So here is the challenge problem statement: https://icpcarchive.ecs.baylor.edu/index.php?option=com_onlinejudge&Itemid=8&page=show_problem&problem=1512
Basically, given a 0/1 matrix, you need to permute the columns so that the first column is fixed and after the permutation of columns, then for each row the 1's in the row occur contiguously (without counting wrap-around). I thought about this problem and came up with a conjecture that would make it easy to solve, but I'm not sure if it's true.
Conjecture: After the first $k$ columns have been chosen, let $S$ be the set of rows that end with a $1$ in the $k$th column, such that there exists an unchosen column that has a $1$ in that row. Then the $k+1$th column can be chosen to be the column that has (i) 1's for all positions in $S$, and (ii) which has a minimum total number of $1$'s.
Is this true? If so an optimal solution could be constructed very quickly. I know that the $k+1$th column has to satisfy condition (i), which reduces the possibilities, but I'm really hoping we can ensure condition (ii) also holds so that the choice of $k+1$th column is essentially unique.