Achieving feasibility for clustered traveling salesman problems using PQ-trees

Let (Figure presented.) be a hypergraph, where (Figure presented.) is a set of vertices and (Figure presented.) is a set of clusters (Figure presented.), (Figure presented.), such that the clusters in (Figure presented.) are not necessarily disjoint. This article considers the feasibility clustered traveling salesman problem, denoted by (Figure presented.). In the (Figure presented.) 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 (Figure presented.). The algorithm uses a PQ-tree data structure and runs in linear time.