Constant Inapproximability for PPA

In the \varepsilon -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 )| \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.