### Abstract

This paper provides the first general technique for proving information lower bounds on two-party unbounded-rounds communication problems. We show that the discrepancy lower bound, which applies to randomized communication complexity, also applies to information complexity. More precisely, if the discrepancy of a two-party function f with respect to a distribution μ is Disc _{μ} f, then any two party randomized protocol computing f must reveal at least Ω(log(1/Disc _{μ} f)) bits of information to the participants. As a corollary, we obtain that any two-party protocol for computing a random function on {0,1} ^{n} × {0,1} ^{n} must reveal Ω(n) bits of information to the participants. In addition, we prove that the discrepancy of the Greater-Than function is Ω(1/√n), which provides an alternative proof to the recent proof of Viola [Vio11] of the Ω(log n) lower bound on the communication complexity of this well-studied function and, combined with our main result, proves the tight Ω(log n) lower bound on its information complexity. The proof of our main result develops a new simulation procedure that may be of an independent interest. In a very recent breakthrough work of Kerenidis et al. [KLL ^{+}12], this simulation procedure was a building block towards a proof that almost all known lower bound techniques for communication complexity (and not just discrepancy) apply to information complexity.

Original language | English (US) |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Proceedings |

Pages | 459-470 |

Number of pages | 12 |

DOIs | |

State | Published - Aug 28 2012 |

Event | 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2012 and the 16th International Workshop on Randomization and Computation, RANDOM 2012 - Cambridge, MA, United States Duration: Aug 15 2012 → Aug 17 2012 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7408 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2012 and the 16th International Workshop on Randomization and Computation, RANDOM 2012 |
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Country | United States |

City | Cambridge, MA |

Period | 8/15/12 → 8/17/12 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Proceedings*(pp. 459-470). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 7408 LNCS). https://doi.org/10.1007/978-3-642-32512-0_39