Domain reduction for the circuit constraint

Abstract

We present an incomplete filtering algorithm for the circuit constraint. The filter removes redundant values by eliminating non-Hamiltonian edges from the associated graph. We prove a necessary condition for an edge to be Hamiltonian, which provides the basis for eliminating edges of a smaller graph defined on a separator of the original graph. The circuit constraint, circuit(y1,⋯,yn}, where yj ∈ {1,⋯,n}, is true if and only if for each j ∈ {1,⋯, n}, y j is the successor of j in some permutation of 1 ⋯ n and yj ∈ Dj, where Dj is the domain of variable j. On a graph of vertices 1,⋯,n, the circuit constraint can be thought as defining a directed Hamiltonian cycle. Nodes of the graph represent the variables. A directed edge (i, j) exists if and only if j is in the domain of variable i. Moreover, elimination of an edge (i, j) from the graph means elimination of the value j from the domain of variable i. With this representation, the problem of domain reduction for the circuit constraint reduces to identifying and eliminating non-Hamiltonian edges on a digraph. In this paper, we present a recursive algorithm that eliminates non-Hamiltonian edges from the graph. A much smaller but denser multi-graph is constructed from a vertex separator S of the original graph by adding certain labelled edges to the subgraph induced by the separator. A directed edge (v, w) with label C is added if C is a connected component separated by S and (v, ci) and (cj, w) are edges of G for some pair of vertices ci, cj in C. We prove that edges that appear in no Hamiltonian cycle containing at least one edge of each component label in the constructed graph are non-Hamiltonian in the original graph. The condition that the constructed graph contains such a Hamiltonian cycle is viewed as a constraint. Global cardinality constraint with vertex degree constraints is a relaxation of this constraint. Then by applying a filtering algorithm for the global cardinality constraint together with in and out-vertex degree constraints, non-Hamiltonian edges are identified and eliminated from the graph. © Springer-Verlag Berlin Heidelberg 2005.