Learning to compute the metric dimension of graphs

Abstract

For distinct vertices x,y,z in graph G=(V,E), we say the pair of vertices x,y can be distinguished by vertex z, if the distance inequation dG(x,z)≠dG(y,z) is true. If any two different vertices in graph G can be distinguished by at least one vertex in the vertex subset S⊂V then we say S⊂V is a resolving set of graph G. The one with minimum number of nodes among all of the resolving sets is called a metric base of graph G, of which the cardinality is called the metric dimension of graph G. The metric dimension problem(MDP) of computing the metric dimension of graphs is a kind of complex combinatorial optimization problem. In this paper, a hybrid algorithm for solving the MDP is developed, through which the metric base of graphs can be learned, and consequently the metric dimension of graphs is estimated. It is particular that the hybrid algorithm combines graph representation learning with greedy repair policy. Extensive experiments show that the hybrid algorithm has higher computational accuracy and efficiency on random graphs. It is also further suggested that the learning process of MDP is greatly influenced by the attributes of complex networks and the designs of graph neural networks themselves.