On the speed of constraint propagation and the time complexity of arc consistency testing
This provides foundational insights into the computational limits of arc consistency algorithms, which are widely used heuristics in constraint satisfaction, but the results are incremental as they refine existing complexity bounds.
The paper determines the time complexity of arc consistency testing for constraint satisfaction problems, showing that the upper bounds O(e(G)v(H)+v(G)e(H)) and O((v(G)+v(H))^3) are tight up to a constant factor, based on examples of slow constraint propagation.
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 (like any algorithm currently known). Our argument for the lower bounds is based on examples of slow constraint propagation. We measure the speed of constraint propagation observed on a pair $G,H$ by the size of a proof, in a natural combinatorial proof system, that Spoiler wins the existential 2-pebble game on $G,H$. The proof size is bounded from below by the game length $D(G,H)$, and a crucial ingredient of our analysis is the existence of $G,H$ with $D(G,H)=Ω(v(G)v(H))$. We find one such example among old benchmark instances for the arc consistency problem and also suggest a new, different construction.