Before revision:  Furthermore, we show that the problem of determining an optimal partition for the rows and columns simultaneously is NP-Hard under a simple linear cost model. To evaluate our algorithm empirically against existing heuristics, we introduce the 1D-VBR format, a specialization of VBRformat where columns are left ungrouped.
After revision:  Furthermore, we show that the problem of determining an optimal partition for the rows and columns for VBR is NP-hard under several reasonable cost models. In light of this finding, we propose a 1-dimensional variant of VBR, called 1D-VBR format, a specialization of VBRformat where columns are left ungrouped.