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arXiv · 1201.2880

Finding a subset of nonnegative vectors with a coordinatewise large sum

Ilya I. Bogdanov·Grigory R. Chelnokov
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Abstract

Given a rational $a=p/q$ and $N$ nonnegative $d$-dimensional real vectors $u_1$, ..., $u_N$, we show that it is always possible to choose $(d-1)+\lceil (pN-d+1)/q\rceil$ of them such that their sum is (componentwise) at least $(p/q)(u_1+...+u_N)$. For fixed $d$ and $a$, this bound is sharp if $N$ is large enough. The method of the proof uses Carathéodory's theorem from linear programming.

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Ilya I. Bogdanov, Grigory R. Chelnokov. 2013-01-04. Finding a subset of nonnegative vectors with a coordinatewise large sum. https://doi.org/10.1016/j.disc.2012.12.006

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