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arXiv · 2502.16305

A Purely Geometric Variant of the Gale-Berlekamp Switching Game

Adrian Dumitrescu·Jeck Lim·János Pach·Ji Zeng
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Abstract

We introduce the following variant of the Gale-Berlekamp switching game. Let $P$ be a set of n noncollinear points in the plane, each of them having weight $+1$ or $-1$. At each step, we pick a line $\ell$ passing through at least two points of $P$, and switch the sign of every point $p \in P\cap\ell$. The objective is to maximize the total weight of the elements of $P$. We show that one can always achieve that this quantity is at least $n - o(n)$, as $n\rightarrow\infty$, and at least $n/3$, for every $n$. Moreover, these can be attained by a polynomial time algorithm.

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Adrian Dumitrescu, Jeck Lim, János Pach, Ji Zeng. 2025-02-22. A Purely Geometric Variant of the Gale-Berlekamp Switching Game. https://arxiv.org/abs/2502.16305

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