### Abstract

We consider two known lower bounds on randomized communication complexity: The smooth rectangle bound and the logarithm of the approximate nonnegative rank. Our main result is that they are the same up to a multiplicative constant and a small additive term. The logarithm of the nonnegative rank is known to be a nearly tight lower bound on the deterministic communication complexity. Our result indicates that proving the analogue for the randomized case, namely that the log approximate nonnegative rank is a nearly tight bound on randomized communication complexity, would imply the tightness of the information cost bound. Another corollary of our result is the existence of a boolean function with a quasipolynomial gap between its approximate rank and approximate nonnegative rank.

Original language | English (US) |
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Title of host publication | Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings |

Publisher | Springer Verlag |

Pages | 701-712 |

Number of pages | 12 |

Edition | PART 1 |

ISBN (Print) | 9783662439470 |

DOIs | |

State | Published - Jan 1 2014 |

Externally published | Yes |

Event | 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 - Copenhagen, Denmark Duration: Jul 8 2014 → Jul 11 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | PART 1 |

Volume | 8572 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 |
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Country | Denmark |

City | Copenhagen |

Period | 7/8/14 → 7/11/14 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings*(PART 1 ed., pp. 701-712). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8572 LNCS, No. PART 1). Springer Verlag. https://doi.org/10.1007/978-3-662-43948-7_58