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  4. An Optimal Space Lower Bound for Approximating MAX-CUT
 
conference paper

An Optimal Space Lower Bound for Approximating MAX-CUT

Kapralov, Michael  
•
Krachun, Dmitry
January 1, 2019
Proceedings Of The 51St Annual Acm Sigact Symposium On Theory Of Computing (Stoc '19)
51st Annual ACM SIGACT Symposium on Theory of Computing (STOC)

We consider the problem of estimating the value of MAX-CUT in a graph in the streaming model of computation. At one extreme, there is a trivial 2-approximation for this problem that uses only O(log n) space, namely, count the number of edges and output half of this value as the estimate for the size of the MAX-CUT. On the other extreme, for any fixed epsilon > 0, if one allows (O) over tilde (n) space, a (1 + epsilon)-approximate solution to the MAX-CUT value can be obtained by storing an (O) over tilde (n)-size sparsifier that essentially preserves MAX-CUT value. Our main result is that any (randomized) single pass streaming algorithm that breaks the 2-approximation barrier requires Omega(n)-space, thus resolving the space complexity of any non-trivial approximations of the MAX-CUT value to within polylogarithmic factors in the single pass streaming model. We achieve the result by presenting a tight analysis of the Implicit Hidden Partition Problem introduced by Kapralov et al.[SODA'17] for an arbitrarily large number of players. In this problem a number of players receive random matchings of Omega(n) size together with random bits on the edges, and their task is to determine whether the bits correspond to parities of some hidden bipartition, or are just uniformly random. Unlike all previous Fourier analytic communication lower bounds, our analysis does not directly use bounds on the l(2) norm of Fourier coefficients of a typical message at any given weight level that follow from hypercontractivity. Instead, we use the fact that graphs received by players are sparse (matchings) to obtain strong upper bounds on the l(1) norm of the Fourier coefficients of the messages of individual players using their special structure, and then argue, using the convolution theorem, that similar strong bounds on the l(1) norm are essentially preserved (up to an exponential loss in the number of players) once messages of different players are combined. We feel that our main technique is likely of independent interest.

  • Details
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Type
conference paper
DOI
10.1145/3313276.3316364
Web of Science ID

WOS:000523199100024

Author(s)
Kapralov, Michael  
Krachun, Dmitry
Date Issued

2019-01-01

Publisher

ASSOC COMPUTING MACHINERY

Publisher place

New York

Published in
Proceedings Of The 51St Annual Acm Sigact Symposium On Theory Of Computing (Stoc '19)
ISBN of the book

978-1-4503-6705-9

Series title/Series vol.

Annual ACM Symposium on Theory of Computing

Start page

277

End page

288

Subjects

Computer Science, Theory & Methods

•

Computer Science

•

streaming lower bounds

•

fourier analysis

•

max-cut

Editorial or Peer reviewed

REVIEWED

Written at

EPFL

EPFL units
THL4  
Event nameEvent placeEvent date
51st Annual ACM SIGACT Symposium on Theory of Computing (STOC)

Phoenix, AZ

Jun 23-26, 2019

Available on Infoscience
April 22, 2020
Use this identifier to reference this record
https://infoscience.epfl.ch/handle/20.500.14299/168310
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