  In the Traveling Salesperson Problem (TSP) we are given a list of locations
and the distances between each pair of them. The goal is to find the shortest
possible tour that visits each location exactly once and returns to the
starting location. Inspired by the fact that general TSP cannot be approximated
in polynomial time within any constant factor, while metric TSP admits a
(slightly better than) $1.5$-approximation in polynomial time, Zhou, Li and Guo
[Zhou et al., ISAAC '22] introduced a parameter that measures the distance of a
given TSP instance from the metric case. They gave an FPT $3$-approximation
algorithm parameterized by $k$, where $k$ is the number of triangles in which
the edge costs violate the triangle inequality. In this paper, we design a
$2.5$-approximation algorithm that runs in FPT time, improving the result of
[Zhou et al., ISAAC '22].
