Computational complexity of the MSTCI problem

Determine the computational complexity class of the Minimum Spanning Tree Cycle Intersection (MSTCI) problem, which asks: given a connected graph G, find a spanning tree T that minimizes the number of non-empty pairwise intersections among the tree-cycles induced by the non-tree edges of G.

Background

The Minimum Spanning Tree Cycle Intersection (MSTCI) problem seeks a spanning tree T of a connected graph G that minimizes the number of non-empty pairwise intersections among tree-cycles induced by non-tree edges. This quantity is called the intersection number and is denoted by cap(G).

While related problems such as the Minimum Fundamental Cycle Basis (MFCB) are known to be NP-Hard, the MSTCI problem does not currently have an established complexity classification. Identifying its complexity class would clarify whether exact algorithms or efficient approximations are feasible in general, and would situate MSTCI among known combinatorial optimization problems in graph theory.