## A characterization of the (natural) graph properties testable with one-sided error (2005)

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Venue: | Proc. of FOCS 2005 |

Citations: | 90 - 18 self |

### BibTeX

@INPROCEEDINGS{Alon05acharacterization,

author = {Noga Alon and Asaf Shapira},

title = {A characterization of the (natural) graph properties testable with one-sided error},

booktitle = {Proc. of FOCS 2005},

year = {2005},

pages = {429--438}

}

### Years of Citing Articles

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### Abstract

The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of property-testing. Our main result in this paper is a solution of an important special case of this general problem; Call a property tester oblivious if its decisions are independent of the size of the input graph. We show that a graph property P has an oblivious one-sided error tester, if and only if P is (almost) hereditary. We stress that any ”natural ” property that can be tested (either with one-sided or with two-sided error) can be tested by an oblivious tester. In particular, all the testers studied thus far in the literature were oblivious. Our main result can thus be considered as a precise characterization of the ”natural” graph properties, which are testable with one-sided error. One of the main technical contributions of this paper is in showing that any hereditary graph property can be tested with one-sided error. This general result contains as a special case all the previous results about testing graph properties with one-sided error. These include the results of [20] and [5] about testing k-colorability, the characterization of [21] of the graph-partitioning problems that are testable with one-sided error, the induced vertex colorability properties of [3], the induced edge colorability properties of [14], a transformation from two-sided to one-sided error testing [21], as well as a recent result about testing monotone graph properties [10]. More importantly, as a special case of our main result, we infer that some of the most well studied graph properties, both in graph theory and computer science, are testable with one-sided error. Some of these properties are the well known graph properties of being Perfect, Chordal, Interval, Comparability, Permutation and more. None of these properties was previously known to be testable. 1

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Citation Context ...es in graph-theory as well as in theoretical and applied computer-science. These properties also arise naturally in Chemistry, Biology, Social Sciences, Statistics as well as in many other areas. See =-=[22]-=-, [24], [26] and their references, where other hereditary properties and their applications are also discussed. To further convey the reader of the power of Theorem 1 we mention that it immediately im... |

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Citation Context ... property can be tested with one-sided error. This general result contains as a special case all the previous results about testing graph properties with one-sided error. These include the results of =-=[23]-=- and [6] about testing k-colorability, the characterization of [25] of the graph-partitioning problems that are testable with one-sided error, the induced vertex colorability properties of [4], the in... |

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Citation Context ...size is a function of ɛ only. Blum, Luby and Rubinfeld [14] were the first to formulate a question of this type, and the general notion of property testing was first formulated by Rubinfeld and Sudan =-=[37]-=-, who were motivated in studying various algebraic properties such as linearity of functions. The main focus of this paper is in testing properties of graphs, where a property of graphs is simply a fa... |

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Citation Context ... Background In this section we discuss the basic notions of regularity, some of the basic applications of regular partitions and state the regularity lemmas that we use in the proof of Theorem 1. See =-=[27]-=- for a comprehensive survey on the regularity-lemma. We start with some basic definitions. For every two nonempty disjoint vertex sets A and B of a graph G, we define e(A, B) to be the number of edges... |

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Citation Context ...ns property testers with one-sided error. For additional results and references on graph property-testing as well as on testing properties of other combinatorial structures, the reader is referred to =-=[18]-=-, [22], [36] and [12]. 2 The New Results 2.1 The main technical result and its immediate applications A graph property is hereditary if it is closed under removal of vertices (and not necessarily unde... |

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Citation Context ...on if |Vi| and |Vj| differ by no more than 1 for all 1 ≤ i < j ≤ k (so in particular each Vi has one of two possible sizes). The Regularity Lemma of Szemerédi can be formulated as follows. Lemma 3.4 (=-=[38]-=-) For every m and ɛ > 0 there exists a number T = T3.4(m, ɛ) with the following property: Any graph G on n ≥ T vertices, has an equipartition A = {Vi | 1 ≤ i ≤ k} of V (G) with m ≤ k ≤ T , for which a... |

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Citation Context ...testers with one-sided error. For additional results and references on graph property-testing as well as on testing properties of other combinatorial structures, the reader is referred to [18], [22], =-=[36]-=- and [12]. 2 The New Results 2.1 The main technical result and its immediate applications A graph property is hereditary if it is closed under removal of vertices (and not necessarily under removal of... |

