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# The Computational Complexity of Integer Programming with Alternations

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## Cite As

Danny Nguyen and Igor Pak. The Computational Complexity of Integer Programming with Alternations. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.CCC.2017.6

## Abstract

We prove that integer programming with three alternating quantifiers is NP-complete, even for a fixed number of variables. This complements earlier results by Lenstra and Kannan, which together say that integer programming with at most two alternating quantifiers can be done in polynomial time for a fixed number of variables. As a byproduct of the proof, we show that for two polytopes P, Q in R^4, counting the projection of integer points in Q\P is #P-complete. This contrasts the 2003 result by Barvinok and Woods, which allows counting in polynomial time the projection of integer points in P and Q separately.
##### Keywords
• Integer Programming
• Alternations
• Projection of Integer Points

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## References

1. S. Arora and B. Barak. Computational complexity: A modern approach. Cambridge University Press, Cambridge, 2009. URL: http://dx.doi.org/10.1017/CBO9780511804090.
2. A. Barvinok. A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. In 34th Annual Symposium on Foundations of Computer Science (Palo Alto, CA, 1993), pages 566-572. IEEE Comput. Soc. Press, Los Alamitos, CA, 1993. URL: http://dx.doi.org/10.1109/SFCS.1993.366830.
3. A. Barvinok. Integer points in polyhedra. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008. URL: http://dx.doi.org/10.4171/052.
4. A. Barvinok and J. E. Pommersheim. An algorithmic theory of lattice points in polyhedra. In New perspectives in algebraic combinatorics (Berkeley, CA, 1996-97), volume 38 of Math. Sci. Res. Inst. Publ., pages 91-147. Cambridge Univ. Press, Cambridge, 1999.
5. A. Barvinok and K. Woods. Short rational generating functions for lattice point problems. J. Amer. Math. Soc., 16(4):957-979, 2003. URL: http://dx.doi.org/10.1090/S0894-0347-03-00428-4.
6. M. Beck and S. Robins. Computing the continuous discretely. Undergraduate Texts in Mathematics. Springer, New York, second edition, 2015. Integer-point enumeration in polyhedra, With illustrations by David Austin. URL: http://dx.doi.org/10.1007/978-1-4939-2969-6.
7. A. Below, U. Brehm, J. A. De Loera, and J. Richter-Gebert. Minimal simplicial dissections and triangulations of convex 3-polytopes. Discrete Comput. Geom., 24(1):35-48, 2000. URL: http://dx.doi.org/10.1007/s004540010058.
8. M. Brion. Points entiers dans les polytopes convexes. Séminaire N. Bourbaki, 36 (1993-1994):145-169, 1995. Talk No. 780. URL: http://www.numdam.org/item?id=SB_1993-1994__36__145_0.
9. J. A. De Loera, J. Rambau, and F. Santos. Triangulations, volume 25 of Algorithms and Computation in Mathematics. Springer-Verlag, Berlin, 2010. Structures for algorithms and applications. URL: http://dx.doi.org/10.1007/978-3-642-12971-1.
10. M. Dyer. On counting lattice points in polyhedra. SIAM J. Comput., 20(4):695-707, 1991. URL: http://dx.doi.org/10.1137/0220044.
11. F. Eisenbrand. Integer programming and algorithmic geometry of numbers. In 50 years of integer programming 1958-2008, pages xx+804. Springer-Verlag, Berlin, 2010. URL: http://dx.doi.org/10.1007/978-3-540-68279-0.
12. F. Eisenbrand and N. Hähnle. Minimizing the number of lattice points in a translated polygon. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1123-1130. SIAM, Philadelphia, PA, 2012.
13. F. Eisenbrand and T. Rothvoß. New hardness results for Diophantine approximation. In Approximation, randomization, and combinatorial optimization, volume 5687 of Lecture Notes in Comput. Sci., pages 98-110. Springer, Berlin, 2009. URL: http://dx.doi.org/10.1007/978-3-642-03685-9_8.
14. E. Grädel. The complexity of subclasses of logical theories. PhD thesis, Universität Basel, 1978.
15. M. Grötschel, L. Lovász, and A. Schrijver. Geometric algorithms and combinatorial optimization, volume 2 of Algorithms and Combinatorics. Springer-Verlag, Berlin, second edition, 1993. URL: http://dx.doi.org/10.1007/978-3-642-78240-4.
16. R. Kannan. Test sets for integer programs, ∀∃ sentences. In Polyhedral combinatorics (Morristown, NJ, 1989), volume 1 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 39-47. Amer. Math. Soc., Providence, RI, 1990.
17. R. Kannan. Lattice translates of a polytope and the Frobenius problem. Combinatorica, 12(2):161-177, 1992. URL: http://dx.doi.org/10.1007/BF01204720.
18. J. Lagarias. The computational complexity of simultaneous Diophantine approximation problems. SIAM J. Comput., 14(1):196-209, 1985. URL: http://dx.doi.org/10.1137/0214016.
19. H. Lenstra. Integer programming with a fixed number of variables. Math. Oper. Res., 8(4):538-548, 1983. URL: http://dx.doi.org/10.1287/moor.8.4.538.
20. C. Moore and S. Mertens. The nature of computation. Oxford University Press, Oxford, 2011. URL: http://dx.doi.org/10.1093/acprof:oso/9780199233212.001.0001.
21. D. Nguyen and I. Pak. Complexity of short generating functions, preprint; arxiv:1702.08660, 2017.
22. D. Nguyen and I. Pak. Complexity of short presburger arithmetic. To appear in STOC'17 - Proceedings of the 2017 ACM Symposium on Theory of Computing, 2017.
24. J. Ruppert and R. Seidel. On the difficulty of triangulating three-dimensional nonconvex polyhedra. Discrete Comput. Geom., 7(3):227-253, 1992. URL: http://dx.doi.org/10.1007/BF02187840.
25. U. Schöning. Complexity of Presburger arithmetic with fixed quantifier dimension. Theory Comput. Syst., 30(4):423-428, 1997. URL: http://dx.doi.org/10.1007/s002240000059.
26. A. Schrijver. Theory of linear and integer programming. Wiley-Interscience Series in Discrete Mathematics. John Wiley &Sons, Ltd., Chichester, 1986. A Wiley-Interscience Publication.
27. P. van Emde Boas. Another NP-complete partition problem and the complexity of computing short vectors in a lattice. Math. Dept. Report, pages 81-04, 1981.
28. K. Woods. Rational Generating Functions and Lattice Point Sets. PhD thesis, University of Michigan, 2004.
29. K. Woods. Presburger arithmetic, rational generating functions, and quasi-polynomials. J. Symb. Log., 80(2):433-449, 2015. URL: http://dx.doi.org/10.1017/jsl.2015.4.
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