Constant Inapproximability for PPA

Argyrios Deligkas; John Fearnley; Alexandros Hollender; Themistoklis Melissourgos

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arXiv · 2201.10011

Constant Inapproximability for PPA

Argyrios Deligkas·John Fearnley·Alexandros Hollender·Themistoklis Melissourgos

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Abstract

In the $\varepsilon$-Consensus-Halving problem, we are given $n$ probability measures $v_1, \dots, v_n$ 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 $|v_i(R^+) - v_i(R^-)| \leq \varepsilon$ 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.

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BibTeX RIS

Argyrios Deligkas, John Fearnley, Alexandros Hollender, Themistoklis Melissourgos. 2022-01-24. Constant Inapproximability for PPA. https://arxiv.org/abs/2201.10011

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