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Citation Context ...ontains as a special case all the previous results about testing graph properties with one-sided error. These include the results of [23] and [6] about testing k-colorability, the characterization of =-=[25]-=- of the graph-partitioning problems that are testable with one-sided error, the induced vertex colorability properties of [4], the induced edge colorability properties of [17], a transformation from t... |

68 | A combinatorial characterization of the testable graph properties: it’s all about regularity
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Citation Context ... and showed that any hereditary property of k-uniform hypergraphs is testable. This proof applies variants of the recently proved hypergraph regularity lemmas. In a joint work with Fischer and Newman =-=[2]-=- we have managed to give a combinatorial characterization of the testable graph properties (recall that in the present paper we mainly deal with one-sided error testers). This characterization implies... |

50 | Testing k-colorability
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Citation Context ... can be tested with one-sided error. This general result contains as a special case all the previous results about testing graph properties with one-sided error. These include the results of [23] and =-=[6]-=- about testing k-colorability, the characterization of [25] of the graph-partitioning problems that are testable with one-sided error, the induced vertex colorability properties of [4], the induced ed... |

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Citation Context ... no copy of H as a (not necessarily induced) subgraph), it is known that whenever H is not bipartite, there is no tester (one-sided or two-sided) whose query complexity is polynomial in 1/ɛ (see [1], =-=[9]-=-). Recall that a hereditary property P is equivalent to being FP-free for a possibly infinite family of graphs FP. The hardness of testing hereditary properties for which FP is finite is (relatively) ... |

45 | Every monotone graph property is testable
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Citation Context ...roperties of [4], the induced edge colorability properties of [17], a transformation from two-sided to one-sided error testing [25], as well as a recent result about testing monotone graph properties =-=[11]-=-. More importantly, as a special case of our main result, we infer that some of the most well studied graph properties, both in graph theory and computer science, are testable with one-sided error. So... |

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Citation Context ...perty testers with one-sided error. For additional results and references on graph property-testing as well as on testing properties of other combinatorial structures, the reader is referred to [18], =-=[22]-=-, [36] and [12]. 2 The New Results 2.1 The main technical result and its immediate applications A graph property is hereditary if it is closed under removal of vertices (and not necessarily under remo... |

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Citation Context ...rom satisfying all the properties of P, then for some i, the graph G is δP(ɛ)-far from satisfying Pi. 2.3.4 An extremal result for all graph property Confirming a conjecture of Erdős, it was shown in =-=[34]-=- that if a graph is ɛ-far from being k-colorable, then it contains a non k-colorable subgraph of size that depends only on ɛ. In [11] this result was extended to any monotone graph property. As we hav... |

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Citation Context ... we study here is the so called dense graph model as defined in [23]. Graph property testing has also been studied in other models, such as the bounded degree model [24] and the general density model =-=[31]-=-. 1.2 Related work The most interesting results in property-testing are those that show that large families of problems are testable. The main result of [23] states that a certain abstract graph parti... |

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Citation Context ...ction property defined by T , if there are n sets S1, . . . , Sn of type T , such that vertices i and j are connected in G if and only if Si ∩ Sj �= ∅. For example, the property of being a d-Box (see =-=[16]-=- and its references) is obtained by letting the “type” of the sets be axis parallel boxes in Rd . See the monograph [29] for more information and examples of intersection graph properties. It is clear... |

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Citation Context ...every fixed graph H let PH be the property of not containing a copy of H, and let P∗ H be the property of not containing an induced copy of H. The property PH was (implicitly) shown to be testable in =-=[3]-=-, and P∗ H was shown to be testable in [4]. 8s• k-colorability: The k-colorability property was (implicitly) shown to be testable already in [34]. In [23], a simplified explicit tester was studied wit... |

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Citation Context ...ve (possibly two-sided) oblivious testers? Note, that the definition of an oblivious tester implicitly assumes that the query complexity of such a tester is a function of ɛ only. • Fischer and Newman =-=[19]-=- have recently shown that every testable graph property is also estimable, namely, for any such property one can estimate how far is a given graph from satisfying 25sthe property (in this paper this q... |

25 |
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Citation Context ...ining no copy of H as a (not necessarily induced) subgraph), it is known that whenever H is not bipartite, there is no tester (one-sided or two-sided) whose query complexity is polynomial in 1/ɛ (see =-=[1]-=-, [9]). Recall that a hereditary property P is equivalent to being FP-free for a possibly infinite family of graphs FP. The hardness of testing hereditary properties for which FP is finite is (relativ... |

