On the Communication Complexity of Approximate Fixed Points

Tim Roughgarden, Omri Weinstein

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

14 Scopus citations

Abstract

We study the two-party communication complexity of finding an approximate Brouwer fixed point of a composition of two Lipschitz functions g ○ f: [0,1]n → [0,1]n, where Alice holds f and Bob holds g. We prove an exponential (in n) lower bound on the deterministic communication complexity of this problem. Our technical approach is to adapt the Raz-McKenzie simulation theorem (FOCS 1999) into geometric settings, thereby 'smoothly lifting' the deterministic query lower bound for finding an approximate fixed point (Hirsch, Papadimitriou and Vavasis, Complexity 1989) from the oracle model to the two-party model. Our results also suggest an approach to the well-known open problem of proving strong lower bounds on the communication complexity of computing approximate Nash equilibria. Specifically, we show that a slightly 'smoother' version of our fixed-point computation lower bound (by an absolute constant factor) would imply that: The deterministic two-party communication complexity of finding an ϵ = Ω(1/log2 N)-approximate Nash equilibrium in an N × N bimatrix game (where each player knows only his own payoff matrix) is at least Nγ for some constant γ > 0. (In contrast, the nondeterministic communication complexity of this problem is only O(log6 N)). The deterministic (Number-In-Hand) multiparty communication complexity of finding an ϵ = Ω(1)-Nash equilibrium in a k-player constant-action game is at least 2Ω(k/log k) (while the nondeterministic communication complexity is only O(k)).

Original languageAmerican English
Title of host publicationProceedings - 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016
PublisherIEEE Computer Society
Pages229-238
Number of pages10
ISBN (Electronic)9781509039333
DOIs
StatePublished - 14 Dec 2016
Externally publishedYes
Event57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016 - New Brunswick, United States
Duration: 9 Oct 201611 Oct 2016

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2016-December
ISSN (Print)0272-5428

Conference

Conference57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016
Country/TerritoryUnited States
CityNew Brunswick
Period9/10/1611/10/16

Bibliographical note

Publisher Copyright:
© 2016 IEEE.

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