Hardness of Approximate Sperner and Applications to Envy-Free Cake Cutting

TL;DR

It is PPAD-complete to find an allocation -even using any constant number of possibly disconnected pieces- that makes just three agents envy-free, and this result has an interesting application for the envy-free cake cutting from fair division.

Abstract

Given a so called ''Sperner coloring'' of a triangulation of the $D$-dimensional simplex, Sperner's lemma guarantees the existence of a rainbow simplex, i.e. a simplex colored by all $D+1$ colors. However, finding a rainbow simplex was the first problem to be proven $\mathsf{PPAD}$-complete in Papadimitriou's classical paper introducing the class $\mathsf{PPAD}$ (1994). In this paper, we prove that the problem does not become easier if we relax ''all $D+1$ colors'' to allow some fraction of missing colors: in fact, for any constant $D$, finding even a simplex with just three colors remains $\mathsf{PPAD}$-complete! Our result has an interesting application for the envy-free cake cutting from fair division. It is known that if agents value pieces of cake using general continuous functions satisfying a simple boundary condition (''a non-empty piece is better than an empty piece of cake''), there exists an envy-free allocation with connected pieces. We show that for any constant number of agents it is $\mathsf{PPAD}$-complete to find an allocation -- even using any constant number of possibly disconnected pieces -- that makes just three agents envy-free. Our results extend to super-constant dimension, number of agents, and number of pieces, as long as they are asymptotically bounded by any $\log^{1-Ω(1)}(ε)$, where $ε$ is the precision parameter (side length for Sperner and approximate envy-free for cake cutting).

Figures (3)

• Figure 1: An example of the core rectangle in the constructions of DBLP:journals/tcs/ChenD09 (left) and its corresponding triangular continuous variant (right) we will consider as the base instance $C$. We present the triangular continuous instance by its projection onto the plane with the first coordinate $x_1=0$. The dashed square presents the core region. We turn the discrete instances into continuous instances by mapping each vertex in the discrete instances to a small square in the continuous instances. Then, we scale the continuous instance, put it inside a triangle, and fill the remaining part of the triangle to ensure certain boundary conditions that avoid creating spurious solutions (\ref{['lem:base-instance-lr-boundaries', 'lem:base-instance-bottom-slice', 'lem:base-instance-ppad-hard']}).
• Figure 2: A demonstration of the first steps for our definition of the coordinate converter. Here, we only focus on the region of the coloring that is at most bichromatic because trichromatic regions (e.g., the shadowed part on the left) are -hard to find. The dashed 1-manifolds represent the "color switches" in each instance. We will find the $\ell_{\infty}$ distance (only in terms of the last two coordinates) of each point to these color switches. The solid 1-manifolds represent the sets of points at distance $\varepsilon^2$ to color switches.
• Figure 3: A demonstration of the second steps for our definition of the coordinate converter. Here, we focus on the region of the coloring that is at most bichromatic. Suppose the colors of the blue / red / white regions are respectively indexed by 1 / 2 / 3. For each "color switch", we draw arrows from the 1-manifold that is $\varepsilon^2$ away from it and has the lower index to the 1-manifold that is $\varepsilon^2$ away from it and has the higher index. The region covered by the arrows are hot regions. Points on the sources of the arrows will have a converted coordinate of $0$, while those on the sinks will have a converted coordinate of $1$. We give converted coordinates for other points on the arrows according to their distances to the sources. Points not on any arrow will be marked as either warm or cold.

Theorems & Definitions (106)

• theorem 1: informal version of \ref{['thm:main']}
• theorem 2: Informal version of \ref{['thm:cake-main']}
• definition 3: End-of-Line
• definition 4
• definition 5
• definition 6
• definition 7: $k$ D- Sperner
• theorem 8: DBLP:journals/jc/HirschPV89DBLP:journals/jcss/Papadimitriou94DBLP:journals/tcs/ChenD09
• definition 9: $3$-out-of-$k\!+\!1$ Approximate Sperner
• remark 10