Computing Connected Resolvability of Graphs Using Binary Enhanced Harris Hawks Optimization

Abstract

In this paper, we consider the NP-hard problem of finding the minimum connected resolving set of graphs. A vertex set B of a connected graph G resolves G if every vertex of G is uniquely identified by its vector of distances to the vertices in B. A resolving set B of G is connected if the subgraph induced by B is a nontrivial connected subgraph of G. The cardinality of the minimal resolving set is the metric dimension of G and the cardinality of minimum connected resolving set is the connected metric dimension of G. The problem is solved heuristically by a binary version of an enhanced Harris Hawk Optimization (BEHHO) algorithm. This is the first attempt to determine the connected resolving set heuristically. BEHHO combines classical HHO with opposition-based learning, chaotic local search and is equipped with an S-shaped transfer function to convert the continuous variable into a binary one. The hawks of BEHHO are binary encoded and are used to represent which one of the vertices of a graph belongs to the connected resolving set. The feasibility is enforced by repairing hawks such that an additional node selected from V\B is added to B up to obtain the connected resolving set. The proposed BEHHO algorithm is compared to binary Harris Hawk Optimization (BHHO), binary opposition-based learning Harris Hawk Optimization (BOHHO), binary chaotic local search Harris Hawk Optimization (BCHHO) algorithms. Computational results confirm the superiority of the BEHHO for determining connected metric dimension.