On the black-box complexity of Sperner's Lemma [article]

Katalin Friedl, Gabor Ivanyos, Miklos Santha, Yves F. Verhoeven
2005 arXiv   pre-print
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
more » ... 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(√(n)) lower bound for its probabilistic, and an Ω(√(n)) lower bound for its quantum query complexity, showing that all these measures are polynomially related.
arXiv:quant-ph/0505185v1 fatcat:h5tf3kizjvaf3nxp5jkmb4uby4