We systematically investigate minimum capacity unbalanced cut problems arising in social networks. Let k be an input parameter. A cut (A, B) is unbalanced if the size of its smaller side is at most k (called k-size) or exactly k (called Ek-size). An s-t cut (A, B) is unbalanced if its s-side is either k-size or Ek-size. In the min k-size cut (s-t cut, resp.) problem, we want to find a k-size cut (s-t cut, resp.) with the minimum capacity. The corresponding min Ek-size cut (and s-t cut) problem is defined in a similar way. While the classical min s-t cut problem has been studied extensively, the minimum capacity unbalanced cut problem has only recently attracted the attention of researchers. In this paper, we prove that the min k-size s-t cut problem is NP-hard, and give O(log n)-approximation algorithms for the min k-size s-t cut problem, the min Ek-size s-t cut problem, and the min Eksize cut problem. These results, together with previous results, complete our research into minimum capacity unbalanced cut problems.