Achieving feasibility for clustered traveling salesman problems using PQ‐trees

Let H=⟨V,𝒮⟩ be a hypergraph, where V$$ V $$ is a set of vertices and 𝒮 is a set of clusters S1,…,Sm$$ {S}_1,\dots, {S}_m $$ , Si⊆V$$ {S}_i\subseteq V $$ , such that the clusters in 𝒮 are not necessarily disjoint. This article considers the feasibility clustered traveling salesman problem, denoted by FCTSP$$ FCTSP $$ . In the FCTSP$$ FCTSP $$ we aim to decide whether a simple path exists that visits each vertex exactly once, such that the vertices of each cluster are visited consecutively. We focus on hypergraphs with no feasible solution path and consider removing vertices from clusters, such that the hypergraph with the new clusters has a feasible solution path for FCTSP$$ FCTSP $$ . The algorithm uses a PQ‐tree data structure and runs in linear time.