The MATRIX PACKING DOWN problem asks to find a row permutation of
a given (0,1)-matrix in such a way that the total sum of the first
non-zero column indexes is maximized. We study the computational
complexity of this problem. We prove that the MATRIX PACKING DOWN
problem is NP-complete even when restricted to zero trace symmetric
(0,1)-matrices or to (0,1)-matrices with at most two 1's per
column. Also, as intermediate results, we introduce several new simple
graph layout problems which are proved to be NP-complete.