RSVP：超越 Weisfeiler-Lehman 图同构测试
RSVP: Beyond Weisfeiler Lehman Graph Isomorphism Test
AI总结：
本文提出多项式时间图同构测试启发式 RSVP，在 O(m^2+mn^2+n^3) 复杂度内超越 Weisfeiler-Lehman 方法，能识别多类 3-WL 无法区分的困难图实例。
摘要：
图同构是一个经典的算法问题，它判断两个输入图在结构上是否完全相同。有趣的是，它是少数几个尚未被证明属于 P 类或 NP 完全类复杂度的问题之一。因此，人们提出了基于智能搜索空间剪枝的策略来开发同构测试求解器，如 nauty 和 bliss，但遗憾的是，这些求解器在最坏情况下仍然是指数级的。因此，基于颜色细化的多项式时间 Weisfeiler-Lehman（WL）同构测试启发式方法在文献中被广泛采用。然而，WL 在多类非同构图实例上失效，例如强正则图、块结构和切换边等。在本文中，我们提出了一种新颖的多项式时间图同构测试启发式方法 RSVP，并展示了与 Weisfeiler-Lehman 方法相比，它在多个具有挑战性的图类上具有更强的判别能力。在运行时间复杂度为 O(m^2+mn^2+n^3) 的界限内（其中 n 和 m 分别是顶点数和边数），我们证明 RSVP 能够在多个“困难”图实例类中识别非同构性，包括 Miyazaki、Paulus、三次 hypohamiltonian、强正则、Latin series 和 Steiner triple system 图，而这些图是 3-WL 测试无法区分的。与 WL 测试类似，我们提出的算法只会产生单侧错误，即同构图永远不会被判定为非同构，尽管反向情况可能发生。
英文摘要：
Graph isomorphism, a classical algorithmic problem, determines whether two input graphs are structurally identical or not. Interestingly, it is one of the few problems that is not yet known to belong to either the P or NP-complete complexity classes. As such, intelligent search-space pruning based strategies were proposed for developing isomorphism testing solvers like nauty and bliss, which are still, unfortunately, exponential in the worst-case scenario. Thus, the polynomial-time Weisfeiler-Lehman (WL) isomorphism testing heuristic, based on colour refinement, has been widely adopted in the literature. However, WL fails for multiple classes of non-isomorphic graph instances such as strongly regular graphs, block structures, and switched edges, among others. In this paper, we propose a novel polynomial-time graph isomorphism testing heuristic, RSVP, and depict its enhanced discriminative power compared to the Weisfeiler-Lehman approach for several challenging classes of graphs. Bounded by a run-time complexity of O(m^2+mn^2+n^3) (where n and m are the number of vertices and edges respectively), we show that RSVP can identify non-isomorphism in several 'hard' graph instance classes including Miyazaki, Paulus, cubic hypohamiltonian, strongly regular, Latin series and Steiner triple system graphs, where the 3-WL test fails. Similar to the WL test, our proposed algorithm is prone to only one-sided errors, where isomorphic graphs will never be determined to be non-isomorphic, although the reverse can happen.
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