Discrete Fixed Points: Models, Complexities, and Applications

We study three discrete fixed point concept (SPERNER, DPZP, BROUWER) under two different models: the polynomial-time function model and the oracle function model. We fully characterize the computational complexities of these three problems. The computational complexity unification of the above problems gives us more choices in the study of different applications. As an example, by a reduction from DPZP, we derive asymptotically equal lower and upper bound for TUCKER in the oracle model. The same reduction also allows us to derive a single proof for the PPAD-completeness of TUCKER in any constant dimension, which is significantly simpler than the recent proofs.