Discrete Fixed Points: Models, Complexities, and Applications

Authors:
Xiaotie Deng;Qi Qi;Amin Saberi;Jie Zhang
Affiliations:
Department of Computer Science, University of Liverpool, Liverpool L1 8ND, United Kingdom;Department of Management Science and Engineering, Stanford University, Stanford, California 94305;Department of Management Science and Engineering, Stanford University, Stanford, California 94305;Department of Computer Science, City University of Hong Kong, Kowloon, Hong Kong SAR, P. R. China
Venue:
Mathematics of Operations Research
Year:
2011

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Abstract

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.