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 ε-GAC for Knapsack and Bounded Knapsack Constraints can be achieved in time $O(n\log n+{n\over{\epsilon^2}})$ or $O(n\log n +{n\over\epsilon^3})$, respectively.

This work was supported by the Intelligent Information Systems Institute, Cornell University (AFOSR grant F49620-01-1-0076).