This is a somewhat subjective question. I am interested in studying the literature on multicommodity flow-cut results, especially the 'positive' results which show that flow is a good approximation to cut (say, within a factor that is constant or polylogarithmic in the number of flows). Examples are:
1) Flow polytope (also called demand polytope) for multicommodity flow in undirected graphs is within O(log k) of cut as shown in
F. Leighton and S. Rao,“An approximate max-flow min-cut theoremfor uniform multicommodity flow problems with applications to approximation algorithms,” in Proc. of 28th Annual Symposium on Foundations of Computer Science, (Los Alamitos, California), 1988.
N. Linial, E. London, and Y. Rabinovich, “The geometry of graphs and some of its algorithmic applications,” Combinatorica, vol. 15, no. 2, pp. 215–245, 1995.
2) Demand polytope of multicommodity flow in directed graphs with symmetric demands is within O(log^2 k) of cut as shown in
P. Klein, S. Plotkin, S. Rao, and E. Tardos, “Bounds on the max-flow min-cut ratio for directed multicommodity flows,” J. Algorithms, no. 22, pp. 241–269, 1997.
3) Max sum-rate in groupcast is within a factor 2 of the multicut. (I don't know a reference for this result. Could someone help me with this? Thanks.)
I would like to know more such positive results affirming flow being close to cut assuming certain structure of the problem (such as undirectedness of graph or symmetric demands, as in above). It would be great if you can give me a one-line summary of the result(s) and a reference of the paper. Thanks.