Exponential lower bounds for finding Brouwer fixed points
Journal of Complexity
On the complexity of the parity argument and other inefficient proofs of existence
Journal of Computer and System Sciences - Special issue: 31st IEEE conference on foundations of computer science, Oct. 22–24, 1990
A sperner lemma complete for PPA
Information Processing Letters
On algorithms for discrete and approximate brouwer fixed points
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
The complexity of computing a Nash equilibrium
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Computing Nash Equilibria: Approximation and Smoothed Complexity
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
On graph-theoretic lemmata and complexity classes
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
Locally 2-dimensional sperner problems complete for the polynomial parity argument classes
CIAC'06 Proceedings of the 6th Italian conference on Algorithms and Complexity
On the black-box complexity of sperner's lemma
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
Discrete Fixed Points: Models, Complexities, and Applications
Mathematics of Operations Research
Algorithmic Solutions for Envy-Free Cake Cutting
Operations Research
Algorithmic Solutions for Envy-Free Cake Cutting
Operations Research
The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions
ACM Transactions on Economics and Computation - Special Issue on Algorithmic Game Theory
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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.