Constant Inapproximability for PPA

TL;DR

It is shown that ε-Consensus-Halving is PPA-complete even when the parameter ε is a constant, and it is proved that this holds for any constant ε < 1/5.

Abstract

In the $\varepsilon$-Consensus-Halving problem, we are given $n$ probability measures $v_1, \dots, v_n$ on the interval $R = [0,1]$, and the goal is to partition $R$ into two parts $R^+$ and $R^-$ using at most $n$ cuts, so that $|v_i(R^+) - v_i(R^-)| \leq \varepsilon$ for all $i$. This fundamental fair division problem was the first natural problem shown to be complete for the class PPA, and all subsequent PPA-completeness results for other natural problems have been obtained by reducing from it. We show that $\varepsilon$-Consensus-Halving is PPA-complete even when the parameter $\varepsilon$ is a constant. In fact, we prove that this holds for any constant $\varepsilon < 1/5$. As a result, we obtain constant inapproximability results for all known natural PPA-complete problems, including Necklace-Splitting, the Discrete-Ham-Sandwich problem, two variants of the pizza sharing problem, and for finding fair independent sets in cycles and paths.

Figures (4)

• Figure 1: A slice of the $(k+1)$D-$\textup{StrongTucker}$ instance $I^*_{ST}$ for fixed coordinates of all dimensions other than the folding dimension $i$ ($x$-axis) and $k+1$ ($y$-axis). $\ell_{j}$ is the label $\lambda$ of the point with coordinate $j$ in dimension $i$ in the $k$D-$\textup{StrongTucker}$ instance $I_{WST}$. By $\mathbf{+1}$ and $\mathbf{-1}$ we denote the $(k+1)$-dimensional vector (label) with all entries $+1$ and $-1$ respectively.
• Figure 2: The density function of the valuation of an agent $\alpha_t^k$ implementing a NOT-gate.
• Figure 3: The density function of the valuation of an agent $\alpha_t^k$ implementing a NAND-gate.
• Figure 4: The valuation function of an agent implementing a modified NAND-gate (top), and the valuation function of an agent implementing a modified NOT-gate reading the output of the NAND-gate (bottom).

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 1: $\varepsilon$-$\textup{Consensus-Halving}$
• Definition 2: $\textup{$2$D-Tucker}$