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

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 10328)

Abstract

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.

Keywords

  • Theoretical Computer Science
  • Number Balance
  • Pigeonhole Principle
  • Symmetric Convex Body
  • Minkowski Problem

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

T. Rothvoss—Supported by NSF grant 1420180 with title “Limitations of convex relaxations in combinatorial optimization”, an Alfred P. Sloan Research Fellowship and a David and Lucile Packard Foundation Fellowship.

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Notes

  1. 1.

    SVP can be defined for any norm, but anywhere the norm is not specified we consider the Euclidean norm.

  2. 2.

    If \(\rho \le 2-\varepsilon \), then one can still obtain an error of \(|\sum _{i=1}^n a_ix_i| \le 2^{-\varTheta (\varepsilon n)}\), but this breaks down if \(\rho \ge 2\).

  3. 3.

    We assume K is given to us by a separation oracle.

  4. 4.

    Here we ignore the dependence of t on n - notice that t is nondecreasing in n, so replacing t(n) by \(t(\sqrt{n})\) only increases \(f_t(n)\).

  5. 5.

    Strictly speaking, the length of axis i is \(2\lambda _i\), but we will continue calling \(\lambda _i\) the “axis length”.

  6. 6.

    Note that there exists an ellipsoid that approximates K within a factor of \(\sqrt{n}\) and if K is a polytope with m facets, then this ellipsoid can be found in time polynomial in n and m. However, if one only has a separation oracle for K, then the best factor achievable in polynomial time is n [GLS12].

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Correspondence to Rebecca Hoberg .

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Hoberg, R., Ramadas, H., Rothvoss, T., Yang, X. (2017). Number Balancing is as Hard as Minkowski’s Theorem and Shortest Vector. In: Eisenbrand, F., Koenemann, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2017. Lecture Notes in Computer Science(), vol 10328. Springer, Cham. https://doi.org/10.1007/978-3-319-59250-3_21

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  • DOI: https://doi.org/10.1007/978-3-319-59250-3_21

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