Minimum cut versus sparsest cut?
The sparsest cut problem, in which one attempts to minimize the ratio between the number of edges cut to the size of the smaller size of the bipartition, is NP-complete. The "last word" on this problem theoretically speaking is the $O(\sqrt{\log n})$ algorithm by Arora, Rao and Vazirani (affectionately knows as ARV), which has found many uses in the area of approximation algorithms.

You might want to start with the simpler algorithm of Leighton and Rao, which gives an $O(\log n)$ approximation and has a relation to flows. See for example these lecture notes, or many others.