了节点替换、删除和插入操作。主要贡献在于建立了ϵ-分配矩阵的等价类与从源节点子集到目标节点的单射映射之间的一一对应关系，从而通过QAP优化实现精确的GED计算。

ABSTRACT

Computing efficiently a robust measure of similarity or dissimilarity between graphs is a major challenge in Pattern Recognition. The Graph Edit Distance (GED) is a flexible measure of dissimilarity between graphs which arises in error-tolerant graph matching. It is defined from an optimal sequence of edit operations (edit path) transforming one graph into an other. Unfortunately, the exact computation of this measure is NP-hard. In the last decade, several approaches have been proposed to approximate the GED in polynomial time, mainly by solving linear programming problems. Among them, the bipartite GED has received much attention. It is deduced from a linear sum assignment of the nodes of the two graphs, which can be efficiently computed by Hungarian-type algorithms. However, edit operations on nodes and edges are not handled simultaneously, which limits the accuracy of the approximation. To overcome this limitation, we propose to extend the linear assignment model to a quadratic one, for directed or undirected graphs having labelized nodes and edges. This is realized through the definition of a family of edit paths induced by assignments between nodes. We formally show that the GED, restricted to the paths in this family, is equivalent to a quadratic assignment problem. Since this problem is NP-hard, we propose to compute an approximate solution by an adaptation of the Integer Projected Fixed Point method. Experiments show that the proposed approach is generally able to reach a more accurate approximation of the optimal GED than the bipartite GED, with a computational cost that is still affordable for graphs of non trivial sizes.

研究动机与目标

• 通过二次分配规划形式化图编辑距离的数学表述。
• 定义一类新矩阵——ϵ-分配矩阵，用于编码图上所有可能的编辑操作（替换、删除、插入）。
• 建立ϵ-分配矩阵等价类与从源节点子集到目标节点的单射函数之间的一一对应关系。
• 通过明确定义的组合优化框架，实现图编辑距离的精确计算。

提出的方法

• 引入一个(n+m)×(m+n)的ϵ-分配矩阵，由四个块矩阵组成：X_sub（替换）、X_rem（删除）、X_ins（插入）和X_ϵ（用于排列约束的辅助矩阵）。
• 施加约束条件，确保X_rem和X_ins为对角矩阵，从而保证每次仅能删除或插入一个节点。
• 基于替换、删除和插入块的相等性，定义ϵ-分配矩阵上的等价关系。
• 利用辅助矩阵X_ϵ，确保整个矩阵X为排列矩阵，从而保持有效编辑路径所需的结构。
• 建立ϵ-分配矩阵等价类与从源节点子集到目标节点的单射函数之间的双射映射。
• 利用排列矩阵结构，将有效编辑序列建模为单射映射，从而实现基于QAP的精确GED计算。

实验结果

研究问题

• RQ1 如何将图编辑距离正式表述为一个二次分配问题？
• RQ2 需要何种矩阵结构才能编码所有可能的编辑操作（替换、删除、插入），同时保持有效性？
• RQ3 是否存在有效编辑路径与节点集之间单射映射之间的一一对应关系？
• RQ4 如何定义ϵ-分配矩阵的等价性，以减少搜索空间中的冗余？
• RQ5 QAP形式化的GED能否建立在具有与编辑路径明确对应关系的明确定义的组合结构之上？

主要发现

• 所有ϵ-分配矩阵的集合An,m被定义为满足排列矩阵约束和特定块结构的(n+m)×(m+n)矩阵。
• 在An,m上的等价关系∼识别出具有相同替换、删除和插入块的矩阵，形成集合An,m∼。
• An,m∼与从X={1,…,n}的子集到Y={1,…,m}的单射函数集合之间存在一一对应关系。
• 由于排列矩阵约束，映射φ_X是单射的，确保没有两个源节点映射到同一个目标节点。
• 该形式化确保所有有效编辑路径均通过单射函数表示被完整捕获。
• QAP形式化为精确图编辑距离计算提供了严谨的代数框架。