Abstract
In the $\varepsilon$-Consensus-Halving problem, we are given $n$ probability measures $v1, \dots, vn$ 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.
We're not able to analyze this paper right now due to high demand.
Please check back later (sorry!).
Generate a summary of this paper on our Pro plan:
We ran into a problem analyzing this paper.