Hardness Results for Consensus-Halving

TL;DR

This paper investigates the computational complexity of the consensus-halving problem, establishing its PPAD-hardness and NP-hardness for various cases, and connecting it to the Necklace Splitting problem.

Contribution

It proves PPAD-hardness of approximate consensus-halving with n cuts and NP-hardness for solutions with fewer cuts, advancing understanding of its computational difficulty.

Findings

Consensus-halving with n cuts is in PPA and PPAD-hard.

Deciding existence of solutions with n-1 cuts is NP-hard.

Approximate Necklace Splitting with two portions is PPAD-hard.

Abstract

We study the consensus-halving problem of dividing an object into two portions, such that each of $n$ agents has equal valuation for the two portions. The $\epsilon$-approximate consensus-halving problem allows each agent to have an $\epsilon$ discrepancy on the values of the portions. We prove that computing $\epsilon$-approximate consensus-halving solution using $n$ cuts is in PPA, and is PPAD-hard, where $\epsilon$ is some positive constant; the problem remains PPAD-hard when we allow a constant number of additional cuts. It is NP-hard to decide whether a solution with $n-1$ cuts exists for the problem. As a corollary of our results, we obtain that the approximate computational version of the Continuous Necklace Splitting Problem is PPAD-hard when the number of portions $t$ is two.