Advertisement

Enumeration of Integer Points in Projections of Unbounded Polyhedra

  • Danny NguyenEmail author
  • Igor Pak
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10328)

Abstract

We extend the Barvinok–Woods algorithm for enumeration of integer points in projections of polytopes to unbounded polyhedra. To achieve this, we employ a new structural result on projections of semilinear subsets of the integer lattice.

Notes

Acknowledgements

We are greatly indebted to Sasha Barvinok and Sinai Robins for introducing us to the subject. We are thankful to Iskander Aliev, Matthias Aschenbrenner, Artëm Chernikov, Jesús De Loera, Lenny Fukshansky, Oleg Karpenkov and Kevin Woods for interesting conversations and helpful remarks. We also thank the anonymous referees for helpful references, which we cannot fully accommodate due to space limit. The second author was partially supported by the NSF.

References

  1. [ADL16]
    Aliev, I., De Loera, J.A., Louveaux, Q.: Parametric polyhedra with at least \(k\) lattice points: their semigroup structure and the \(k\)-Frobenius problem. In: Beveridge, A., Griggs, J.R., Hogben, L., Musiker, G., Tetali, P. (eds.) Recent Trends in Combinatorics. Springer, Switzerland (2016)Google Scholar
  2. [B+12]
    Baldoni, V., Berline, N., De Loera, J.A., Köppe, M., Vergne, M.: Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra. Found. Comput. Math. 12, 435–469 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [Bar93]
    Barvinok, A.: A polynomial time algorithm for counting integral points in polyhedra when the fimension is fixed. In: Proceedings of the 34th FOCS, IEEE, Los Alamitos, CA, pp. 566–572 (1993)Google Scholar
  4. [Bar08]
    Barvinok, A.: Integer Points in Polyhedra. EMS, Zürich (2008)CrossRefzbMATHGoogle Scholar
  5. [BP99]
    Barvinok, A., Pommersheim, J.E.: An algorithmic theory of lattice points in polyhedra. In: New Perspectives in Algebraic Combinatorics, pp. 91–147. Cambridge University Press, Cambridge (1999)Google Scholar
  6. [BW03]
    Barvinok, A., Woods, K.: Short rational generating functions for lattice point problems. J. Amer. Math. Soc. 16, 957–979 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [BV07]
    Berline, N., Vergne, M.: Local Euler-Maclaurin formula for polytopes. Mosc. Math. J. 7, 355–386 (2007)MathSciNetzbMATHGoogle Scholar
  8. [CH16]
    Chistikov, D., Haase, C.: The taming of semi-linear set. In: Proceedings of the ICALP 2016, pp. 127:1–127:13 (2016)Google Scholar
  9. [D+04]
    De Loera, J.A., Haws, D., Hemmecke, R., Huggins, P., Sturmfels, B., Yoshida, R.: Short rational functions for toric algebra and applications. J. Symbolic Comput. 38, 959–973 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [D+04]
    De Loera, J.A., Hemmecke, R., Tauzer, J., Yoshida, R.: Effective lattice point counting in rational convex polytopes. J. Symbolic Comput. 38, 1273–1302 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [DK97]
    Dyer, M., Kannan, R.: On Barvinok’s algorithm for counting lattice points in fixed dimension. Math. Oper. Res. 22, 545–549 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Eis03]
    Eisenbrand, F.: Fast integer programming in fixed dimension. In: Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 196–207. Springer, Heidelberg (2003). doi: 10.1007/978-3-540-39658-1_20 CrossRefGoogle Scholar
  13. [ES08]
    Eisenbrand, F., Shmonin, G.: Parametric integer programming in fixed dimension. Math. Oper. Res. 33, 839–850 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [FT87]
    Frank, A., Tardos, É.: An application of simultaneous Diophantine approximation in combinatorial optimization. Combinatorica 7, 49–65 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [Gin66]
    Ginsburg, S.: The Mathematical Theory of Context Free Languages. McGraw-Hill, New York (1966)zbMATHGoogle Scholar
  16. [GS64]
    Ginsburg, S., Spanier, E.: Bounded ALGOL-like languages. Trans. Amer. Math. Soc. 113, 333–368 (1964)MathSciNetzbMATHGoogle Scholar
  17. [Kan90]
    Kannan, R.: Test sets for integer programs, \(\forall \exists \) sentences. In: Polyhedral Combinatorics, pp. 39–47. AMS, Providence (1990)Google Scholar
  18. [Köp07]
    Köppe, M.: A primal Barvinok algorithm based on irrational decompositions. SIAM J. Discrete Math. 21, 220–236 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [KV08]
    Köppe, M., Verdoolaege, S.: Computing parametric rational generating functions with a primal Barvinok algorithm. Electron. J. Combin. 15(1), 19 (2008). RP 16Google Scholar
  20. [Len83]
    Lenstra, H.: Integer programming with a fixed number of variables. Math. Oper. Res. 8, 538–548 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [Ler05]
    Leroux, J.: A polynomial time presburger criterion and synthesis for number decision diagrams. In: Proceedings of the 20th LICS, IEEE, Chicago, IL, pp. 147–156 (2005)Google Scholar
  22. [Mei93]
    Meiser, S.: Point location in arrangement of hyperplanes. Inform. Comput. 106, 286–303 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Par66]
    Parikh, R.: On context-free languages. J. Assoc. Comput. Mach. 13, 570–581 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [Sch86]
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986)zbMATHGoogle Scholar
  25. [V+07]
    Verdoolaege, S., Seghir, R., Beyls, K., Loechner, V., Bruynooghe, M.: Counting integer points in parametric polytopes using Barvinok’s rational functions. Algorithmica 48, 37–66 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [W04]
    Woods, K.: Rational Generating Functions and Lattice Point Sets, Ph.D. thesis, University of Michigan, 112 p. (2004)Google Scholar
  27. [W15]
    Woods, K.: Presburger arithmetic, rational generating functions, and quasi-polynomials. J. Symb. Log. 80, 433–449 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA

Personalised recommendations