Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements

2026-02-11

AI Summary

This work investigates the problem of simultaneously partitioning multiple point sets in high-dimensional space by a single hyperplane according to prescribed ratios—a non-uniform variant of the Ham-Sandwich cut. We present two novel proofs of the α-Ham-Sandwich theorem: a purely combinatorial proof that reveals a deep connection to unique sink orientations on grids, and a topological proof based on duality and the Poincaré–Miranda theorem, which introduces the new notion of “rainbow permutations.” Furthermore, we extend the theorem to matroidal structures and establish that both the realizability of rainbow permutations and that of unique sink orientations are ∃ℝ-complete, thereby highlighting the intrinsic computational complexity underlying these geometric and combinatorial phenomena.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Interdisciplinary science discovery with web data mining
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The famous Ham-Sandwich theorem states that any $d$ point sets in $\mathbb{R}^d$ can be simultaneously bisected by a single hyperplane. The $\alpha$-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 $\alpha$-Ham-Sandwich theorem and Unique Sink Orientations of grids. The second proof uses point-hyperplane duality and the Poincar\'e-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 $\exists \mathbb{R}$-complete, which also implies that the realizability problem for grid Unique Sink Orientations is $\exists \mathbb{R}$-complete.

Problem

• Ham-Sandwich theorem
• biased cuts
• Unique Sink Orientations
• rainbow arrangements
• realizability problem

Innovation

• Unique Sink Orientations
• Rainbow Arrangements
• α-Ham-Sandwich Theorem
• Oriented Matroids
• ∃ℝ-completeness