Almost square packing

Abstract

The almost square rectangle packing problem involves packing all rectangles with sizes 1 ×2 to n ×(n+1) (almost squares) into an enclosing rectangle of minimal area. This extends the previously studied square packing problem by adding an additional degree of freedom for each rectangle, deciding in which orientation the item should be packed. We show how to extend the model and search strategy that worked well for square packing to solve the new problem. Some adapted versions of known redundant constraints improve overall search times. Based on a visualization of the search tree, we derive a decomposition method that initially only looks at the subproblem given by one of the cumulative constraints. This decomposition leads to further modest improvements in execution times. We find a solution for problem size 26 for the first time and dramatically improve best known times for finding solutions for smaller problem sizes by up to three orders of magnitude. © 2011 Springer-Verlag Berlin Heidelberg.