Flow-Based Combinatorial Chance Constraints

Abstract

Abstract. We study stochastic variants of flow-based global constraints as combinatorial chance constraints. As a specific case study, we focus on the stochastic weighted alldifferent constraint. We first show that determining the consistency of this constraint is NP-hard. We then show how the combinatorial structure of the alldifferent constraint can be used to define chance-based filtering, and to compute a policy. Our propagation algorithm can be extended immediately to related flow-based constraints such as the weighted cardinality constraint. The main benefits of our approach are that our chance-constrained global constraints can be integrated naturally in classical deterministic CP systems, and are more scalable than existing approaches for stochastic constraint programming.

The complexity of the Kth largest subset problem and related problems

“…Finally, the Kth largest m-tuple problem was used by Cire et al [16] to show NP-hardness of another probabilistic problem, namely determining whether a so-called "chance-alldifferent constraint" has a solution. This is a weighted extension of stochastic constraint programming.…”

“…This is a weighted extension of stochastic constraint programming. Cire et al [16] remarked: "Also, we are not aware if the problem of deciding whether there exists a feasible solution to chance-alldifferent is in NP." But this seems unlikely, as their reduction combined with Theorem 3 proves PP-hardness of this problem.…”