I'm thinking about a problem in graphs, one part of this problem is as described:
We have a graph G=(V,E), and its spanning tree T=(V,F) (F is subset of E), for each min cut in G (on E), which partitions the graph into two subgraphs with nodes (U,U') (not necessary for each subgraph to be connected) we check the size of this cut in F, name size of them G(U,U') and T(U,U'), I want to find:
ratio = max{T(U,U')/G(U,U')} for all possible U,U'
I think this is NP-Hard, but I cannot prove it. Something is obvious here, that is if we have a vertex in T with same degree as G, ratio is 1, Also It is obvious 0 < ratio <= 1.
U intersect U' = null, U union U' = V, and none of the U and U' are empty.
This is the non-uniform densest cut problem in a unit-weight tree with general unit demands. A 2011 paper by Bonsma et al. states as an open problem the complexity of non-uniform sparsest cut in unit-weight graphs of bounded treewidth with general unit demands, so I suspect that your problem is open too—sparsest and densest cut are very closely related (basically the same for uniform demands).