On the black-box complexity of Sperner's Lemma

Abstract

We present several results on the complexity of various forms of Sperner's Lemma in the black-box model of computing. We give a deterministic algorithm for Sperner problems over pseudo-manifolds of arbitrary dimension. The query complexity of our algorithm is linear in the separation number of the skeleton graph of the manifold and the size of its boundary. As a corollary we get an 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 Ω(√n) the deterministic lower bound of Crescenzi and Silvestri. The tightness of this bound was not known before. In another 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. © Springer Science+Business Media, LLC 2008.