Geometric Give and Take
🤖 AI Summary
This study addresses a geometric special case of Spencer’s balancing game under a planar straight-line arrangement: Alice must prevent Bob from emptying any cell by sequentially selecting lines, using the minimum possible initial number of stones. Focusing on the cell decomposition induced by $n$ lines in general position, the work integrates tools from computational geometry, combinatorial game theory, and algorithmic complexity. By leveraging dual graphs and a carefully designed potential function, the authors construct an effective strategy for Alice. The main contribution is the first proof that the minimum initial number of stones required is $\Theta(n^3)$, accompanied by a polynomial-time algorithm to compute this value, thereby fully characterizing the complexity of this geometric balancing game.
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
balancing game
geometric arrangement
pebbling
line arrangement
combinatorial game
Innovation
Methods, ideas, or system contributions that make the work stand out.
geometric balancing game
line arrangement
pebbling strategy
polynomial-time computability
combinatorial geometry
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