Constant Inapproximability for PPA

<jats:p>Abstract.</jats:p> <jats:p>In the [Formula: see text]-Consensus-Halving problem, we are given [Formula: see text] probability measures [Formula: see text] on the interval [Formula: see text], and the goal is to partition [Formula: see text] into two parts [Formula: see text] and [Formula: see text] using at most [Formula: see text] cuts, so that [Formula: see text] for all [Formula: see text]. 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 [Formula: see text]-Consensus-Halving is PPA -complete even when the parameter [Formula: see text] is a constant. In fact, we prove that this holds for any constant [Formula: see text]. 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.</jats:p>