Theory of Computing Systems

, Volume 62, Issue 3, pp 481–509 | Cite as

Tropically Convex Constraint Satisfaction

  • Manuel Bodirsky
  • Marcello Mamino


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 N Pc o N P, 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 N Pc o N P. 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 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 N P-hard.


Tropical convexity Semi-linear relations Max-closure Constraint satisfaction Max-plus-average inequalities Stochastic games Piecewise linear constraints Computational complexity 



Both authors have received funding from the European Research Council (grant agreement number 257039 and 681988), and from the German Research Foundation (DFG, project number 622397).


  1. 1.
    Akian, M., Gaubert, S., Guterman, A.: Tropical polyhedra are equivalent to mean payoff games. Int. Algebra Comput. 22(1), 125001 (2012). (43 pages)MathSciNetMATHGoogle Scholar
  2. 2.
    Andersson, D., Miltersen, P.B.: The complexity of solving stochastic games on graphs. In Algorithms and Computation, 20th International Symposium, ISAAC 2009, Honolulu, Hawaii, USA, December 16-18, 2009. Proceedings, pages 112–121 (2009)Google Scholar
  3. 3.
    Atserias, A., Maneva, E.: Mean-payoff games and propositional proofs. Inf. Comput. 209(4), 664–691 (2011). A preliminary version appeared in the Proceedings of 37th International Colloquium on Automata, Languages and Programming (ICALP), volume 6198 of Lecture Notes in Computer Science, Springer-Verlag, pages 102–113, 2010MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Barto, L., Kozik, M.: Absorbing subalgebras, cyclic terms and the constraint satisfaction problem. Logical Methods Comput. Sci. 8/1(07), 1–26 (2012)MathSciNetMATHGoogle Scholar
  5. 5.
    Bezem, M., Nieuwenhuis, R., Rodríguez-Carbonell, E.: The max-atom problem and its relevance. In LPAR, pages 47–61 (2008)Google Scholar
  6. 6.
    Bodirsky, M., Jonsson, P., von Oertzen, T.: Semilinear program feasibility. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S.E., Thomas, W. (eds.) Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Computer Science, pp 79–90. Springer Verlag (2009)Google Scholar
  7. 7.
    Bodirsky, M., Jonsson, P., von Oertzen, T.: Essential convexity and complexity of semi-algebraic constraints. Logical Methods Comput. Sci. 8(4) (2012). An extended abstract about a subset of the results has been published under the titleSemilinear Program Feasibility at ICALP’10Google Scholar
  8. 8.
    Bodirsky, M., Martin, B., Antoine, M.: Constraint satisfaction problems over the integers with successor. In Proceedings of ICALP, arXiv:1503.08572 (2015)
  9. 9.
    Boros, E., Elbassioni, K., Gurvich, V., Kazuhisa M.: Every Stochastic Game with Perfect Information Admits a Canonical form. RRR-09-2009, RUTCOR, Rutgers University (2009)Google Scholar
  10. 10.
    Bulatov, A.A., Krokhin, A.A., Jeavons, P.G.: Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34, 720–742 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    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
  12. 12.
    Develin, M., Sturmfels, B.: Tropical convexity. Doc. Math. 9, 1–27 (2004)MathSciNetMATHGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Filar, J., Vrieze, K.: Competitive Markov Decision Processes. Springer, New York (1996)CrossRefMATHGoogle Scholar
  16. 16.
    Garey, M., Johnson, D.: A guide NP-completeness. CSLI Press, Stanford (1978)MATHGoogle Scholar
  17. 17.
    Gillette, D.: Stochastic games with zero probabilities. Contrib. Theory Games 3, 179–187 (1957)MathSciNetMATHGoogle 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, March 4-7, 2015, Garching, Germany, pages 379–391 (2015)Google Scholar
  19. 19.
    Helton, J.W., Nie, J.: Semidefinite representation of convex sets. Math Program 122(1), 21–64 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hodges, W.: A shorter model theory. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
  21. 21.
    Jeavons, P.G., Cooper, M.C.: Tractable constraints on ordered domains. Artif. Intell. 79(2), 327–339 (1995)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. J. ACM 44(4), 527–548 (1997)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Jonsson, P., Bäckström, C.: A unifying approach to temporal constraint reasoning. Artif. Intell. 102(1), 143–155 (1998)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Jonsson, P., Lööw, T.: Computation complexity of linear constraints over the integers. Artif. Intell. 195, 44–62 (2013)CrossRefMATHGoogle Scholar
  25. 25.
    Jonsson, P., Thapper, J.: Constraint satisfaction and semilinear expansions of addition over the rationals and the reals. J. Comput. Syst. Sci. 82(5), 912–928 (2016)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Khachiyan, L.: A polynomial algorithm in linear programming. Dokl. Akad. Nauk SSSR 244, 1093–1097 (1979)MathSciNetMATHGoogle Scholar
  27. 27.
    Liggett, T.M., Lippman, S.A.: Stochastic games with perfect information and time average payoff. SIAM Rev. 11(4), 604–607 (1969)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Möhring, R.H., Skutella, M., Stork, F.: Scheduling with and/or precedence constraints. SIAM J. Comput. 33(2), 393–415 (2004)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Pöschel, R.: A general galois theory for operations and relations and concrete characterization of related algebraic structures. Tech Report of Akademie der Wissenschaften der DDR (1980)Google Scholar
  30. 30.
    Scowcroft, P.: A representation of convex semilinear sets. Algebra Univers. 62(2–3), 289–327 (2009)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    van den Dries, L.: Tame topology and o-minimal structures, volume 248 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  32. 32.
    Zwick, U., Paterson, M.: The complexity of mean payoff games on graphs. Theor. Comput. Sci. 158(1&2), 343–359 (1996)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institut für AlgebraTU DresdenDresdenGermany

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