The integrality gap of the subtour elimination relaxation for the Traveling Salesman Problem is a longstanding open problem, epitomized by the \(\frac{4}{3}\)-conjecture.
Understanding this gap requires a detailed analysis of the extreme points of the subtour elimination polytope.
In this work, we introduce a new perspective for studying the integrality gap through what we call the Asymptotic Framework.
Rather than focusing on individual vertices, we consider entire families of vertices and analyze their behavior in the limit.
This approach allows us to capture structural properties that might otherwise remain hidden at the level of isolated instances.
We develop tools for systematically constructing and analyzing such families, and we demonstrate how this framework yields new insights into the broader study of the integrality gap.
As a first application of the framework, we prove the \(frac{4}{3}\)-conjecture for a class of instances that we call fractional negligible.
Furthermore, we demonstrate how the framework can be used to identify infinitely many families of vertices with high integrality gap
and show that such families are in some sense pervasive.