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Unfairly Splitting Separable Necklaces

Authors Patrick Schnider , Linus Stalder, Simon Weber

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LIPIcs.STACS.2025.71.pdf
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Author Details

Patrick Schnider
  • Department of Computer Science, ETH Zürich, Switzerland
Linus Stalder
  • Department of Computer Science, ETH Zürich, Switzerland
Simon Weber
  • Department of Computer Science, ETH Zürich, Switzerland

Funding

  • Weber, Simon: Swiss National Science Foundation under project no. 204320.

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Patrick Schnider, Linus Stalder, and Simon Weber. Unfairly Splitting Separable Necklaces. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 71:1-71:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.71

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 α-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 α-Ham Sandwich. We consider the more general unfair necklace splitting problem on n-separable necklaces, i.e., the problem of splitting these necklaces such that each thief gets a desired fraction of each type of jewels. This more general problem is the natural necklace-splitting-type version of α-Ham Sandwich, and its complexity status is one of the main open questions posed by Borzechowski, Schnider and Weber. We show that the unfair splitting problem is also polynomial-time solvable, and can thus also not be used to show UEOPL-hardness for α-Ham Sandwich.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Theory of computation → Computational geometry
Keywords
  • Necklace splitting
  • n-separability
  • well-separation
  • Ham Sandwich
  • alpha-Ham Sandwich
  • unfair splitting
  • fair division

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