On the Speed of Constraint Propagation and the Time Complexity of Arc Consistency Testing

Establishing arc consistency on two relational structures is one of the most popular heuristics for the constraint satisfaction problem. We aim at determining the time complexity of arc consistency testing. The input structures G and H can be supposed to be connected colored graphs, as the general problem reduces to this particular case. We first observe the upper bound O(e(G)v(H) + v(G)e(H)), which implies the bound O(e(G)e(H)) in terms of the number of edges and the bound O((v(G) + v(H))3) in terms of the number of vertices. We then show that both bounds are tight up to a constant factor as long as an arc consistency algorithm is based on constraint propagation (as all current algorithms are).