  The Metric Traveling Salesman Problem (TSP) is a classical NP-hard
optimization problem. The double-tree shortcutting method for Metric TSP yields
an exponentially-sized space of TSP tours, each of which approximates the
optimal solution within at most a factor of 2. We consider the problem of
finding among these tours the one that gives the closest approximation, i.e.\
the \emph{minimum-weight double-tree shortcutting}. Previously, we gave an
efficient algorithm for this problem, and carried out its experimental
analysis. In this paper, we address the related question of the worst-case
approximation ratio for the minimum-weight double-tree shortcutting method. In
particular, we give lower bounds on the approximation ratio in some specific
metric spaces: the ratio of 2 in the discrete shortest path metric, 1.622 in
the planar Euclidean metric, and 1.666 in the planar Minkowski metric. The
first of these lower bounds is tight; we conjecture that the other two bounds
are also tight, and in particular that the minimum-weight double-tree method
provides a 1.622-approximation for planar Euclidean TSP.
