I came across this problem and am struggling to find a way to approach it. Any thoughts would be greatly appreciated! As input we have a matrix $\{-1, 0, 1\}^{n\ \times\ k} $ , for example:
 
$\begin{bmatrix}
    1 & 0 & 1 & 0  & -1 \\
    -1 & 0 & 0 & 0  & 1 \\
    0 & 1 & 1 & 0 & -1 \\
    -1 & -1 & 0 & 1  & 1 \\
    1 & 0 & 0 & 0  & -1
\end{bmatrix}$

Without trying every single permutation, find an ordering of columns $c_i$ that maximises the number of rows for which the first non-zero element is $1$ . 

For the example above, one such ordering (it's not unique!) is $(c_3, c_4, c_1, c_2, c_5)$, i.e. : 
 
$\begin{bmatrix}
    1 & 0 & 1 & 0  & -1 \\
    0 & 0 & -1 & 0  & 1 \\
    1 & 0 & 0 & 1 & -1 \\
    0 & 1 & 1 & -1  & 1 \\
    0 & 0 & 1 & 0  & -1
\end{bmatrix}$

Here, for 4 out of 5 rows the first non-zero element is 1.