The graph isomorphism problem looks deceptively simple, but although polynomial-time algorithms exist for certain types of graphs such as planar graphs and graphs with bounded degree or eigenvalue multiplicity, its complexity class is still unknown. Information about potential isomorphisms between two graphs is contained in the eigenvalues and eigenvectors of their adjacency matrices. However, symmetries of graphs often lead to repeated eigenvalues so that associated eigenvectors are determined only up to basis rotations, which complicates graph isomorphism testing. We consider orthogonal and doubly stochastic relaxations of the graph isomorphism problem, analyze the geometric properties of the resulting solution spaces, and show that their complexity increases significantly if repeated eigenvalues exist. By restricting the search space to suitable subspaces, we derive an efficient Frank-Wolfe based continuous optimization approach for detecting isomorphisms. We illustrate the efficacy of the algorithm with the aid of various highly symmetric graphs.

翻译：图同构问题看似简单，但尽管对于平面图、有界度或特征值重数的图等特定类型，存在多项式时间算法，其复杂性类别至今未知。两个图之间潜在同构的信息蕴含在邻接矩阵的特征值和特征向量中。然而，图的对称性常导致特征值重复，使得相关特征向量仅在基旋转下确定，这增加了图同构检测的复杂性。我们考虑图同构问题的正交与双随机松弛，分析所得解空间的几何性质，并证明若存在重复特征值，则其复杂度显著增加。通过将搜索空间限制在合适的子空间，我们推导出一种基于Frank-Wolfe的高效连续优化方法来检测同构。借助多种高度对称的图，我们展示了该算法的有效性。