FPT status of MAD Spanning Tree parameterized by treewidth

Determine whether the MAD Spanning Tree (MADST) problem, which asks for a spanning tree T of a connected graph G with Wiener index at most b, is fixed-parameter tractable when parameterized by the treewidth of G; that is, establish whether there exists an algorithm running in time f(tw(G))·|V(G)|^{O(1)} for some computable function f.

Background

The paper provides a dynamic program that solves MADST in time 2^{O(2^k)} n^{O(k)} on graphs of treewidth k, placing the problem in XP but not FPT for this parameter. After presenting FPT results for other parameters (modular width, vertex integrity, and an above-guarantee parameter), the authors explicitly note that the fixed-parameter tractability of MADST with respect to treewidth remains unresolved.

Clarifying the FPT status for treewidth would align MADST with many classical graph problems where treewidth is a central parameter enabling efficient algorithms.