The Complexity of Splitting Necklaces and Bisecting Ham Sandwiches
This work addresses fundamental problems in computational complexity theory, settling the status of PPA as a class for natural problems without circuits, which is incremental but important for theoretical computer science.
The paper resolves the computational complexity of NECKLACE-SPLITTING and DISCRETE HAM SANDWICH problems, showing they are PPA-complete, specifically for the two-thief case in NECKLACE-SPLITTING, and strengthens a prior result on CONSENSUS-HALVING to PPA-completeness for approximate versions.
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 Möbius 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.