A Purely Geometric Variant of the Gale-Berlekamp Switching Game

TL;DR

This paper introduces a geometric variant of the Gale-Berlekamp switching game involving points in the plane, providing bounds on achievable total weights and polynomial-time algorithms for near-optimal solutions.

Contribution

The paper presents a new geometric variant of the Gale-Berlekamp game, establishes bounds on maximum total weight, and offers polynomial-time algorithms to approximate these bounds.

Findings

Achieves total weight at least n - o(n) for large n

Guarantees at least n/3 total weight for all n

Provides polynomial-time algorithms for near-optimal solutions

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.