Sperner’s Colorings and Optimal Partitioning of the Simplex
We discuss coloring and partitioning questions related to Sperner’s Lemma, originally motivated by an application in hardness of approximation. Informally, we call a partitioning of the (k − 1)-dimensional simplex into k parts, or a labeling of a lattice inside the simplex by k colors, “Sperner-admissible” if color i avoids the face opposite to vertex i. The questions we study are of the following flavor: What is the Sperner-admissible labeling/partitioning that makes the total area of the boundary between different colors/parts as small as possible?
First, for a natural arrangement of “cells” in the simplex, we prove an optimal lower bound on the number of cells that must be non-monochromatic in any Sperner-admissible labeling. This lower bound is matched by a simple labeling where each vertex receives the minimum admissible color.
Second, we show for this arrangement that in contrast to Sperner’s Lemma, there is a Sperner-admissible labeling such that every cell contains at most 4 colors.
Finally, we prove a geometric variant of the first result: For any Sperner-admissible partition of the regular simplex, the total surface area of the boundary shared by at least two different parts is minimized by the Voronoi partition (A1∗, …, A k ∗) where A i ∗ contains all the points whose closest vertex is e i . We also discuss possible extensions of this result to general polytopes and some open questions.
The second author is indebted in many ways to Jirka Matoušek, who introduced him to Sperner’s Lemma in an undergradute course at Charles University a long time ago.
- 1.C. Chekuri, A. Ene, Submodular cost allocation problem and applications, in Proceedings of the ICALP, 2011, pp. 354–366Google Scholar
- 4.A. Ene, J. Vondrák, Hardness of submodular cost allocation: lattice matching and a simplex coloring conjecture, in Proceedings of the APPROX, 2014, pp. 144–159Google Scholar
- 5.A. Ene, J. Vondrák, Y. Wu, Local distribution and the symmetry gap: approximability of multiway partitioning problems, in Proceedings of the ACM-SIAM SODA, 2013, pp. 306–325Google Scholar
- 9.R. Manokaran, J. Naor, P. Raghavendra, R. Schwartz, SDP gaps and UGC hardness for multiway cut, 0-extension, and metric labeling, in Proceedings of the ACM STOC, 2008, pp. 11–20Google Scholar
- 10.M. Mirzakhani, J. Vondrák, Sperner’s colorings, hypergraph labeling problems and fair division, in Proceedings of the ACM-SIAM SODA, 2015, pp. 873–886Google Scholar