Approximated consistency for knapsack constraints

Abstract

While global constraints give a broader view on the entire problem and therefore allow more effective constraint propagation, the development of efficient generalized arc-consistency (GAC) algorithms for global constraints is frequently prevented by the fact that the associated decision problems are NP-hard. A prominent example for this is the Knapsack Constraint. On the other hand, there exist approximation algorithms for many NP-hard problems. By introducing the concept of approximated consistency for a special class of global constraints, so-called optimization constraints, we show how existing approximation algorithms can be exploited for the development of efficient filtering algorithms for Knapsack Constraints. As our main result; we show how eGAC for Knapsack and Bounded Knapsack Constraints can be achieved in time O(n log n + n/ε2) or O (n log n + n/ε3), respectively. © Springer-Verlag 2003.