Consensus Halving is PPA-Complete

TL;DR

This paper proves that the consensus halving problem is PPA-complete, establishing its computational complexity and linking it to other problems like necklace splitting, with implications for understanding the difficulty of fair division tasks.

Contribution

It is the first to show consensus halving is PPA-complete without explicit circuit definitions and connects its approximate version to necklace splitting, indicating similar complexity.

Findings

Consensus halving is PPA-complete.

Approximate consensus halving is polynomial-time equivalent to necklace splitting.

PPAD-hardness is established for necklace splitting.

Abstract

We show that the computational problem CONSENSUS-HALVING is PPA-complete, the first PPA-completeness result for a problem whose definition does not involve an explicit circuit. We also show that an approximate version of this problem is polynomial-time equivalent to NECKLACE SPLITTING, which establishes PPAD-hardness for NECKLACE SPLITTING, and suggests that it is also PPA-complete.