A Filter for the Circuit Constraint

We present an incomplete filtering algorithm for the circuit constraint. The filter removes redundant values by eliminating nonhamiltonian edges from the associated graph. We identify nonhamiltonian edges by analyzing a smaller graph with labeled edges that is defined on a separator of the original graph. The complexity of the procedure for each separator S is approximately O(|S|). We found that it identified all infeasible instances and eliminated about one-third of the redundant domain elements in feasible instances. The circuit constraint can be written circuit(x1, . . . , xn) where the domain of each xi is Di ⊂ {1, . . . , n}. The constraint requires that y1, . . . , yn be a cyclic permutation of 1, . . . , n, where yi+1 = xyi , i = 1, . . . , n− 1 y1 = xyn Let directed graph G contain an edge (i, j) if and only if j belongs to the domain of xi. If edge (i, j) is selected when xi = j, the circuit constraint requires that the selected edges form a hamiltonian circuit or tour of G. One approach to filtering the circuit constraint is to make use of necessary conditions for hamiltonicity of G. Chvátal [2, 3] analyzes several conditions, one of which (1-toughness) is a very restricted case of a condition we develop for filtering nonhamiltonian edges. Some elementary techniques for filtering domains after a partial tour has been constructed are described by Shufelt and Berliner [4], whose analysis relies on the special structure of a chessboard problem, and Caseau and Laburthe [1], who solve small traveling salesman problems. Neither approach is intended for the general filtering problem in which arbitrary variable domains are given.