Constant Inapproximability for PPA

TL;DR

This paper proves that the $ ext{Consensus-Halving}$ problem remains PPA-complete even when the approximation parameter is a fixed constant, leading to inapproximability results for several other natural PPA-complete problems.

Contribution

It establishes the constant inapproximability of $ ext{Consensus-Halving}$ for any $ ext{ε} < 1/5$, extending PPA-completeness to fixed approximation parameters and impacting related problems.

Findings

$ ext{Consensus-Halving}$ is PPA-complete for any constant $ ext{ε} < 1/5$

Constant inapproximability results for multiple PPA-complete problems

Implications for fair division and geometric problems

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.