New filtering for the cumulative constraint in the context of non-overlapping rectangles

Abstract

This paper describes new filtering methods for the constraint. The first method introduces bounds for the so called longest cumulative hole problem and shows how to use these bounds in the context of the non-overlapping constraint. The second method introduces balancing knapsack constraints which relate the total height of the tasks that end at a specific time-point with the total height of the tasks that start at the same time-point. Experiments on tight rectangle packing problems show that these methods drastically reduce both the time and the number of backtracks for finding all solutions as well as for finding the first solution. For example, we found without backtracking all solutions to 66 perfect square instances of order 23-25 and sizes ranging from 332 ×332 to 661 ×661. © 2008 Springer-Verlag Berlin Heidelberg.