### Abstract

Recent works of Alon-Shapira (A characterization of the (natural) graph properties testable with one-sided error. Available at: http://www.math.tau.ac.il/~nogaa/PDFS/heredit2.pdf) and Rödl-Schacht (Generalizations of the removal lemma, Available at: http://www.informatik.huberlin.de/~schacht/pub/preprints/gen_removal.pdf) have demonstrated that every hereditary property of undirected graphs or hypergraphs is testable with one-sided error; informally, this means that if a graph or hypergraph satisfies that property "locally" with sufficiently high probability, then it can be perturbed (or "repaired") into a graph or hypergraph which satisfies that property "globally."In this paper we make some refinements to these results, some of which may be surprising. In the positive direction, we strengthen the results to cover hereditary properties of multiple directed polychromatic graphs and hypergraphs. In the case of undirected graphs, we extend the result to continuous graphs on probability spaces and show that the repair algorithm is "local" in the sense that it only depends on a bounded amount of data;in particular,the graph can be repaired in a time linear in the number of edges. We also show that local repairability also holds for monotone or partite hypergraph properties (this latter result is also implicitly in (Ishigamis work Removal lemma for infinitelymany forbidden hypergraphs and property testing. Available at: arXiv.org: math.CO/0612669)). In the negative direction, we show that local repairability breaks down for directed graphs or for undirected 3-uniform hypergraphs.The reason for this contrast in behavior stems from(the limitations of)Ramsey theory.

Original language | English (US) |
---|---|

Pages (from-to) | 373-463 |

Number of pages | 91 |

Journal | Random Structures and Algorithms |

Volume | 36 |

Issue number | 4 |

DOIs | |

State | Published - Jul 2010 |

### Fingerprint

### Keywords

- de Finetti's theorem
- Exchangeable processes
- Hypergraph regularity lemma
- Local testability

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics

### Cite this

*Random Structures and Algorithms*,

*36*(4), 373-463. https://doi.org/10.1002/rsa.20300

**Testability and repair of hereditary hypergraph properties.** / Austin, Tim; Tao, Terence.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 36, no. 4, pp. 373-463. https://doi.org/10.1002/rsa.20300

}

TY - JOUR

T1 - Testability and repair of hereditary hypergraph properties

AU - Austin, Tim

AU - Tao, Terence

PY - 2010/7

Y1 - 2010/7

N2 - Recent works of Alon-Shapira (A characterization of the (natural) graph properties testable with one-sided error. Available at: http://www.math.tau.ac.il/~nogaa/PDFS/heredit2.pdf) and Rödl-Schacht (Generalizations of the removal lemma, Available at: http://www.informatik.huberlin.de/~schacht/pub/preprints/gen_removal.pdf) have demonstrated that every hereditary property of undirected graphs or hypergraphs is testable with one-sided error; informally, this means that if a graph or hypergraph satisfies that property "locally" with sufficiently high probability, then it can be perturbed (or "repaired") into a graph or hypergraph which satisfies that property "globally."In this paper we make some refinements to these results, some of which may be surprising. In the positive direction, we strengthen the results to cover hereditary properties of multiple directed polychromatic graphs and hypergraphs. In the case of undirected graphs, we extend the result to continuous graphs on probability spaces and show that the repair algorithm is "local" in the sense that it only depends on a bounded amount of data;in particular,the graph can be repaired in a time linear in the number of edges. We also show that local repairability also holds for monotone or partite hypergraph properties (this latter result is also implicitly in (Ishigamis work Removal lemma for infinitelymany forbidden hypergraphs and property testing. Available at: arXiv.org: math.CO/0612669)). In the negative direction, we show that local repairability breaks down for directed graphs or for undirected 3-uniform hypergraphs.The reason for this contrast in behavior stems from(the limitations of)Ramsey theory.

AB - Recent works of Alon-Shapira (A characterization of the (natural) graph properties testable with one-sided error. Available at: http://www.math.tau.ac.il/~nogaa/PDFS/heredit2.pdf) and Rödl-Schacht (Generalizations of the removal lemma, Available at: http://www.informatik.huberlin.de/~schacht/pub/preprints/gen_removal.pdf) have demonstrated that every hereditary property of undirected graphs or hypergraphs is testable with one-sided error; informally, this means that if a graph or hypergraph satisfies that property "locally" with sufficiently high probability, then it can be perturbed (or "repaired") into a graph or hypergraph which satisfies that property "globally."In this paper we make some refinements to these results, some of which may be surprising. In the positive direction, we strengthen the results to cover hereditary properties of multiple directed polychromatic graphs and hypergraphs. In the case of undirected graphs, we extend the result to continuous graphs on probability spaces and show that the repair algorithm is "local" in the sense that it only depends on a bounded amount of data;in particular,the graph can be repaired in a time linear in the number of edges. We also show that local repairability also holds for monotone or partite hypergraph properties (this latter result is also implicitly in (Ishigamis work Removal lemma for infinitelymany forbidden hypergraphs and property testing. Available at: arXiv.org: math.CO/0612669)). In the negative direction, we show that local repairability breaks down for directed graphs or for undirected 3-uniform hypergraphs.The reason for this contrast in behavior stems from(the limitations of)Ramsey theory.

KW - de Finetti's theorem

KW - Exchangeable processes

KW - Hypergraph regularity lemma

KW - Local testability

UR - http://www.scopus.com/inward/record.url?scp=77953248503&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953248503&partnerID=8YFLogxK

U2 - 10.1002/rsa.20300

DO - 10.1002/rsa.20300

M3 - Article

VL - 36

SP - 373

EP - 463

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 4

ER -