3 research outputs found
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 and can be supposed to be connected colored graphs, as the
general problem reduces to this particular case. We first observe the upper
bound , which implies the bound in terms of
the number of edges and the bound 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
by the size of a proof, in a natural combinatorial proof system, that
Spoiler wins the existential 2-pebble game on . The proof size is bounded
from below by the game length , and a crucial ingredient of our
analysis is the existence of with . We find one
such example among old benchmark instances for the arc consistency problem and
also suggest a new, different construction.Comment: 19 pages, 5 figure