alt Hacker NewsI think it's NP hard, maybe from Sparsest Cut. But you could probably find the min-cut and then iterate by adding capacity on edges in the min cut until you find a cut of the right size. (if the desired cut-size is close to the min cut size at least).
It's NP-hard from Minimum s–t Cut with at least k Vertices. That's the edge version, but since the grid graph is 4-regular(-ish), the problem is trivially convertible to the vertex version.
Edit: apex-4-regular