## Abstract

This paper studies lower bounds for fundamental optimization problems in the CONGEST model. We show that solving problems exactly in this model can be a hard task, by providing tildemega (n^{2}) lower bounds for cornerstone problems, such as minimum dominating set (MDS), Hamiltonian path, Steiner tree and max-cut. These are almost tight, since all of these problems can be solved optimally in O(n^{2}) rounds. Moreover, we show that even in bounded-degree graphs and even in simple graphs with maximum degree 5 and logarithmic diameter, it holds that various tasks, such as finding a maximum independent set (MaxIS) or a minimum vertex cover, are still difficult, requiring a near-tight number of tilde (n) rounds. Furthermore, we show that in some cases even approximations are difficult, by providing an tilde (n^{2}) lower bound for a (7/8+)-approximation for MaxIS, and a nearly-linear lower bound for an O(log n )-approximation for the k-MDS problem for any constant k geq 2, as well as for several variants of the Steiner tree problem. Our lower bounds are based on a rich variety of constructions that leverage novel observations, and reductions among problems that are specialized for the CONGEST model. However, for several additional approximation problems, as well as for exact computation of some central problems in P, such as maximum matching and max flow, we show that such constructions cannot be designed, by which we exemplify some limitations of this framework.

Original language | English |
---|---|

Title of host publication | PODC 2019 - Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing |

Publisher | Association for Computing Machinery |

Pages | 238-247 |

Number of pages | 10 |

ISBN (Electronic) | 9781450362177 |

DOIs | |

State | Published - 16 Jul 2019 |

Externally published | Yes |

Event | 38th ACM Symposium on Principles of Distributed Computing, PODC 2019 - Toronto, Canada Duration: 29 Jul 2019 → 2 Aug 2019 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Principles of Distributed Computing |
---|

### Conference

Conference | 38th ACM Symposium on Principles of Distributed Computing, PODC 2019 |
---|---|

Country/Territory | Canada |

City | Toronto |

Period | 29/07/19 → 2/08/19 |

### Bibliographical note

Publisher Copyright:© 2019 ACM.

## Keywords

- Approximation algorithms
- Communication complexity
- Congest
- Distributed computing
- Optimization problems

## ASJC Scopus subject areas

- Software
- Hardware and Architecture
- Computer Networks and Communications