Matching algorithmic bounds for finding a Brouwer fixed point

Abstract

We prove a new discrete fixed point theorem for direction-preserving functions defined on integer points, based on a novel characterization of boundary conditions for the existence of fixed points. The theorem allows us to derive an improved algorithm for finding such a fixed point. We also develop a new lower bound proof technique. Together, they allow us to derive an asymptotic matching bound for the problem of finding a fixed point in a hypercube of any constantly bounded finite dimension. Exploring a linkage with the approximation version of the continuous fixed point problem, we obtain asymptotic matching bounds for the complexity of the approximate Brouwer fixed point problem in the continuous case for Lipschitz functions. It settles a fifteen-years-old open problem of Hirsch, Papadimitriou, and Vavasis by improving both the upper and lower bounds. Our characterization for the existence of a fixed point is also applicable to functions defined on nonconvex domains, which makes it a potentially useful tool for the design and analysis of algorithms for fixed points in general domains.