Conjunctions of Among Constraints

Many existing global constraints can be encoded as a conjunction of among constraints. An among constraint holds if the number of the variables in its scope whose value belongs to a prespecified set, which we call its range, is within some given bounds. It is known that domain filtering algorithms can benefit from reasoning about the interaction of among constraints so that values can be filtered out taking into consideration several among constraints simultaneously. The present pa- per embarks into a systematic investigation on the circumstances under which it is possible to obtain efficient and complete domain filtering algorithms for conjunctions of among constraints. We start by observing that restrictions on both the scope and the range of the among constraints are necessary to obtain meaningful results. Then, we derive a domain flow-based filtering algorithm and present several applications. In particular, it is shown that the algorithm unifies and generalizes several previous existing results.Comment: 15 pages plus appendi

On the implementation of a primal-dual algorithm for second order time-dependent Mean Field Games with local couplings

We study a numerical approximation of a time-dependent Mean Field Game (MFG) system with local couplings. The discretization we consider stems from a variational approach described in [14] for the stationary problem and leads to the finite difference scheme introduced by Achdou and Capuzzo-Dolcetta in [3]. In order to solve the finite dimensional variational problems, in [14] the authors implement the primal-dual algorithm introduced by Chambolle and Pock in [20], whose core consists in iteratively solving linear systems and applying a proximity operator. We apply that method to time-dependent MFG and, for large viscosity parameters, we improve the linear system solution by replacing the direct approach used in [14] by suitable preconditioned iterative algorithms

Undirected Forest Constraints

Abstract. We present two constraints that partition the vertices of an undirected n-vertex, m-edge graph G = (V, E) into a set of vertex-disjoint trees. The first is the resource-forest constraint, where we assume that a subset R ⊆ V of the vertices are resource vertices. The constraint specifies that each tree in the forest must contain at least one resource vertex. This is the natural undirected counterpart of the tree constraint [1], which partitions a directed graph into a forest of directed trees where only certain vertices can be tree roots. We describe a hybrid-consistency algorithm that runs in O(m + n) time for the resource-forest constraint, a sharp improvement over the O(mn) bound that is known for the directed case. The second constraint is proper-forest. In this variant, we do not have the requirement that each tree contains a resource, but the forest must contain only proper trees, i.e., trees that have at least two vertices each. We develop a hybrid-consistency algorithm for this case whose running time is O(mn) in the worst case, and O(m √ n) in many (typical) cases.

A Circuit Constraint for Multiple Tours Problems

International audienceRouting problems appear in many practical applications. In the context of Constraint Programming, circuit constraints have been successfully developed to handle problems like the well-known Traveling Salesman Problem or the Vehicle Routing Problem. These kind of constraints are linked to the search for a Hamiltonian circuit in a graph. In this paper we consider a more general multiple tour problem that consists in covering a part of the graph with a set of minimal cost circuits. We define a new global constraint WeightedSubCircuits that generalizes the WeightedCircuit constraint by releasing the need to obtain a Hamil-tonian circuit. It enforces multiple disjoint circuits of bounded total cost to partially cover a weighted graph, the subsets of vertices to be covered being induced by external constraints. We show that enforcing Bounds Consistency for WeightedSubCircuits is NP-hard. We propose an incomplete but polynomial filtering method based on the search for a lower bound of a weighted Steiner circuit