Geometric Give and Take

TL;DR

This paper analyzes a geometric balancing game involving line arrangements and pebble distribution, establishing the minimum initial pebbles needed for Alice to prevent Bob from emptying any box, with a focus on arrangements of lines in general position.

Contribution

It determines the polynomial-time computable minimum number of pebbles needed for Alice to win, showing it scales as Θ(n^3) for general position line arrangements.

Findings

Minimum pebbles needed is Θ(n^3) for general position arrangements.

The value of f(𝓛) can be computed in polynomial time.

The game dynamics relate to geometric and combinatorial properties of line arrangements.

Abstract

We consider a special, geometric case of a balancing game introduced by Spencer in 1977. Consider any arrangement $\mathcal{L}$ of $n$ lines in the plane, and assume that each cell of the arrangement contains a box. Alice initially places pebbles in each box. In each subsequent step, Bob picks a line, and Alice must choose a side of that line, remove one pebble from each box on that side, and add one pebble to each box on the other side. Bob wins if any box ever becomes empty. We determine the minimum number $f(\mathcal L)$ of pebbles, computable in polynomial time, for which Alice can prevent Bob from ever winning, and we show that $f(\mathcal L)=\Theta(n^3)$ for any arrangement $\mathcal{L}$ of $n$ lines in general position.