Abstract
We study constraint satisfaction problems (CSPs) that are k-consistent in the sense that any k input constraints can be simultaneously satisfied. In this setting, we focus on constraint languages with a single binary constraint type. Such a constraint satisfaction problem is equivalent to the question whether there is a homomorphism from an input digraph G to a fixed target digraph H. The instance corresponding to G is k-consistent if every subgraph of G of size at most k is homomorphic to H. Let ρ k (H) be the largest ρ such that every k-consistent G contains a subgraph G′ of size at least ρ ∥ E(G)∥ that is homomorphic to H. The ratio ρ k (H) reflects the fraction of constraints of a k-consistent instance that can be always satisfied. We determine ρ k (H) for all digraphs H that are not acyclic and show that lim k→ ∞ ρ k (H) = 1 if and only if H has tree duality. In addition, for graphs H with tree duality, we design an algorithm that computes in linear time for a given input graph G either a homomorphism from almost the entire graph G to H, or a subgraph of G of bounded size that is not homomorphic to H.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bulatov, A., Krokhin, A., Jeavons, P.: The Complexity of Maximal Constraint Languages. In: Proc. 33rd Symp. on Theory of Computation, STOC, pp. 667–674 (2001)
Cook, S., Mitchell, D.: Finding Hard Instances of the Satisfiability Problem: A Survey. In: Satisfiability Problem: Theory and Applications. DIMACS Series in DMTCS, vol. 35. AMS (1997)
Dalmau, V., Pearson, J.: Closure Functions and Width 1 Problems. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 159–173. Springer, Heidelberg (1999)
Dechter, R., van Beek, P.: Local and Global Relational Consistency. Theor. Comput. Sci. 173, 283–308 (1997)
Dvořák, Z.: personal communication
Dvořák, Z., Král’, D., Pangrác, O.: Locally Consistent Constraint Satisfaction Problems. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 469–480. Springer, Heidelberg (2004)
Dvořák, Z., Král’, D., Pangrác, O.: Locally Consistent Constraint Satisfaction Problems. To appear in Theor. Comput. Sci.
Eppstein, D.: Improved Algorithms for 3-coloring, 3-edge-coloring and Constraint Satisfaction. In: Proc. 12th ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 329–337 (2001)
Feder, T., Motwani, R.: Worst-case Time Bounds for Coloring and Satisfiability Problems. J. Algorithms 45(2), 192–201 (2002)
Feder, T., Vardi, M.: Monotone monadic SNP and constraint satisfaction. In: Proc. 25th Symposium on the Theory of Computation, STOC, pp. 612–622 (1993)
Freuder, E.C.: A sufficient condition for backtrack-free search. J. ACM 29, 24–32 (1982)
Hell, P., Nešetřil, J.: Graphs and homomorphisms. Oxford University Press, Oxford (2004)
Hell, P., Nešetřil, J., Zhu, X.: Duality and polynomial testing of tree homomorphisms. Trans. Amer. Math. 348(4), 1281–1297 (1996)
Huang, M.A., Lieberherr, K.: Implications of Forbidden Structures for Extremal Algorithmic Problems. Theor. Comput. Sci. 40, 195–210 (1985)
Janson, S., Łuczak, T., Ruciński, A.: Random Graphs. Wiley, New York (2000)
Jukna, S.: Extremal Combinatorics with Applications in Computer Science. Springer, Heidelberg (2001)
Král’, D.: Locally Satisfiable Formulas. In: Proc. 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 323–332. SIAM, Philadelphia (2004)
Král’, D., Pangrác, O.: An Asymptotically Optimal Linear-Time Algorithm for Locally Consistent Constraint Satisfaction Problems (submitted)
Lieberherr, K., Specker, E.: Complexity of Partial Satisfaction. J. of the ACM 28(2), 411–422 (1981)
Lieberherr, K., Specker, E.: Complexity of Partial Satisfaction II. Technical Report 293, Dept. of EECS, Princeton University (1982)
Usiskin, Z.: Max-min Probabilities in the Voting Paradox. Ann. Math. Stat. 35, 857–862 (1963)
Trevisan, L.: On Local versus Global Satisfiability. SIAM J. Discrete Math. 17(4), 541–547 (2004); A preliminary version available as ECCC report TR97-12
Woeginger, G.J.: Exact Algorithms for NP-hard Problems: A Survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003)
Yannakakis, M.: On the Approximation of Maximum Satisfiability. J. Algorithms 17, 475–502 (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bodirsky, M., Král’, D. (2005). Locally Consistent Constraint Satisfaction Problems with Binary Constraints. In: Kratsch, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2005. Lecture Notes in Computer Science, vol 3787. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11604686_26
Download citation
DOI: https://doi.org/10.1007/11604686_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-31000-6
Online ISBN: 978-3-540-31468-4
eBook Packages: Computer ScienceComputer Science (R0)