On algorithms for discrete and approximate brouwer fixed points

  • Authors:

  • Xi Chen;Xiaotie Deng

  • Affiliations:

  • Tsinghua University, Beijing, P.R. China;City University of Hong Kong, Hong Kong SAR, P.R. China

  • Venue:
  • Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
  • Year:
  • 2005

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Abstract

We study the algorithmic complexity of the discrete fixed point problem and develop an asymptotic matching bound for a cube in any constantly bounded finite dimension. To obtain our upper bound, we derive a new fixed point theorem, based on a novel characterization of boundary conditions for the existence of fixed points.In addition, exploring a linkage with the approximation problem of the continuous fixed point problem, we obtain asymptotic matching bounds for complexity of the approximate Brouwer fixed point problem in the continuous case for Lipschitz functions that close a previous exponential gap. It settles a fifteen years old open problem of Hirsch, Papadimitriou and Vavasis by improving both the upper and lower bounds.Our new characterization for existence of a fixed point is also applicable to functions defined on non-convex domain and makes it a potentially useful tool for design and analysis of algorithms for fixed points in general domain.