Motivation: In standard augmenting path maxflow algorithms, the inner loop requires finding paths from source to sink in a directed, weighted graph. Theoretically, it is well-known that in order for the algorithm to even terminate when there are irrational edge capacities, we need to put restrictions on the paths that we find. The Edmonds-Karp algorithm, for example, tells us to find shortest paths.
Empirically, it has been observed that we might also want to find fat (is there a better term for this?) paths. For example, when using capacity scaling, we find shortest paths that can bear at least $\epsilon$ amount of flow. There is no restriction on how long the path can be. When we can no longer find any paths, we decrease $\epsilon$ and repeat.
I am interested in optimizing the choice of augmenting paths for a very specific application of max-flow, and I want to explore this tradeoff between short and fat paths. (Note: it is not necessary for me to always solve the problem. I am most interested in finding the largest lower bound on flow in the shortest amount of wall time.)
Question: Is there a standard way to interpolate between the shortest path approach and the capacity-scaling approach? That is, is there an algorithm for finding paths that are both short and fat, where ideally some parameter would control how much length in the path we are willing to trade off for fatness? At the extremes, I'd like to be able to recover shortest paths on one end and capacity scaling-style paths on the other.