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 (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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Atserias, A., Bulatov, A.A., Dalmau, V.: On the power of k -consistency. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 279–290. Springer, Heidelberg (2007)
Atserias, A., Dalmau, V.: A combinatorial characterization of resolution width. J. Comput. Syst. Sci. 74(3), 323–334 (2008)
Atserias, A., Kolaitis, P.G., Vardi, M.Y.: Constraint propagation as a proof system. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 77–91. Springer, Heidelberg (2004)
Berkholz, C.: Lower bounds for existential pebble games and k-consistency tests. In: LICS 2012, pp. 25–34. IEEE Computer Society, Los Alamitos (2012)
Berkholz, C.: On the complexity of finding narrow proofs. In: FOCS 2012, pp. 351–360. IEEE Computer Society, Los Alamitos (2012)
Berkholz, C., Verbitsky, O.: On the speed of constraint propagation and the time complexity of arc consistency testing. E-print: arxiv.org/abs/1303.7077 (2013)
Bessière, C.: Constraint Propagation. In: Handbook of Constraint Programming. Elsevier, Amsterdam (2006)
Bessière, C., Régin, J.C., Yap, R.H.C., Zhang, Y.: An optimal coarse-grained arc consistency algorithm. Artificial Intelligence 165(2), 165–185 (2005)
Dechter, R., Pearl, J.: A problem simplification approach that generates heuristics for constraint-satisfaction problems. Tech. rep., Cognitive Systems Laboratory, Computer Science Department, University of California, Los Angeles (1985)
Feder, T., Vardi, M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM Journal on Computing 28(1), 57–104 (1998)
Kasif, S.: On the parallel complexity of discrete relaxation in constraint satisfaction networks. Artificial Intelligence 45(3), 275–286 (1990)
Kolaitis, P.G., Panttaja, J.: On the complexity of existential pebble games. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 314–329. Springer, Heidelberg (2003)
Kolaitis, P.G., Vardi, M.Y.: On the expressive power of datalog: Tools and a case study. J. Comput. Syst. Sci. 51(1), 110–134 (1995)
Kolaitis, P.G., Vardi, M.Y.: A game-theoretic approach to constraint satisfaction. In: Kautz, H.A., Porter, B.W. (eds.) AAAI/IAAI 2000, pp. 175–181. AAAI Press/The MIT Press, California (2000)
Mackworth, A.K.: Consistency in networks of relations. Artificial Intelligence 8(1), 99–118 (1977)
McConnell, R.M., Mehlhorn, K., Näher, S., Schweitzer, P.: Certifying algorithms. Computer Science Review 5(2), 119–161 (2011)
Mohr, R., Henderson, T.C.: Arc and path consistency revisited. Artificial Intelligence 28(2), 225–233 (1986)
Régin, J.-C.: AC-*: A configurable, generic and adaptive arc consistency algorithm. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 505–519. Springer, Heidelberg (2005)
Samal, A., Henderson, T.: Parallel consistent labeling algorithms. International Journal of Parallel Programming 16, 341–364 (1987)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Berkholz, C., Verbitsky, O. (2013). On the Speed of Constraint Propagation and the Time Complexity of Arc Consistency Testing. In: Chatterjee, K., Sgall, J. (eds) Mathematical Foundations of Computer Science 2013. MFCS 2013. Lecture Notes in Computer Science, vol 8087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40313-2_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-40313-2_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40312-5
Online ISBN: 978-3-642-40313-2
eBook Packages: Computer ScienceComputer Science (R0)