Novel two-level hybrid genetic algorithms based on different Cayley-type encodings for solving the clustered shortest-path tree problem

This paper investigates the clustered shortest-path tree (CluSPT) problem, a generalized network design problem that has several applications in various areas such as network design, agricultural irrigation settings, distribution of goods and products, etc. The main characteristic of the investigated problem is its hierarchical structure with the nodes of the underlying graph split into a given number of clusters. Its aim is finding a shortest-path spanning tree from a predefined source node to all the other nodes of the graph, with the property that every cluster should generate a connected subgraph. In this paper, we present two different two-level genetic algorithms (GAs) for solving the clustered shortest-path tree problem, in which, a framework with a macro-level sub-problem is used to generate trees spanning the clusters using two different Cayley encodings. The encodings in question are Prufer and Dandelion codes. This sub-problem is coupled with a micro-level sub-problem that determines the best corresponding feasible solution of the CluSPT problem associated to a given tree spanning the clusters by means of a dynamic programming algorithm. Extensive computational results are stated and interpreted on the two sets of benchmark instances available from the literature and containing 452 non-euclidean instances. The achieved results prove that our novel two-level based genetic algorithms are highly competitive against the existing solution approaches from the specialized literature, more precisely, 146 best known solutions were equaled and new improved solutions were found for 292 out of 452 benchmark instances. Between our proposed solution approaches the differences are not remarkable for small instances, but when the size of the problem grows, the GA based on Dandelion code outperforms the GA based on Prufer code, being better suited to evolutionary search due to its locality and heritability properties.