On the blackbox complexity of Sperner's Lemma
Abstract
We present several results on the complexity of various forms of Sperner's Lemma in the blackbox model of computing. We give a deterministic algorithm for Sperner problems over pseudomanifolds 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 $O(\sqrt{n})$ deterministic query algorithm for the blackbox version of the problem {\bf 2DSPERNER}, a well studied member of Papadimitriou's complexity class PPAD. This upper bound matches the $\Omega(\sqrt{n})$ 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 $\Omega(\sqrt[4]{n})$ lower bound for its probabilistic, and an $\Omega(\sqrt[8]{n})$ lower bound for its quantum query complexity, showing that all these measures are polynomially related.
 Publication:

arXiv eprints
 Pub Date:
 May 2005
 DOI:
 10.48550/arXiv.quantph/0505185
 arXiv:
 arXiv:quantph/0505185
 Bibcode:
 2005quant.ph..5185F
 Keywords:

 Quantum Physics
 EPrint:
 16 pages with 1 figure