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Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements

Authors Michaela Borzechowski, Sebastian Haslebacher , Hung P. Hoang , Patrick Schnider , Simon Weber

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Discrete mathematics
Keywords
  • α-Ham-Sandwich Theorem
  • Pseudo-Hyperplanes
  • Arrangements
  • Unique Sink Orientations
  • Oriented Matroids

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Abstract

The famous Ham-Sandwich theorem states that any d point sets in ℝ^d can be simultaneously bisected by a single hyperplane. The α-Ham-Sandwich theorem gives a sufficient condition for the existence of biased cuts, i.e., hyperplanes that do not cut off half but some prescribed fraction of each point set. We give two new proofs for this theorem. The first proof is completely combinatorial and highlights a strong connection between the α-Ham-Sandwich theorem and Unique Sink Orientations of grids. The second proof uses point-hyperplane duality and the Poincaré-Miranda theorem and allows us to generalize the result to and beyond oriented matroids. For this we introduce a new concept of rainbow arrangements, generalizing colored pseudo-hyperplane arrangements. Along the way, we also show that the realizability problem for rainbow arrangements is ∃ℝ-complete, which also implies that the realizability problem for grid Unique Sink Orientations is ∃ℝ-complete.

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Michaela Borzechowski, Sebastian Haslebacher, Hung P. Hoang, Patrick Schnider, and Simon Weber. Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.19

Author Details

Michaela Borzechowski
  • Department of Mathematics and Computer Science, Freie Universität Berlin, Germany
Sebastian Haslebacher
  • Department of Computer Science, ETH Zürich, Switzerland
Hung P. Hoang
  • Algorithms and Complexity Group, Faculty of Informatics, TU Wien, Austria
Patrick Schnider
  • Department of Mathematics and Computer Science, University of Basel, Switzerland
  • Department of Computer Science, ETH Zürich, Switzerland
Simon Weber
  • Department of Computer Science, ETH Zürich, Switzerland

Funding

  • Borzechowski, Michaela: DFG within GRK 2434 Facets of Complexity.
  • Hoang, Hung P.: Austrian Science Foundation (FWF, projects 10.55776/Y1329 and ESP1136425).

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