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

The Complexity of Splitting Necklaces and Bisecting Ham Sandwiches