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Max-Closed Semilinear Constraint Satisfaction

  • Manuel Bodirsky
  • Marcello MaminoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)

Abstract

A semilinear relation \(S \subseteq {\mathbb Q}^n\) is max-closed if it is preserved by taking the componentwise maximum. The constraint satisfaction problem for max-closed semilinear constraints is at least as hard as determining the winner in Mean Payoff Games, a notorious problem of open computational complexity. Mean Payoff Games are known to be in \(\mathsf {NP}\cap \mathsf {co}\text {-}\mathsf {NP}\), which is not known for max-closed semilinear constraints. Semilinear relations that are max-closed and additionally closed under translations have been called tropically convex in the literature. One of our main results is a new duality for open tropically convex relations, which puts the CSP for tropically convex semilinear constraints in general into \(\mathsf {NP}\cap \mathsf {co}\text {-}\mathsf {NP}\). This extends the corresponding complexity result for scheduling under and-or precedence constraints, or equivalently the max-atoms problem. To this end, we present a characterization of max-closed semilinear relations in terms of syntactically restricted first-order logic, and another characterization in terms of a finite set of relations L that allow primitive positive definitions of all other relations in the class. We also present a subclass of max-closed constraints where the CSP is in \(\mathsf {P}\); this class generalizes the class of max-closed constraints over finite domains, and the feasibility problem for max-closed linear inequalities. Finally, we show that the class of max-closed semilinear constraints is maximal in the sense that as soon as a single relation that is not max-closed is added to L, the CSP becomes \(\mathsf {NP}\)-hard.

Notes

Acknowledgements

Both authors have received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement no. 257039), and the German Research Foundation (DFG, project number 622397).

References

  1. 1.
    Akian, M., Gaubert, S., Guterman, A.: Tropical polyhedra are equivalent to mean payoff games. Int. Algebra Comput. 22(1), 43 (2012). 125001MathSciNetzbMATHGoogle Scholar
  2. 2.
    Atserias, A., Maneva, E.: Mean-payoff games and propositional proofs. Inf. Comput. 209(4), 664–691 (2011). In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010, Part I. LNCS, vol. 6198, pp. 102–113. Springer, Heidelberg (2010)Google Scholar
  3. 3.
    Barto, L., Kozik, M.: Absorbing subalgebras, cyclic terms and the constraint satisfaction problem. Logical Methods Comput. Sci. 8(1:07), 1–26 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bezem, M., Nieuwenhuis, R., Rodríguez-Carbonell, E.: The max-atom problem and its relevance. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS, vol. 5330, pp. 47–61. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Bodirsky, M., Jonsson, P., von Oertzen, T.: Semilinear program feasibility. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part II. LNCS, vol. 5556, pp. 79–90. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Bodirsky, M., Jonsson, P., von Oertzen, T.: Essential convexity and complexity of semi-algebraic constraints. Logical Methods Comput. Sci. 8(4), 1–25 (2012). An extended abstract about a subset of the results has been published under the title Semilinear Program Feasibility at ICALP 2010MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bodirsky, M., Mamino, M.: Max-closed semilinear constraint satisfaction (2015). arXiv:1506.04184
  8. 8.
    Bodirsky, M., Martin, B., Mottet, A.: Constraint satisfaction problems over the integers with successor. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015, Part I. LNCS, vol. 9134, pp. 256–267. Springer, Heidelberg (2015)Google Scholar
  9. 9.
    Boros, E., Elbassioni, K., Gurvich, V., Makino, K.: Every stochastic game with perfect information admits a canonical form. RRR-09-2009, RUTCOR, Rutgers University (2009)Google Scholar
  10. 10.
    Boros, E., Elbassioni, K., Gurvich, V., Makino, K.: A pumping algorithm for ergodic stochastic mean payoff games with perfect information. In: Eisenbrand, F., Shepherd, F.B. (eds.) IPCO 2010. LNCS, vol. 6080, pp. 341–354. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Bulatov, A.A., Krokhin, A.A., Jeavons, P.G.: Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34, 720–742 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Andradas, C., Rubio, R., Vélez, M.P.: An algorithm for convexity of semilinear sets over ordered fields. Real Algebraic and Analytic Geometry Preprint Server, No. 12Google Scholar
  13. 13.
    Develin, M., Sturmfels, B.: Tropical convexity. Doc. Math. 9, 1–27 (2004)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Feder, T., Vardi, M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: a study through Datalog and group theory. SIAM J. Comput. 28, 57–104 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ferrante, J., Rackoff, C.: A decision procedure for the first order theory of real addition with order. SIAM J. Comput. 4(1), 69–76 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Filar, J., Vrieze, K.: Competitive Markov Decision Processes. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  17. 17.
    Gillette, D.: Stochastic games with zero probabilities. Contrib. Theor. Games 3, 179–187 (1957)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Grigoriev, D., Podolskii, V.V.: Tropical effective primary and dual nullstellensätze. In: 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, Garching, Germany, 4–7 March 2015, pp. 379–391 (2015)Google Scholar
  19. 19.
    Hodges, W.: A Shorter Model Theory. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  20. 20.
    Jeavons, P.G., Cooper, M.C.: Tractable constraints on ordered domains. Artif. Intell. 79(2), 327–339 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. J. ACM 44(4), 527–548 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jonsson, P., Bäckström, C.: A unifying approach to temporal constraint reasoning. Artif. Intell. 102(1), 143–155 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jonsson, P., Lööw, T.: Computation complexity of linear constraints over the integers. Artif. Intell. 195, 44–62 (2013)CrossRefzbMATHGoogle Scholar
  24. 24.
    Jonsson, P., Thapper, J.: Constraint satisfaction and semilinear expansions of addition over the rationals and the reals (2015). arXiv:1506.00479
  25. 25.
    Khachiyan, L.: A polynomial algorithm in linear programming. Dokl. Akad. Nauk SSSR 244, 1093–1097 (1979)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Liggett, T.M., Lippman, S.A.: Stochastic games with perfect information and time average payoff. SIAM Rev. 11(4), 604–607 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Möhring, R.H., Skutella, M., Stork, F.: Scheduling with and/or precedence constraints. SIAM J. Comput. 33(2), 393–415 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pöschel, R.: A general galois theory for operations and relations and concrete characterization of related algebraic structures. Technical report of Akademie der Wissenschaften der DDR (1980)Google Scholar
  29. 29.
    Scowcroft, P.: A representation of convex semilinear sets. Algebra Univers. 62(2–3), 289–327 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zwick, U., Paterson, M.: The complexity of mean payoff games on graphs. Theor. Comput. Sci. 158(1&2), 343–359 (1996)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institut für AlgebraTU DresdenDresdenGermany

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