The Complexity of Splitting Necklaces and Bisecting Ham Sandwiches

Understand

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.