On the complexity of 2D discrete fixed point problem

We study a computational complexity version of the 2D Sperner problem, which states that any three coloring of vertices of a triangulated triangle, satisfying some boundary conditions, will have a trichromatic triangle. In introducing a complexity class PPAD, Papadimitriou [C.H. Papadimitriou, On graph-theoretic lemmata and complexity classes, in: Proceedings of the 31st Annual Symposium on Foundations of Computer Science, 1990, 794–801] proved that its 3D analogue is PPAD-complete about fifteen years ago. The complexity of 2D-SPERNER itself has remained open since then.We settle this open problem with a PPAD-completeness proof. The result also allows us to derive the computational complexity characterization of a discrete version of the 2D Brouwer fixed point problem, improving a recent result of Daskalakis, Goldberg and Papadimitriou [C. Daskalakis, P.W. Goldberg, C.H. Papadimitriou, The complexity of computing a Nash equilibrium, in: Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), 2006]. Those hardness results for the simplest version of those problems provide very useful tools to the study of other important problems in the PPAD class.