Sounds like a case for Branch and Bound.

I would partition the problem space by adding the next free category to either last used column or the next empty column. The root problem would be "category 1 in column 1", so the children would "category 2 in column 1" and "category 2 in column 2", etc. The low boundary for each solution is the height of the highest column, and the solution's high boundary is maximum(sum of heights of free categories, lower boundary). There is a constraint (free categories > empty columns) that has to be fulfilled to assure there will not be empty columns.

It's probably best to generate the solution tree breadth-first as you will descend far down the left side very quickly if the global lower and upper boundaries have not been narrowed down. Breadth-first would garantuee that you limit them soon. On the other hand, for larger problems the breadth-first approach will also require more memory to hold the solution tree.

I tried to write code yesterday but it was 3am and my mind wouldn't cooperate. Now I've just gotten up way waay too late into the day and it still doesn't. When I'm feeling a bit fresher I will update here with some code (or capitulation *g*). The tricky problem is coming up with a convenient data structure - I tried anonymous arrays for the columns, but it's inconvenient to have to deep copy to generate the child problems.

Makeshifts last the longest.