Push relabel why return back to source
Inspecting the correctness proof and/or executing any non-trivial example, you'll note that push-relabel algorithms are different from Ford-Fulkerson and variants in that they do not maintain a flow; that's just not part of the invariant, and it's blatantly violated right at the start.

This whole "push flow to the sink, then push the excess back to the source" idea is only that: an idea. The algorithm does not really work like that, always. But it's still useful.

Once the "wave" reaches the sink $t$, it saturates the cut $V \setminus \{t\}$. You still have vertices with excess left¹, though; the conservation of flow property is violated. This has to be remedied before we can terminate with a flow -- hence the "push flow back to the source" phase.




Whenever the cut $V \setminus \{s\}$ is not by coincidence a min-cut.