The Complexity of the Path-following Solutions of Two-dimensional Sperner/Brouwer Functions

TL;DR

This paper proves that computing the specific solutions obtained by path-following algorithms for certain PPAD-complete problems, like Sperner and Brouwer functions, is PSPACE-complete, highlighting the complexity of these solutions.

Contribution

It establishes PSPACE-completeness for the path-following solutions of two-dimensional Sperner and Brouwer problems, supporting a broader conjecture about their computational difficulty.

Findings

Path-following solutions are PSPACE-complete to compute.

Supports the conjecture that all such solutions are PSPACE-complete.

Provides new complexity results for Sperner and Brouwer problems.

Abstract

There are a number of results saying that for certain "path-following" algorithms that solve PPAD-complete problems, the solution obtained by the algorithm is PSPACE-complete to compute. We conjecture that these results are special cases of a much more general principle, that all such algorithms compute PSPACE-complete solutions. Such a general result might shed new light on the complexity class PPAD. In this paper we present a new PSPACE-completeness result for an interesting challenge instance for this conjecture. Chen and Deng~\cite{CD} showed that it is PPAD-complete to find a trichromatic triangle in a concisely-represented Sperner triangulation. The proof of Sperner's lemma --- that such a solution always exists --- identifies one solution in particular, that is found via a natural "path-following" approach. Here we show that it is PSPACE-complete to compute this specific solution, together with a similar result for the computation of the path-following solution of two-dimensional discrete Brouwer functions.