On the Black-Box 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 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) 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. 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. Classification: computational and structural complexity, quantum computation and in...

• Type: Conference
• Year: 2005
• Where: FCT
• Authors: Katalin Friedl, Gábor Ivanyos, Miklos Santha, Yves F. Verhoeven
• Keywords: Deterministic Lower Bound, FCT 2005, Lower Bound, Query Complexity