Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements

TL;DR

The paper addresses the problem of biasing cuts in the Ham-Sandwich setting for well-separated $d$-dimensional point sets by introducing two independent proofs of the discrete $\alpha$-Ham-Sandwich theorem: a combinatorial approach via grid Unique Sink Orientations and a dual, topological approach via rainbow arrangements and the Poincaré-Miranda theorem. The first proof builds a grid graph and shows the induced subgrids are USOs, yielding a bijection that guarantees a unique $(\alpha_1,\dots,\alpha_d)$-cut; the second proof uses point-hyperplane duality to formulate rainbow arrangements and applies Poincaré-Miranda to locate common intersection points corresponding to the required level counts, with extensions to oriented matroids. The authors then generalize well-separation to $(\beta,\gamma)$-separation and establish a broader rainbow-arrangement framework, including a lower-level,$\, ext{∃R}$-completeness hardness result for realizability. A key linkage is drawn between the $\alpha$-Ham-Sandwich theorem, grid USOs, and realizability questions, culminating in a connection to bicolored stretchability and a demonstration of $\exists\mathbb{R}$-hardness for related realizability problems. Overall, the work deepens the relationship between geometric partitioning, combinatorial orientations, and topological methods, while providing new generalizations and complexity boundaries for realizability questions in higher dimensions and matroidal settings.

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 $α$-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 $\exists \mathbb{R}$-complete, which also implies that the realizability problem for grid Unique Sink Orientations is $\exists \mathbb{R}$-complete.

Figures (8)

• Figure 1: A family of well-separated point sets $\mathcal{F} = (P_1, P_2, P_3)$ in $\mathbb{R}^3$. The hyperplane $h$ separates $P_1\cup P_2$ from $P_3$.
• Figure 2: Well-separated point sets in $\mathbb{R}^2$ with two $\alpha$-cuts.
• Figure 3: Well-separated point sets in $\mathbb{R}^2$ with the corresponding grid orientation. The line drawn on the left is represented by the highlighted vertex on the right.
• Figure 4: A rainbow arrangement with colors red and blue. The red 3-level and the blue 2-level are highlighted and intersect in a unique point.
• Figure 5: A well-separated rainbow arrangement in $\mathbb{R}^2$. Red pseudo-lines are oriented from left to right, blue pseudo-lines from top to bottom. Note that in order to intersect the blue pseudo-lines in this order, the two red pseudo-lines need to cross twice. In particular, this means that there is no combinatorially equivalent arrangement using straight lines, i.e., this is a NO-instance of bicolored stretchability.

Theorems & Definitions (25)

• Definition 1
• Definition 2
• Theorem 3
• Theorem 4
• Lemma 5
• Lemma 6
• Theorem 7: Bárány, Hubard, and Jerónimo baranySlicingConvexSets2008
• Theorem 7
• Definition 8: Generalized Arrangement, Rainbow Arrangement
• Definition 9: Well-Separated