The communication complexity of local search

Yakov Babichenko, Shahar Dobzinski, Noam Nisan

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

7 Scopus citations

Abstract

We study a communication variant of local search. There is some fixed, commonly known graph G. Alice holds fA and Bob holds fB, both are functions that specify a value for each vertex. The goal is to find a local maximum of fA + fB with respect to G, i.e., a vertex v for which (fA + fB)(v) ≥ (fA + fB)(u) for each neighbor u of v. Our main result is that finding a local maximum requires polynomial (in the number of vertices) bits of communication. The result holds for the following families of graphs: three dimensional grids, hypercubes, odd graphs, and degree 4 graphs. Moreover, we prove an optimal communication bound of Ω(⌋N) for the hypercube, and for a constant dimension grid, where N is the number of vertices in the graph. We provide applications of our main result in two domains, exact potential games and combinatorial auctions. Each one of the results demonstrates an exponential separation between the non-deterministic communication complexity and the randomized communication complexity of a total search problem. First, we show that finding a pure Nash equilibrium in 2-player N-action exact potential games requires poly(N) communication. We also show that finding a pure Nash equilibrium in n-player 2-action exact potential games requires exp(n) communication. The second domain that we consider is combinatorial auctions, in which we prove that finding a local maximum in combinatorial auctions requires exponential (in the number of items) communication even when the valuations are submodular.

Original languageAmerican English
Title of host publicationSTOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing
EditorsMoses Charikar, Edith Cohen
PublisherAssociation for Computing Machinery
Pages650-661
Number of pages12
ISBN (Electronic)9781450367059
DOIs
StatePublished - 23 Jun 2019
Event51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 - Phoenix, United States
Duration: 23 Jun 201926 Jun 2019

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019
Country/TerritoryUnited States
CityPhoenix
Period23/06/1926/06/19

Bibliographical note

Publisher Copyright:
© 2019 Association for Computing Machinery.

Keywords

  • Communication Complexity
  • Congestion Games
  • Local Search

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