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

  • Christoph Berkholz
  • Oleg Verbitsky
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8087)


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).

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.


Time Complexity Binary Relation Constraint Propagation Proof System Colored Graph 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Christoph Berkholz
    • 1
  • Oleg Verbitsky
    • 2
  1. 1.RWTH Aachen UniversityAachenGermany
  2. 2.Humboldt-University of BerlinBerlinGermany

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