Explore a groundbreaking lecture on the s-t-path graph Traveling Salesman Problem (TSP) that surpasses the known integrality ratio of 3/2. Delve into a polynomial-time algorithm with an improved approximation ratio of 1.497, presented as part of the Combinatorial Optimization workshop at the Hausdorff Center for Mathematics. Discover novel techniques introduced by the speaker, including a new type of ear-decomposition, an enhanced ear induction linked to matroid union, a stronger lower bound, and a method to reduce general instances to those where s and t have small distances. Gain insights into this collaborative work that refines previous approximation algorithms and pushes the boundaries of combinatorial optimization in graph theory.