Name of graph partition that balances edges between sets with edges remaining within sets Is there a common name for this problem:

Let G=(V, E) be an undirected graph.  Partition V into sets $S_1$, $S_2$, ..., $S_k$, such that (the number of edges between sets) + (the number of "non"-edges within sets) is minimized.

To be clear: define "extra edges" to be the set of edges with endpoints in different sets; define "missing edges" to be the set of edges in the complement of G with endpoints in the same set.  The goal is to find a partition that minimizes the sum of "extra edges" and "missing edges".

(The sets need not be of similar size.)
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.