The Complexity of Splitting Necklaces and Bisecting Ham Sandwiches

TL;DR

This paper proves that the problems NECKLACE-SPLITTING and DISCRETE HAM SANDWICH are PPA-complete, establishing their computational difficulty and connecting them to the complexity class PPA through novel reductions and embeddings.

Contribution

It demonstrates PPA-completeness for NECKLACE-SPLITTING and DISCRETE HAM SANDWICH, including an approximate consensus-halving problem, advancing understanding of their computational complexity.

Findings

NECKLACE-SPLITTING is PPA-complete for the case with two thieves.

DISCRETE HAM SANDWICH is also shown to be PPA-complete.

The results include a smooth embedding of a high-dimensional M"obius strip in consensus-halving.

Abstract

We resolve the computational complexity of two problems known as NECKLACE-SPLITTING and DISCRETE HAM SANDWICH, showing that they are PPA-complete. For NECKLACE SPLITTING, this result is specific to the important special case in which two thieves share the necklace. We do this via a PPA-completeness result for an approximate version of the CONSENSUS-HALVING problem, strengthening our recent result that the problem is PPA-complete for inverse-exponential precision. At the heart of our construction is a smooth embedding of the high-dimensional M\"obius strip in the CONSENSUS-HALVING problem. These results settle the status of PPA as a class that captures the complexity of "natural" problems whose definitions do not incorporate a circuit.