Abstract
We establish optimal bounds on the number of nested propagation steps in k-consistency tests. It is known that local consistency algorithms such as arc-, path- and k-consistency are not efficiently parallelizable. Their inherent sequential nature is caused by long chains of nested propagation steps, which cannot be executed in parallel. This motivates the question “What is the minimum number of nested propagation steps that have to be performed by k-consistency algorithms on (binary) constraint networks with n variables and domain size d?”
It was known before that 2-consistency requires Ω(nd) and 3-consistency requires Ω(n 2) sequential propagation steps. We answer the question exhaustively for every k ≥ 2: there are binary constraint networks where any k-consistency procedure has to perform Ω(n k − 1 d k − 1) nested propagation steps before local inconsistencies were detected. This bound is tight, because the overall number of propagation steps performed by k-consistency is at most n k − 1 d k − 1.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
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.: The Propagation Depth of Local Consistency. ArXiv e-prints (2014), http://arxiv.org/abs/1406.4679
Berkholz, C.: Lower bounds for existential pebble games and k-consistency tests. Logical Methods in Computer Science 9(4) (2013), http://arxiv.org/abs/1205.0679
Berkholz, C., Verbitsky, O.: On the speed of constraint propagation and the time complexity of arc consistency testing. In: Chatterjee, K., Sgall, J. (eds.) MFCS 2013. LNCS, vol. 8087, pp. 159–170. Springer, Heidelberg (2013)
Cooper, M.C.: An optimal k-consistency algorithm. Artificial Intelligence 41(1), 89–95 (1989)
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)
Freuder, E.C.: Synthesizing constraint expressions. Commun. ACM 21, 958–966 (1978)
Gaspers, S., Szeider, S.: The parameterized complexity of local consistency. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 302–316. Springer, Heidelberg (2011)
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: Proc AAAI/IAAI 2000, pp. 175–181 (2000)
Ladkin, P.B., Maddux, R.D.: On binary constraint problems. J. ACM 41(3), 435–469 (1994), http://doi.acm.org/10.1145/176584.176585
Samal, A., Henderson, T.: Parallel consistent labeling algorithms. International Journal of Parallel Programming 16, 341–364 (1987)
Susswein, S., Henderson, T., Zachary, J., Hansen, C., Hinker, P., Marsden, G.: Parallel path consistency. International Journal of Parallel Programming 20(6), 453–473 (1991), http://dx.doi.org/10.1007/BF01547895
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Berkholz, C. (2014). The Propagation Depth of Local Consistency. In: O’Sullivan, B. (eds) Principles and Practice of Constraint Programming. CP 2014. Lecture Notes in Computer Science, vol 8656. Springer, Cham. https://doi.org/10.1007/978-3-319-10428-7_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-10428-7_14
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10427-0
Online ISBN: 978-3-319-10428-7
eBook Packages: Computer ScienceComputer Science (R0)