  Among various variants of the traveling salesman problem, the s-t-path graph
TSP has the special feature that we know the exact integrality ratio, 3/2, and
an approximation algorithm matching this ratio. In this paper, we go below this
threshold: we devise a polynomial-time algorithm for the s-t-path graph TSP
with approximation ratio 1.497. Our algorithm can be viewed as a refinement of
the 3/2-approximation algorithm by Seb\H{o} and Vygen [2014], but we introduce
several completely new techniques. These include a new type of
ear-decomposition, an enhanced ear induction that reveals a novel connection to
matroid union, a stronger lower bound, and a reduction of general instances to
instances in which s and t have small distance (which works for general
metrics).
