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

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

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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.