Propagating Knapsack Constraints in Sublinear Time

We develop an efficient incremental version of an existing cost-based filtering algorithm for the knapsack constraint. On a universe of n elements, m invocations of the algorithm require a total ofO(n log n+mk log(n/k)) time, where k ≤ n depends on the specific knapsack instance. We show that the expected value of k is significantly smaller than n on several interesting input distributions, hence while keeping the same worst-case complexity, on expectation the new algorithm is faster than the previously best method which runs in amortized linear time. After a theoretical study, we introduce heuristic enhancements and demonstrate the new algorithm’s performance experimentally.