Abstract
Given an instance \(\mathcal{I}\) of a CSP, a tester for \(\mathcal{I}\) distinguishes assignments satisfying \(\mathcal{I}\) from those which are far from any assignment satisfying \(\mathcal{I}\). The efficiency of a tester is measured by its query complexity, the number of variable assignments queried by the algorithm. In this paper, we characterize the hardness of testing Boolean CSPs in terms of the algebra generated by the relations used to form constraints. In terms of computational complexity, we show that if a non-trivial Boolean CSP is sublinear-query testable (resp., not sublinear-query testable), then the CSP is in NL (resp., P-complete, ⊕L-complete or NL-complete) and that if a sublinear-query testable Boolean CSP is constant-query testable (resp., not constant-query testable), then counting the number of solutions of the CSP is in P (resp., \(\sharp\) P-complete).
Also, we conjecture that a CSP instance is testable in sublinear time if its Gaifman graph has bounded treewidth. We confirm the conjecture when a near-unanimity operation is a polymorphism of the CSP.
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
Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM J. on Comput. 25(2), 252–271 (1996)
Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. J. ACM 45(4), 653–750 (1998)
Ben-Sasson, E., Harsha, P., Raskhodnikova, S.: Some 3CNF properties are hard to test. SIAM J. on Comput. 35(1), 1–21 (2006)
Fischer, E., Lehman, E., Newman, I., Raskhodnikova, S., Rubinfeld, R., Samorodnitsky, A.: Monotonicity testing over general poset domains. In: Proc. 48th Annual IEEE Symposium on Foundations of Computer Science, pp. 474–483 (2002)
Yoshida, Y.: Testing list H-homomorphisms. In: Proc. 27th Annual IEEE Conference on Computational Complexity, pp. 85–95 (2012)
Bulatov, A.A., Krokhin, A.A., Jeavons, P.G.: Constraint satisfaction problems and finite algebras. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, p. 272. Springer, Heidelberg (2000)
Jeavons, P.: On the algebraic structure of combinatorial problems. Theoretical Computer Science 200(1-2), 185–204 (1998)
Schaefer, T.: The complexity of satisfiability problems. In: Proc. 10th Annual ACM Symposium on the Theory of Computing, pp. 216–226 (1978)
Alon, N., Shapira, A.: A characterization of the (natural) graph properties testable with one-sided error. SIAM J. on Comput. 37(6), 1703–1727 (2008)
Alon, N., Fischer, E., Newman, I., Shapira, A.: A combinatorial characterization of the testable graph properties: it’s all about regularity. In: Proc. 38th Annual ACM Symposium on Theory of Computing, pp. 251–260 (2006)
Borgs, C., Chayes, J.T., Lovász, L., Sós, V.T., Szegedy, B., Vesztergombi, K.: Graph limits and parameter testing. In: Proc. 36th Annual ACM Symposium on the Theory of Computing, pp. 261–270 (2006)
Kaufman, T., Sudan, M.: Algebraic property testing: the role of invariance. In: Proc. 40th Annual ACM Symposium on the Theory of Computing, pp. 403–412 (2008)
Bhattacharyya, A., Grigorescu, E., Shapira, A.: A unified framework for testing linear-invariant properties. In: Proc. 51st Annual IEEE Symposium on Foundations of Computer Science, pp. 478–487 (2010)
Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54(1), 1–24 (2007)
Post, E.: The two-valued iterative systems of mathematical logic. Annals of Mathematics Studies, vol. 5. Princeton Univ. Pr. (1941)
Allender, E., Bauland, M., Immerman, N., Schnoor, H., Vollmer, H.: The complexity of satisfiability problems: Refining Schaefer’s theorem. J. Comp. Sys. Sci. 75(4), 245–254 (2009)
Creignou, N., Hermann, M.: Complexity of generalized satisfiability counting problems. Inform. and Comput. 125(1), 1–12 (1996)
Chakraborty, S., Fischer, E., Lachish, O., Matsliah, A., Newman, I.: Testing st-connectivity. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) RANDOM 2007 and APPROX 2007. LNCS, vol. 4627, pp. 380–394. Springer, Heidelberg (2007)
Newman, I.: Property testing of massively parametrized problems - A survey. In: Goldreich, O. (ed.) Property Testing. LNCS, vol. 6390, pp. 142–157. Springer, Heidelberg (2010)
Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4) (2008)
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
Bhattacharyya, A., Yoshida, Y. (2013). An Algebraic Characterization of Testable Boolean CSPs. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7965. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39206-1_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-39206-1_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39205-4
Online ISBN: 978-3-642-39206-1
eBook Packages: Computer ScienceComputer Science (R0)