Splitting Planar Necklaces with Few Straight Cuts

Abstract

In the classical necklace splitting problem, beads of different colors lie on a line, and the goal is to divide them among $k$ agents using as few cuts as possible so that every agent receives the same number of beads of each color. We study the planar analog in which a cut is a full straight line and the objective is minimizing the number of lines: $m$ points colored red and blue must be separated by straight-line cuts into regions that can be allocated to $k$ agents so that each agent receives the same number of points of each color. Projection to a line gives $2(k-1)$ cuts, and $k-1$ are sometimes necessary; we give algorithms, static and dynamic, that provably close part of this gap. We develop several approaches that capitalize on the extra degree of freedom: a ``star'' search that produces solutions with as few as $\lceil \frac{k}{2} \rceil$ cuts and never more than $k$ whenever a fair contiguous circular partition exists, with further guarantees for evenly spaced input, and a general algorithm that produces at most $k + \log_2(k+1) - 2$ cuts in $Θ(m \log k)$ time, built on a one-dimensional necklace splitting subroutine that requires no presorting. We then adapt the general algorithm to a dynamic setting, maintaining exact fairness and the cut bound under arbitrary relocation of points, with expected update time that scales with the total distance moved.