Unfairly Splitting Separable Necklaces
Patrick Schnider, Linus Stalder, Simon Weber

TL;DR
This paper studies the unfair splitting of n-separable necklaces, a generalization of the classical Necklace Splitting problem, and proves it can be solved in polynomial time, impacting complexity classifications related to the $ ext{α}$-Ham Sandwich problem.
Contribution
It introduces and analyzes the unfair necklace splitting problem for n-separable necklaces, showing it is polynomial-time solvable, thus clarifying its complexity status.
Findings
Unfair splitting of n-separable necklaces is polynomial-time solvable.
The problem cannot be used to prove UEOPL-hardness for $ ext{α}$-Ham Sandwich.
This extends understanding of necklace splitting and its relation to computational complexity.
Abstract
The Necklace Splitting problem is a classical problem in combinatorics that has been intensively studied both from a combinatorial and a computational point of view. It is well-known that the Necklace Splitting problem reduces to the discrete Ham Sandwich problem. This reduction was crucial in the proof of PPA-completeness of the Ham Sandwich problem. Recently, Borzechowski, Schnider and Weber [ISAAC'23] introduced a variant of Necklace Splitting that similarly reduces to the -Ham Sandwich problem, which lies in the complexity class UEOPL but is not known to be complete. To make this reduction work, the input necklace is guaranteed to be n-separable. They showed that these necklaces can be fairly split in polynomial time and thus this subproblem cannot be used to prove UEOPL-hardness for -Ham Sandwich. We consider the more general unfair necklace splitting problem on…
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