Constant Inapproximability for PPA
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In the ε-Consensus-Halving problem, we are given n probability measures v_1, …, 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^-)| ≤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.
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