Direction preserving zero point computing and applications

Abstract

We study the connection between the direction preserving zero point and the discrete Brouwer fixed point in terms of their computational complexity. As a result, we derive a PPAD-completeness proof for finding a direction preserving zero point, and a matching oracle complexity bound for computing a discrete Brouwer's fixed point. Building upon the connection between the two types of combinatorial structures for Brouwer's continuous fixed point theorem, we derive an immediate proof that TUCKER is PPAD-complete for all constant dimensions, extending the results of Pálvölgyi for 2D case [20] and Papadimitriou for 3D case [21]. In addition, we obtain a matching algorithmic bound for TUCKER in the oracle model. © 2009 Springer-Verlag Berlin Heidelberg.