On the complexity of Sperner’s Lemma

We present several results on the complexity of various forms of Sperner’s Lemma. In the black-box model of computing, we exhibit a deterministic algorithm for Sperner problems over pseudo-manifolds of arbitrary dimension. The query complexity of our algorithm is essentially linear in the separation number of the skeleton graph of the manifold and the size of its boundary. As a corollary we get an O( √ n) deterministic query algorithm for the black-box version of the problem 2D-SPERNER, a well studied member of Papadimitriou’s complexity class PPAD. This upper bound matches the Ω( √ n) deterministic lower bound of Crescenzi and Silvestri. In another black-box result we prove for the same problem an Ω( 4 √ n) lower bound for its probabilistic, and an Ω( 8 √ n) lower bound for its quantum query complexity, showing that all these measures are polynomially related. Finally we explicit Sperner problems on a 2-dimensional pseudo-manifold and prove that they are complete respectively for the classes PPAD, PPADS and PPA. This is the first time that a 2-dimensional Sperner problem is proved to be complete for any of the polynomial parity argument classes.