The Multi-Inter-Distance Constraint

We introduce the MULTI-INTER-DISTANCE constraint that ensures no more than m variables are assigned to values lying in a window of p consecutive values. This constraint is useful for modelling scheduling problems where tasks of processing time p compete for m identical resources. We present a propagator that achieves bounds consistency in cubic time. Experiments show that this new constraint offers a much stronger filtering than an edge-finder and that it allows to solve larger instances of the runway scheduling problem. Properties of the Scheduling Graph Theorem : The MULTIINTER-DISTANCE constraint is satisfiable if and only if the scheduling graph has no negative cycles. Definition: The altered graph Gi is the scheduling graph of the original problem where the upper bound of the domain of Xi is set to v. For instance, the graph G3 is obtained by decrementing the weights in blue in the graph above by one. Corollary: If the altered graph Gi has a negative cycle then Xi ≥ v. Theorem: If Xi = v has no interval support and the domain of Xj has a larger upper bound than the domain of Xi , then Xj = v has no interval support. Theorem: Let u∗ be the smallest domain upper bound that is larger than min(dom(Xi)). Let t be the largest value s.t. the shortest path in Gu ∗ i from min(dom(Xi)) to t is null, then Xi = t is the smallest interval support for Xi. The Runway Scheduling Problem In this problem, n planes need to land in an airport with m runways. The planes fly around the airport while waiting for their authorization to land. A plane can only land within specific time intervals in which it is in position for landing. Outside these time intervals, the plane cannot land. The goal is to dispatch the planes on the m runways while maximizing the minimum time between two successive landings. time Landing time intervals Summary: n planes m runways Each plane has a set of time intervals within which landing is possible. The goal is to maximize the time between each landing on a same runway. The Constraint The constraint MULTI-INTER-DISTANCE is satisfied when no more than m variables are assigned within a window of p consecutive values. Multi-Inter-Distance([X1, . . . , Xn],m, p) ⇐⇒ ∀v |{i | Xi ∈ [v, v + p)}| ≤ m ⇐⇒ ∀i |{j | 0 ≤ Xj −Xi ≤ p}| ≤ m This constraint models a scheduling problem where n tasks with processing time p compete for m identical resource.