25 | Functions that have read-twice constant width branching programs are not necessarily testable, Random Structures and Algorithms
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Citation Context ...sides testing graph properties. In [5] it is proved that every regular language is testable. This result was extended to any read-once branching program in [30]. On the other hand, it was 2sproved in =-=[20]-=-, that there are read-twice branching programs that are not-testable. The main result of [8] states that any constraint satisfaction problem is testable. With this abundance of general testability res... |

24 | A characterization of easily testable induced subgraphs
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Citation Context ... being FP-free for a possibly infinite family of graphs FP. The hardness of testing hereditary properties for which FP is finite is (relatively) well understood, as it follows from the main result of =-=[7]-=- that if FP has a graph on at least 5 vertices, then there is no tester (one-sided or two-sided) for P, whose query complexity is polynomial in 1/ɛ. When FP is infinite the situation is much more comp... |

22 | Queries. Testing of function that have small width branching programs
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Citation Context ...nd non-testability results in other areas besides testing graph properties. In [5] it is proved that every regular language is testable. This result was extended to any read-once branching program in =-=[30]-=-. On the other hand, it was 2sproved in [20], that there are read-twice branching programs that are not-testable. The main result of [8] states that any constraint satisfaction problem is testable. Wi... |

19 | Testing graphs for colorability properties
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Citation Context ... the characterization of [25] of the graph-partitioning problems that are testable with one-sided error, the induced vertex colorability properties of [4], the induced edge colorability properties of =-=[17]-=-, a transformation from two-sided to one-sided error testing [25], as well as a recent result about testing monotone graph properties [11]. More importantly, as a special case of our main result, we i... |

17 | Graph limits and testing hereditary graph properties
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Citation Context ... of [11], the upper bounds of Theorems 1 and 2 cannot be generally improved even for monotone graph properties. See the precise statement in [11]. 9s2.5 Recent results Recently, Lovász and B. Szegedy =-=[28]-=- found an alternative proof of Theorem 1 using the method of convergent graph sequences. Their result is slightly weaker than ours as it does not give any explicit upper bound on the query complexity ... |

16 | Testing satisfiability
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Citation Context ...This result was extended to any read-once branching program in [30]. On the other hand, it was 2sproved in [20], that there are read-twice branching programs that are not-testable. The main result of =-=[8]-=- states that any constraint satisfaction problem is testable. With this abundance of general testability results, a natural question is what makes a combinatorial property testable. As graphs are the ... |

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Citation Context ...es. Our main tool for the proof of Theorem 3 is the following result, which is a strengthened version of a result of Frankl and Füredi in [21] and can be deduced, for example, from the main result of =-=[39]-=-. Theorem 7 ([21, 39]) For any graph F = (R, T ), with |T | = t > 0 edges there is a constant δ = δ(F ) with the following property: For any integer n there is a graph Gn = (V, E) on n vertices, which... |

12 | What is the furthest graph from a hereditary property? Random Structures Algorithms
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Citation Context ...characterization implies that the regularity lemma is in some sense essential to graph property testing. Finally, the main techniques developed in this paper have been applied in another recent study =-=[13]-=- of the family of hereditary graph properties. 2.6 Organization Our main tool in the proof of Theorem 1 is a novel application of a powerful variant of Szemerédi’s Regularity Lemma proved in [4]. In S... |

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5 |
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Citation Context ...e a query complexity that is bounded by a function of ɛ but one that depends on the size of the graph (e.g. Q(ɛ, n) = 1/ɛ + (−1) n ). Though this seems like an annoying technicality, it was proved in =-=[10]-=- that this subtlety may have non-trivial ramifications. The second, seemingly more severe, restriction on an oblivious tester is that it cannot use the size of the input in order to make its decisions... |

3 |
Colored packings of sets in combinatorial design theory
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(Show Context)
Citation Context ... Moreover, finding this independent set requires Ω(n 2 ) queries. Our main tool for the proof of Theorem 3 is the following result, which is a strengthened version of a result of Frankl and Füredi in =-=[21]-=- and can be deduced, for example, from the main result of [39]. Theorem 7 ([21, 39]) For any graph F = (R, T ), with |T | = t > 0 edges there is a constant δ = δ(F ) with the following property: For a... |

1 |
Generalizations of the removal lemma, manuscript
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(Show Context)
Citation Context ...Their result is slightly weaker than ours as it does not give any explicit upper bound on the query complexity of testing even simple hereditary properties such as triangle-freeness. Rödl and Schacht =-=[35]-=- have generalized Theorem 1 and showed that any hereditary property of k-uniform hypergraphs is testable. This proof applies variants of the recently proved hypergraph regularity lemmas. In a joint wo... |