Marking Vertices to Find Graph Isomorphism Mapping Based on Continuous-Time Quantum Walk

Abstract

The isomorphism problem involves judging whether two graphs are topologically the same and producing structure-preserving isomorphism mapping. It is widely used in various areas. Diverse algorithms have been proposed to solve this problem in polynomial time, with the help of quantum walks. Some of these algorithms, however, fail to find the isomorphism mapping. Moreover, most algorithms have very limited performance on regular graphs which are generally difficult to deal with due to their symmetry. We propose IsoMarking to discover an isomorphism mapping effectively, based on the quantum walk which is sensitive to topological structures. Firstly, IsoMarking marks vertices so that it can reduce the harmful influence of symmetry. Secondly, IsoMarking can ascertain whether the current candidate bijection is consistent with existing bijections and eventually obtains qualified mapping. Thirdly, our experiments on 1585 pairs of graphs demonstrate that our algorithm performs significantly better on both ordinary graphs and regular graphs.

MapEff: An Effective Graph Isomorphism Agorithm Based on the Discrete-Time Quantum Walk

“…The verification method was to test all the adjoint nodes and to see whether their mapping nodes were adjacent. This is the same with the verification method in [ 13 , 14 ]. If the result of isomorphism mapping passed the verification, it was deemed to be correct, and vice versa.…”

“…In our experiments, 16 groups of graphs with 1585 pairs isomorphic graphs were used in total, and the datasets are the same ones referenced in [ 14 ]. We concentrated on the accuracy performance of discovering isomorphism mapping since accuracy is the key indicator of approximation algorithm.…”

“…Some source codes about the baselines are available from the thesis [ 13 ]. And we also implemented the other codes, including Emms-D, Emms-C, and IsoMarking, according to related work [ 1 , 12 , 14 ].…”

“…However, these methods have limited performance for regular graph. Recently, we adopted the idea of vertex marking in this algorithm to reduce the impact of symmetry on a regular graph and designed a new algorithm named IsoMarking [ 14 ]. Consequently, it performs better compared with Qiang’s methods.…”

“…Therefore, the strongly similar graphs affect the accuracy of this kind of algorithm. In the last year, we proposed a novel algorithm named IsoMarking [ 14 ] by marking vertices to distinguish the symmetric nodes to reduce the harmful effect of symmetry in regular graphs. However, marking process for each node introduced a heavy complexity, thus a low efficiency, especially in large number of nodes.…”