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COMPLEXITY OF SHORT GENERATING FUNCTIONS

Part of: Theory of computing

Published online by Cambridge University Press:  14 February 2018

DANNY NGUYEN  and
IGOR PAK
DANNY NGUYEN
Affiliation:
Department of Mathematics, University of California, Los Angeles, USA; ldnguyen@math.ucla.edu, pak@math.ucla.edu
IGOR PAK
Affiliation:
Department of Mathematics, University of California, Los Angeles, USA; ldnguyen@math.ucla.edu, pak@math.ucla.edu

Abstract

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We give complexity analysis for the class of short generating functions. Assuming #P$\not \subseteq$FP/poly, we show that this class is not closed under taking many intersections, unions or projections of generating functions, in the sense that these operations can increase the bit length of coefficients of generating functions by a super-polynomial factor. We also prove that truncated theta functions are hard for this class.

MSC classification

Primary: 68Q17: Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
Secondary: 68Q15: Complexity classes (hierarchies, relations among complexity classes, etc.)
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 6 , 2018 , e1
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2018

References

Aaronson, S., ‘ $\mathsf{P}\stackrel{?}{=}\mathsf{N}\mathsf{P}$ ’, in Open Problems in Mathematics (Springer, New York, 2016), 1–122.Google Scholar
Arora, S. and Barak, B., Computational Complexity: A Modern Approach (Cambridge University Press, Cambridge, UK, 2009).Google Scholar
Bach, E., Miller, G. and Shallit, J., ‘Sum of divisors, perfect numbers and factoring’, SIAM J. Comput. 15 (1986), 1143–1154.Google Scholar
Barvinok, A., ‘A polynomial time algorithm for counting integral points in polyhedra when the fimension is fixed’, inProc. 34th FOCS (IEEE, Los Alamitos, CA, 1993), 566–572.Google Scholar
Barvinok, A., ‘The complexity of generating functions for integer points in polyhedra and beyond’, inProc. ICM, Vol. 3 (EMS, Zürich, 2006), 763–787.Google Scholar
Barvinok, A., Integer Points in Polyhedra (EMS, Zürich, 2008).CrossRefGoogle Scholar
Barvinok, A. and Pommersheim, J. E., ‘An algorithmic theory of lattice points in polyhedra’, inNew Perspectives in Algebraic Combinatorics (Cambridge University Press, Cambridge, UK, 1999), 91–147.Google Scholar
Barvinok, A. and Woods, K., ‘Short rational generating functions for lattice point problems’, J. Amer. Math. Soc. 16 (2003), 957–979.Google Scholar
Bombieri, E., Granville, A. and Pintz, J., ‘Squares in arithmetic progressions’, Duke Math. J. 66 (1992), 369–385.Google Scholar
Cai, J.-Y., ‘ S 2 P ⊆Z P P N P ’, J. Comput. System Sci. 73 (2007), 25–35.CrossRefGoogle Scholar
Eisenbrand, F., ‘Integer programming and algorithmic geometry of numbers’, in50 Years of Integer Programming (Springer, Berlin, 2010), 505–560.Google Scholar
Gasarch, W. I., ‘The second $\mathsf{P}\stackrel{?}{=}\mathsf{N}\mathsf{P}$ poll’, ACM SIGACT News 43(2) (2012), 53–77.CrossRefGoogle Scholar
Grädel, E., ‘The complexity of subclasses of logical theories’, Dissertation, Universität Basel, 1987.Google Scholar
Guy, R. K., Unsolved Problems in Number Theory, 3rd edn (Springer, New York, 2004).CrossRefGoogle Scholar
Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers (Oxford University Press, Oxford, UK, 2008).Google Scholar
Hough, B., ‘Solution of the minimum modulus problem for covering systems’, Ann. of Math. (2) 181 (2015), 361–382.Google Scholar
Kannan, R., ‘Lattice translates of a polytope and the Frobenius problem’, Combinatorica 12 (1992), 161–177.Google Scholar
Klivans, A. and Spielman, D., ‘Randomness efficient identity testing of multivariate polynomials’, inProc. 33rd FOCS (ACM, New York, 2001), 216–223.Google Scholar
Lagarias, J. and Odlyzko, A., ‘Computing 𝜋(x): an analytic method’, J. Algorithms 8 (1987), 173–191.CrossRefGoogle Scholar
Manders, K. and Adleman, L., ‘ NP -complete decision problems for binary quadratics’, J. Comput. System Sci. 16 (1978), 168–184.Google Scholar
Moore, C. and Mertens, S., The Nature of Computation (Oxford University Press, Oxford, UK, 2011).Google Scholar
Nguyen, D. and Pak, I., ‘Complexity of short Presburger arithmetic’, inProc. 49th STOC (ACM, New York, 2017), 812–820.Google Scholar
Nguyen, D. and Pak, I., ‘Enumeration of integer points in projections of unbounded polyhedra’, inProc. IPCO 2017, Lecture Notes in Computer Science, 10328 (Springer, New York, 2017), 417–429.Google Scholar
Nguyen, D. and Pak, I., ‘The computational complexity of integer programming with alternations’, inProc. 32nd CCC, 2017 (LIPICS, Dagstuhl, Germany, 2017), Art. 6, 18 pp.Google Scholar
Nguyen, D. and Pak, I., ‘Short Presburger arithmetic is hard’, inProc. 58th FOCS (IEEE, Los Alamitos, CA, 2017), 37–48.Google Scholar
Papadimitriou, C. H., Computational Complexity (Addison-Wesley, Reading, MA, 1994).Google Scholar
Schöning, U., ‘Complexity of Presburger arithmetic with fixed quantifier dimension’, Theory Comput. Syst. 30 (1997), 423–428.Google Scholar
Szemerédi, E., ‘The number of squares in an arithmetic progression’, Studia Sci. Math. Hungar. 9(3–4) (1974), 417.Google Scholar
Tao, T. and Vu, V. H., Additive Combinatorics (Cambridge University Press, Cambridge, UK, 2006).CrossRefGoogle Scholar
Weil, A., Number Theory. An Approach Through History (Birkhäuser, Boston, MA, 1984).Google Scholar
Woods, K., ‘Rational generating functions and lattice point sets’, PhD Thesis, University of Michigan, 2004, 112 pp.Google Scholar
Woods, K., ‘Presburger arithmetic, rational generating functions, and quasi-polynomials’, J. Symbolic Logic 80 (2015), 433–449.Google Scholar