Number Balancing is as Hard as Minkowski’s Theorem and Shortest Vector

  • Rebecca HobergEmail author
  • Harishchandra Ramadas
  • Thomas Rothvoss
  • Xin Yang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10328)


The number balancing (NBP) problem is the following: given real numbers \(a_1,\ldots ,a_n \in [0,1]\), find two disjoint subsets \(I_1,I_2 \subseteq [n]\) so that the difference \(|\sum _{i \in I_1}a_i - \sum _{i \in I_2}a_i|\) of their sums is minimized. An application of the pigeonhole principle shows that there is always a solution where the difference is at most \(O(\frac{\sqrt{n}}{2^n})\). Finding the minimum, however, is NP-hard. In polynomial time, the differencing algorithm by Karmarkar and Karp from 1982 can produce a solution with difference at most \(n^{-\varTheta (\log n)}\), but no further improvement has been made since then.

In this paper, we show a relationship between NBP and Minkowski’s Theorem. First we show that an approximate oracle for Minkowski’s Theorem gives an approximate NBP oracle. Perhaps more surprisingly, we show that an approximate NBP oracle gives an approximate Minkowski oracle. In particular, we prove that any polynomial time algorithm that guarantees a solution of difference at most \(2^{\sqrt{n}}/2^{n}\) would give a polynomial approximation for Minkowski as well as a polynomial factor approximation algorithm for the Shortest Vector Problem.


Theoretical Computer Science Number Balance Pigeonhole Principle Symmetric Convex Body Minkowski Problem 
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  1. [Ajt96]
    Ajtai, M.: Generating hard instances of lattice problems. In: Proceedings of the 28th STOC, pp. 99–108. ACM (1996)Google Scholar
  2. [AR05]
    Aharonov, D., Regev, O.: Lattice problems in NP cap conp. J. ACM 52(5), 749–765 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [Boh96]
    Bohman, T.: A sum packing problem of erdös and the conway-guy sequence. Proc. AMS 124(12), 3627–3636 (1996)CrossRefzbMATHGoogle Scholar
  4. [GJ97]
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1997)zbMATHGoogle Scholar
  5. [GLS12]
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, vol. 2, pp. 122–125. Springer, Heidelberg (2012)zbMATHGoogle Scholar
  6. [HR07]
    Haviv, I., Regev, O.: Tensor-based hardness of the shortest vector problem to within almost polynomial factors, pp. 469–477 (2007)Google Scholar
  7. [Joh48]
    John, F.: Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays Presented to R. Courant on his 60th Birthday, 8 January 1948, pp. 187–204. Interscience Publishers Inc., New York (1948)Google Scholar
  8. [KK82]
    Karmarkar, N., Karp, R.: The differencing method of set partitioning. Technical report, CS Division, UC Berkeley (1982).
  9. [KPP04]
    Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Heidelberg (2004)CrossRefzbMATHGoogle Scholar
  10. [LLL82]
    Lenstra, A., Lenstra, H., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261(4), 515–534 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Lov86]
    Lovász, L.: An Algorithmic Theory of Numbers, Graphs and Convexity. SIAM (1986)Google Scholar
  12. [Lov90]
    Lovász, L.: Geometric Algorithms and Algorithmic Geometry. American Mathematical Society (1990)Google Scholar
  13. [Lun88]
    Lunnon, W.: Integer sets with distinct subset-sums. Math. Comput. 50(181), 297–320 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [LY11]
    Lev, V., Yuster, R.: On the size of dissociated bases. Electr. J. Comb. 18(1), P117 (2011)MathSciNetzbMATHGoogle Scholar
  15. [Mat02]
    Matousek, J.: Lectures on Discrete Geometry. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  16. [Mer06]
    Mertens, S.: The easiest hard problem: number partitioning. Comput. Complex. Stat. Phys. 125(2), 125–139 (2006)MathSciNetzbMATHGoogle Scholar
  17. [MR09]
    Micciancio, D., Regev, O.: Lattice-based cryptography. In: Post-quantum Cryptography, pp. 147–191. Springer (2009)Google Scholar
  18. [MT90]
    Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations. Wiley Inc., New York (1990)zbMATHGoogle Scholar
  19. [NV10]
    Nguyen, P., Vallée, B.: The lll algorithm. In: Nguyen, P., Vallée, B. (eds.) Information Security and Cryptography. Springer, Heidelberg (2010)Google Scholar
  20. [Pap94]
    Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48(3), 498–532 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [Sch87]
    Schnorr, C.: A hierarchy of polynomial time lattice basis reduction algorithms. Theor. Comput. Sci. 53, 201–224 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [WY92]
    Woeginger, G., Yu, Z.: On the equal-subset-sum problem. Inf. Process. Lett. 42(6), 299–302 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Rebecca Hoberg
    • 1
    Email author
  • Harishchandra Ramadas
    • 1
  • Thomas Rothvoss
    • 1
  • Xin Yang
    • 1
  1. 1.University of WashingtonSeattleUSA

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