Before revision:  For-Each Sparsification: If each cut value needs to be individually preserved ( Andoni et al., ITCS 2016) , then the situation is more interesting. Here, we give a cut sketch whose size scales with \sqrt{\beta We also show that this result is tight by exhibiting a matching lower bound of \sqrt{\beta .
After revision:  For-Each Sparsification: If each cut value needs to be individually preserved ( Andoni et al., ITCS 2016) cut sparsifiers/sketches as a function of cut balance, defined the maximum ratio of the cut value in the two directions of a directed graph (Ene et al., STOC 2016). We also show that this result is tight by exhibiting a matching lower bound of \sqrt{\beta .