Given a set of integer points $S$, one is often interested in finding $\operatorname{conv}(S)$ or characterizing certain cases, where $\operatorname{conv}(S)$ is described by few inequalities. Examples would be stable set polytope on perfect graphs or the min-cost flow polytope.
There are certain techniques to prove this; for example, total unimodularity, and total dual integrality (TDI). I am looking for examples, where given a point in the relaxation, there is an algorithm which retrieves the convex combinators to write this point as a convex combination of integral points.
Can you point to some examples? It would be great if you can also link to a paper or something where the algorithm is described.