On the Black-Box Complexity of Sperner's Lemma

摘要

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 O(n~(1/2)) 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~(1/2)) 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(n~(1/4)) lower bound for its probabilistic, and an Ω(n~(1/8)) lower bound for its quantum query complexity, showing that all these measures are polynomially related.