Constant Inapproximability for PPA

In the ϵ-Consensus-Halving problem, we are given n probability measures v1, ..., vn 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 |vi(R+)-vi(R-)| ≤ ϵ 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 ϵ-Consensus-Halving is PPA-complete even when the parameter ϵ is a constant. In fact, we prove that this holds for any constant ϵ < 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.