## Abstract

We study a communication variant of local search. There is some fixed, commonly known graph G. Alice holds f_{A} and Bob holds f_{B}, both are functions that specify a value for each vertex. The goal is to find a local maximum of f_{A} + f_{B} with respect to G, i.e., a vertex v for which (f_{A} + f_{B})(v) ≥ (f_{A} + f_{B})(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 language | English |
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Title of host publication | STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Moses Charikar, Edith Cohen |

Publisher | Association for Computing Machinery |

Pages | 650-661 |

Number of pages | 12 |

ISBN (Electronic) | 9781450367059 |

DOIs | |

State | Published - 23 Jun 2019 |

Event | 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 - Phoenix, United States Duration: 23 Jun 2019 → 26 Jun 2019 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 |
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Country/Territory | United States |

City | Phoenix |

Period | 23/06/19 → 26/06/19 |

### Bibliographical note

Publisher Copyright:© 2019 Association for Computing Machinery.

## Keywords

- Communication Complexity
- Congestion Games
- Local Search