Abstract
We study digraphs preserved by a Maltsev operation, Maltsev digraphs. We show that these digraphs retract either onto a directed path or to the disjoint union of directed cycles, showing that the constraint satisfaction problem for Maltsev digraphs is in logspace, L. (This was observed in [19] using an indirect argument.) We then generalize results in [19] to show that a Maltsev digraph is preserved not only by a majority operation, but by a class of other operations (e.g., minority, Pixley) and obtain a O(V G 4)-time algorithm to recognize Maltsev digraphs. We also prove analogous results for digraphs preserved by conservative Maltsev operations which we use to establish that the list homomorphism problem for Maltsev digraphs is in L. We then give a polynomial time characterisation of Maltsev digraphs admitting a conservative 2-semilattice operation. Finally, we give a simple inductive construction of directed acyclic digraphs preserved by a Maltsev operation.
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References
Barto, L., Bulin, J.: CSP dichotomy for special polyads (2011) (submitted)
Barto, L., Kozik, M., Maróti, M., Niven, T.: CSP dichotomy for special triads. Proceedings of the AMS 137, 2921–2934 (2009)
Barto, L., Kozik, M., Niven, T.: Graphs, polymorphisms and the complexity of homomorphism problems. In: Proceedings of the 40th annual ACM Symposium on Theory of Computing, STOC 2008, pp. 789–796 (2008)
Bulatov, A., Dalmau, V.: A simple algorithm for Mal’tsev constraints. SIAM Journal on Computing 36(1), 16–27 (2006)
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)
Bulatov, A., Jeavons, P., Krokhin, A.: Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing 34(3), 720–742 (2005)
Bulatov, A.A., Krokhin, A.A., Larose, B.: Dualities for constraint satisfaction problems. In: Creignou, N., Kolaitis, P.G., Vollmer, H. (eds.) Complexity of Constraints. LNCS, vol. 5250, pp. 93–124. Springer, Heidelberg (2008)
Dalmau, V.: Linear Datalog and bounded path duality of relational structures. Logical Methods in Computer Science 1, 1–32 (2005)
Dalmau, V., Krokhin, A.: Majority constraints have bounded pathwidth duality. European Journal of Combinatorics 29(4), 821–837 (2008)
Dalmau, V., Larose, B.: Maltsev + Datalog ⇒ Symmetric Datalog. In: Proceedings of the 23rd IEEE Symposium on Logic in Computer Science, LICS 2008, pp. 297–306 (2008)
Dyer, M.E., Richerby, D.: On the complexity of #CSP. In: Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, pp. 725–734 (2010)
Egri, L.: On CSPs below P and the NL ≠ P conjecture (2011) (submitted)
Egri, L., Krokhin, A.A., Larose, B., Tesson, P.: The complexity of the list homomorphism problem for graphs. In: Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science, STACS 2010, pp. 335–346 (2010)
Egri, L., Larose, B., Tesson, P.: Symmetric Datalog and constraint satisfaction problems in logspace. In: Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science, LICS 2007, pp. 193–202 (2007)
Hagemann, J., Mitschke, A.: On n-permutable congruences. Algebra Universalis 3, 8–12 (1973)
Hell, P., Nešetřil, J.: On the complexity of H-coloring. Journal of Combinatorial Theory, Series B 48, 92–110 (1990)
Hell, P., Rafiey, A.: The dichotomy of list homomorphisms for digraphs. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, pp. 1703–1713 (2011)
Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. J. ACM 44(4), 527–548 (1997)
Kazda, A.: Maltsev digraphs have a majority polymorphism. European Journal of Combinatorics 32, 390–397 (2011)
Maróti, M., McKenzie, R.: Existence theorems for weakly symmetric operations. Algebra Universalis 59(3-4), 463–489 (2008)
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Carvalho, C., Egri, L., Jackson, M., Niven, T. (2011). On Maltsev Digraphs. In: Kulikov, A., Vereshchagin, N. (eds) Computer Science – Theory and Applications. CSR 2011. Lecture Notes in Computer Science, vol 6651. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20712-9_14
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DOI: https://doi.org/10.1007/978-3-642-20712-9_14
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