## Abstract

Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least εn^{2} edges have to be deleted from it to make it H-free. We show that in this case G contains at least f(ε,H)n^{h} copies of H. This is proved by establishing a directed version of Szemerédi's regularity lemma, and implies that for every H there is a one-sided error property tester whose query complexity is bounded by a function of ε only for testing the property P_{H} of being H-free. As is common with applications of the undirected regularity lemma, here too the function 1/f(ε,H) is an extremely fast growing function in ε. We therefore further prove a precise characterization of all the digraphs H, for which f(ε,H) has a polynomial dependency on ε. This implies a characterization of all the digraphs H, for which the property of being H-free has a one-sided error property tester whose query complexity is polynomial in 1/ε. We further show that the same characterization also applies to two-sided error property testers as well. A special case of this result settles an open problem raised by the first author in (Alon, Proceedings of the 42nd IEEE FOCS, IEEE, New York, 2001, pp. 434-441). Interestingly, it turns out that if P_{H} has a polynomial query complexity, then there is a two-sided ε-tester for P_{H} that samples only O(1/ε) vertices, whereas any one-sided tester for P_{H} makes at least (1/ε)^{d/2} queries, where d is the average degree of H. We also show that the complexity of deciding if for a given directed graph H, P _{H} has a polynomial query complexity, is NP-complete, marking an interesting distinction from the case of undirected graphs. For some special cases of directed graphs H, we describe very efficient one-sided error property-testers for testing P_{H}. As a consequence we conclude that when H is an undirected bipartite graph, we can give a one-sided error property tester with query complexity O((1/ε)^{h/2}), improving the previously known upper bound of O((1/ε)^{h2}). The proofs combine combinatorial, graph theoretic and probabilistic arguments with results from additive number theory.

Original language | English (US) |
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Pages (from-to) | 354-382 |

Number of pages | 29 |

Journal | Journal of Computer and System Sciences |

Volume | 69 |

Issue number | 3 SPEC. ISS. |

DOIs | |

State | Published - Nov 2004 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Networks and Communications
- Computational Theory and Mathematics
- Applied Mathematics

## Keywords

- Core
- Directed graphs
- Property testing
- Regularity