Direction preserving zero point computing and applications

Deng, Xiaotie, Qi, Qi and Zhang, Jie (2009) Direction preserving zero point computing and applications. pp. 410-421 . (doi:10.1007/978-3-642-10841-9_37).

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.

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