On the speed of constraint propagation and the time complexity of arc consistency testing

TL;DR

This paper analyzes the time complexity of arc consistency testing in constraint satisfaction problems, establishing tight bounds and demonstrating the speed of constraint propagation through combinatorial proof systems and game-theoretic measures.

Contribution

It provides tight upper and lower bounds for the time complexity of arc consistency algorithms based on constraint propagation, using combinatorial and game-theoretic analysis.

Findings

Upper bounds: $O(e(G)v(H)+v(G)e(H))$ and $O((v(G)+v(H))^3)$

Lower bounds are tight up to a constant factor, based on slow constraint propagation examples

Existence of graph pairs with $D(G,H)= ext{Omega}(v(G)v(H))$ demonstrating slow propagation

Abstract

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 $D(G,H)=\Omega(v(G)v(H))$. We find one such example among old benchmark instances for the arc consistency problem and also suggest a new, different construction.