TY - GEN

T1 - The communication complexity of local search

AU - Babichenko, Yakov

AU - Dobzinski, Shahar

AU - Nisan, Noam

N1 - Publisher Copyright:
© 2019 Association for Computing Machinery.

PY - 2019/6/23

Y1 - 2019/6/23

N2 - 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.

AB - 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.

KW - Communication Complexity

KW - Congestion Games

KW - Local Search

UR - http://www.scopus.com/inward/record.url?scp=85068781029&partnerID=8YFLogxK

U2 - 10.1145/3313276.3316354

DO - 10.1145/3313276.3316354

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AN - SCOPUS:85068781029

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 650

EP - 661

BT - STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

A2 - Charikar, Moses

A2 - Cohen, Edith

T2 - 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019

Y2 - 23 June 2019 through 26 June 2019

ER -