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.

翻译：$varepsilon$- consensus-Halving road $ $ varepsolon $ 1,\ dots, v_n$, v_n$, v_n$, v_n$ $ = [0, 1]$, 目标是将美元分成两部分, 美元和美元- 美元以最多削减美元计算, 这样, 美元- v_ i( R__) - v_i( R__)\\leq\ leq\ varepsilon- $ 所有美元, 这个根本的公平分化问题是该类PPA的第一个自然问题已经证明是完整的, 而此后所有关于其他自然问题PPA的完全性结果都是通过减少而取得的。 我们显示, $\varepslonon- Cons- $- salversional $ (R) - varepeprelon) - von to all $ (Rive $) < 1/5$.