Name of graph partition that balances edges between sets with edges remaining within sets
This is the Min-Disagreements version of the correlation clustering problem (on complete graphs), defined by Bansal, Blum, and Chawla (full version). They give a (huge) constant factor approximation for the problem and prove it's NP-hard. Charikar, Guruswami, and Wirth show the problem is APX-hard, and improve the approximation factor to 4 via region growing. There is a beautiful and simple combinatorial algorithm (a variant of QuickSort), due to Ailon, Charikar, and Newman, that gives a factor 3 approximation. I believe the current best known approximation ratio is due to Chawla, Makarychev, Schramm, and Yaroslavtsev, and is around 2.06.