Consistency Check for the Bin Packing Constraint Revisited

The bin packing problem (BP) consists in finding the minimum number of bins necessary to pack a set of items so that the total size of the items in each bin does not exceed the bin capacity C. The bin capacity is common for all the bins. This problem can be solved in Constraint Programming (CP) by introducing one placement variable xi for each item and one load variable lj for each bin. The Pack([x1, . . . , xn], [w1, . . . , wn], [l1, . . . , lm]) constraint introduced by Shaw [1] links the placement variables x1, . . . , xn of n items having weights w1, . . . , wn with the load variables of m bins l1, . . . , lm with domains {0, . . . , C}. More precisely the constraint ensures that ∀j ∈ {1, . . . ,m} : lj = ∑n i=1(xi = j) · wi where xi = j is reified to 1 if the equality holds and to 0 otherwise. The Pack constraint was successfully used in several applications. In addition to the decomposition constraints ∀j ∈ {1, . . . ,m} : lj = ∑n i=1(xi = j) · wi and the redundant constraint ∑n i=1 wi = ∑n j=1 lj , Shaw introduced: 1. a filtering algorithm based on a knapsack reasoning inside each bin, and 2. a failure detection algorithm based on a reduction of the partial solution to a bin packing problem.