Variable neighborhood search for metric dimension and minimal doubly resolving set problems

In this paper, two similar NP-hard optimization problems on graphs are considered: the metric dimension problem and the problem of determining a doubly resolving set with the minimum cardinality. Both are present in many diverse areas, including network discovery and verification, robot navigation, and chemistry. For each problem, a new mathematical programming formulation is proposed. For solving more realistic large-size instances, a variable neighborhood search based heuristic is designed. An extensive experimental comparison on five different types of instances indicates that the VNS approach consistently outperforms a genetic algorithm, the only existing heuristic in the literature designed for solving those problems.