## Abstract

We consider the question of whether adaptivity can improve the complexity of property testing algorithms in the dense graphs model. It is known that there can be at most a quadratic gap between adaptive and non-adaptive testers in this model, but it was not known whether any gap indeed exists. In this work we reveal such a gap. Specifically, we focus on the well studied property of bipartiteness. Bogdanov and Trevisan (IEEE Symposium on Computational Complexity, 2004) proved a lower bound of Ω(1/ε^{2}) on the query complexity of non-adaptive testing algorithms for bipartiteness. This lower bound holds for graphs with maximum degree O(εn). Our main result is an adaptive testing algorithm for bipartiteness of graphs with maximum degree O(en) whose query complexity is^{1} Õ(1/ε^{3/2}). A slightly modified version of our algorithm can be used to test the combined property of being bipartite and having maximum degree O(εn). Thus we demonstrate that adaptive testers are stronger than non-adaptive testers in the dense graphs model. We note that the upper bound we obtain is tight up-to polylogarithmic factors, in view of the Ω(1/ε^{3/2}) lower bound of Bogdanov and Trevisan for adaptive testers. In addition we show that Õ(1/ε^{3/2}) queries also suffice when (almost) all vertices have degree Ω(√ε · n). In this case adaptivity is not necessary.

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

Subtitle of host publication | Algorithms and Techniques - 10th International Workshop, APPROX 2007 and 11th International Workshop, RANDOM 2007, Proceedings |

Pages | 525-539 |

Number of pages | 15 |

DOIs | |

State | Published - 2007 |

Externally published | Yes |

Event | 10th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2007 and 11th International Workshop on Randomization and Computation, RANDOM 2007 - Princeton, NJ, United States Duration: 20 Aug 2007 → 22 Aug 2007 |

### 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 | 4627 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 10th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2007 and 11th International Workshop on Randomization and Computation, RANDOM 2007 |
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Country/Territory | United States |

City | Princeton, NJ |

Period | 20/08/07 → 22/08/07 |