On the Solvability Problem for Restricted Classes of Word Equations

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9840)

Abstract

We investigate the complexity of the solvability problem for restricted classes of word equations with and without regular constraints. For general word equations, the solvability problem remains \({{\mathrm{\mathsf {NP}}}}\)-hard, even if the variables on both sides are ordered, and for word equations with regular constraints, the solvability problems remains \({{\mathrm{\mathsf {NP}}}}\)-hard for variable disjoint (i. e., the two sides share no variables) equations with two variables, only one of which is repeated. On the other hand, word equations with only one repeated variable (but an arbitrary number of variables) and at least one non-repeated variable on each side, can be solved in polynomial-time.

Keywords

Word equations Regular constraints NP-hardness 

Notes

Acknowledgements

We are indebted to Artur Jeż for valuable discussions. Markus L. Schmid gratefully acknowledges partial support for this research from DFG, that in particular enabled his visit at the University of Kiel.

References

  1. 1.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer International Publishing AG, Cham (2015)CrossRefMATHGoogle Scholar
  2. 2.
    Da̧browski, R., Plandowski, W.: Solving two-variable word equations. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 408–419. Springer, Heidelberg (2004)Google Scholar
  3. 3.
    Fernau, H., Manea, F., Mercaş, R., Schmid, M.L.: Pattern matching with variables: fast algorithms and new hardness results. In: Proceedings of 32nd Symposium on Theoretical Aspects of Computer Science, STACS 2015, Leibniz International Proceedings in Informatics (LIPIcs), vol. 30, pp. 302–315 (2015)Google Scholar
  4. 4.
    Fernau, H., Schmid, M.L.: Pattern matching with variables: a multivariate complexity analysis. Inf. Comput. 242, 287–305 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fernau, H., Schmid, M.L., Villanger, Y.: On the parameterised complexity of string morphism problems. Theory of Computing Systems (2015). http://dx.doi.org/10.1007/s00224-015-9635-3
  6. 6.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63, 512–530 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Jeż, A.: One-variable word equations in linear time. Algorithmica 74, 1–48 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Jeż, A.: Recompression: a simple and powerful technique for word equations. J. ACM 63(1), 4:1–4:51 (2016)MathSciNetGoogle Scholar
  9. 9.
    Kratochvíl, J., Kr̆ivánek, M.: On the computational complexity of codes in graphs. In: Chytil, M.P., Koubek, V., Janiga, L. (eds.) MFCS 1988. LNCS, vol. 324, pp. 396–404. Springer, Heidelberg (1988)Google Scholar
  10. 10.
    Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)CrossRefMATHGoogle Scholar
  11. 11.
    Makanin, G.: The problem of solvability of equations in a free semigroup. Matematicheskii Sbornik 103, 147–236 (1977)MathSciNetMATHGoogle Scholar
  12. 12.
    Plandowski, W.: An efficient algorithm for solving word equations. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, STOC 2006, pp. 467–476 (2006)Google Scholar
  13. 13.
    Reidenbach, D., Schmid, M.L.: Patterns with bounded treewidth. Inf. Comput. 239, 87–99 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Robson, J.M., Diekert, V.: On quadratic word equations. STACS 1999. LNCS, vol. 1563, pp. 217–226. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  15. 15.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of 10th Annual ACM Symposium on Theory of Computing, STOC 1978, pp. 216–226. ACM (1978)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceKiel UniversityKielGermany
  2. 2.Fachbereich IV – Abteilung InformatikwissenschaftenTrier UniversityTrierGermany

Personalised recommendations