Decomposing the feasibility of Clustered Spanning Tree by Paths

Let H=〈V,S〉 be a hypergraph, where V is a set of vertices and S is a set of clusters S₁,…,S_m, S_i⊆V, such that the clusters in S are not necessarily disjoint. This paper considers the Clustered Spanning Tree by Paths problem, denoted by CSTP. This problem aims to decide whether a feasible solution tree exists, spanning all vertices of V, such that each cluster induces a path in the solution tree. We introduce the idea of a minimum cardinality feasible removal list and a minimum cardinality feasible insertion list, which removes or inserts vertices from or into clusters, such that using each one of the lists separately creates hypergraphs with feasible solution trees for CSTP. In addition, we decompose the intersection graph of H into smaller instances, for those cases where the intersection graph contains a cut node or a separating edge. We demonstrate how to compose a minimum cardinality feasible removal list or a minimum cardinality feasible insertion list of a given hypergraph from the smaller minimum cardinality feasible removal lists or minimum cardinality feasible insertion lists of the smaller instances. This approach may reduce the size and intricacy of the instances.