Hardness of Approximate Sperner and Applications to Envy-Free Cake Cutting
<|reference_start|>Hardness of Approximate Sperner and Applications to Envy-Free Cake Cutting: Given a so called ''Sperner coloring'' of a triangulation of the $D$-dimensional simplex, Sperner's lemma guarantees the existence of a rainbow simplex, i.e. a simplex colored by all $D+1$ colors. However, finding a rainbow simplex was the first problem to be proven $\mathsf{PPAD}$-complete in Papadimitriou's classical paper introducing the class $\mathsf{PPAD}$ (1994). In this paper, we prove that the problem does not become easier if we relax ''all $D+1$ colors'' to allow some fraction of missing colors: in fact, for any constant $D$, finding even a simplex with just three colors remains $\mathsf{PPAD}$-complete!   Our result has an interesting application for the envy-free cake cutting from fair division. It is known that if agents value pieces of cake using general continuous functions satisfying a simple boundary condition (''a non-empty piece is better than an empty piece of cake''), there exists an envy-free allocation with connected pieces. We show that for any constant number of agents it is $\mathsf{PPAD}$-complete to find an allocation -- even using any constant number of possibly disconnected pieces -- that makes just three agents envy-free. Our results extend to super-constant dimension, number of agents, and number of pieces, as long as they are asymptotically bounded by any $\log^{1-\Omega(1)}(\epsilon)$, where $\epsilon$ is the precision parameter (side length for Sperner and approximate envy-free for cake cutting).<|reference_end|>
@article{gao2024hardness,
    title={Hardness of Approximate Sperner and Applications to Envy-Free Cake
  Cutting},
    author={Ruiquan Gao, Mohammad Roghani, Aviad Rubinstein, Amin Saberi},
    journal={arXiv preprint arXiv:2409.15713},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.15713},
    primaryClass={cs.CC cs.GT}
}