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Every hereditary permutation property is testable
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Electronic Notes in Discrete Mathematics 38 (2011) 123–128 www.elsevier.com/locate/endm
Every hereditary permutation property is testable Antˆonio J. O. Bastos a,2 Carlos Hoppen b,2 Yoshiharu Kohayakawa c,2 Rudini M. Sampaio a,2 a
b
Universidade Federal do Cear´ a, Fortaleza, Brazil Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil c Universidade de S˜ ao Paulo, S˜ ao Paulo, Brazil
Abstract In 2010 Moreira and three of the current authors introduced a notion of property testing for permutations based on a distance that measures the randomness of a permutation. They proved that every hereditary permutation property is testable in this sense, or weakly testable in their terminology. In this paper, we show that hereditary permutation properties are testable in a stronger sense, namely when testability is defined by means of the usual 1 -distance for permutations. This is the permutation analogue of the well-known Alon-Shapira result on the testability of hereditary graph properties. Keywords: Property testing, permutations, hereditary properties.
Given a combinatorial structure S and a property P, there has been great interest in determining whether one can decide, with high accuracy, if an object s ∈ S satisfies P, or if it is ‘far’ from satisfying it, based on a randomly chosen substructure of s with sufficiently large but constant size. A property P whose occurrence can be estimated in this way is called a testable S-property. 1
The second author acknowledges the support by FAPERGS (Proc. 10/0388-2) and CNPq (Proc. 484154/2010-9). The third author was partially supported by CNPq (Proc. 308509/2007-2, 484154/2010-9). The fourth author was partially supported by Funcap (Proc. 07.013.00/09) and CNPq (Proc. 484154/2010-9). 2 Email: {ajob,rudini}@lia.ufc.br, [email protected], [email protected] 1571-0653/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2011.09.021
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A rich theory has been developed in the case where S is the set of finite dense graphs. Following the seminal paper of Goldreich, Goldwasser and Ron [3], a graph property P is said to be testable if, for every ε > 0, there exist a positive integer k and a randomized algorithm T , called a tester, which has the ability to query whether a desired pair of vertices in the input graph are adjacent or not. Moreover, the following properties are satisfied for any input graph G with at least k vertices: (i) the number of queries made by T is bounded above by a function that depends only on the error bound ε, and not on the size of the input; (ii) if G satisfies P, then the tester identifies this with probability at least 1 − ε; (iii) if G is “ε-far” from satisfying P, then, with probability at least 1 − ε, the tester confirms that G does not satisfy P, where, by being ε-far, we mean that no graph obtained from G by the addition or removal of at most εn2 edges satisfies P. There is a long list of surprisingly general results characterizing testable graph properties. For instance, it is known that, for graphs, the class of testable properties remains unchanged if we consider only testers of a particular form, namely those that randomly choose a k-subset X of vertices in G and then verify whether the induced subgraph G[X] satisfies some related property P . Moreover, using a variant of the Szemer´edi Regularity Lemma, Alon and Shapira [1] proved that every hereditary graph property is testable, that is, every graph property that is inherited by induced subgraphs, is testable. More recently, three of the current authors considered property testing for permutations [4]. Here, a permutation σ of length |σ| = n is a bijective function on [n] = {1, 2, . . . , n}, and it is represented by σ = (σ(1), . . . , σ(n)). In parallel with the graph case, the testability for permutation parameters has been defined in terms of subpermutations: given permutations τ on [m] and σ on [n], we say that a strictly increasing m-tuple (x1 , . . . , xm ) ∈ [n]m induces the subpermutation τ in σ if τ (i) < τ (j) if and only if σ(xi ) < σ(xj ) for every (i, j) ∈ [m]2 . The set S = {x1 , . . . , xm } is the index set of this subpermutation, which is called the subpermutation of σ induced by S and is denoted by σ[S]. As an illustration, if τ = (3, 1, 4, 2) and σ = (5, 6, 2, 4, 7, 1, 3), then τ = σ[{1, 3, 5, 7}]. With this, if k < n are integers and σ is a permutation on [n], then the random subpermutation sub(k, σ) of σ is the induced subpermutation of σ of size k whose index set S = {s1 < · · · < sk } is chosen uniformly at random in [n]. Following the graph case, in order to define property testing for permutations, it remains to introduce a metric with respect to
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which the ‘proximity’ to satisfying some given property is measured. Tothis end, let σ and π be permutations on [n]. We say that a set S ⊆ [n] re2 moves all distinct subpermutations of σ and π if, whenever a set of indices {a1 < · · · < ak } ⊂ [n] induces different subpermutations of σ and π, there are indices i < j ∈ [k] with {ai , aj } ∈ S. It is easy to see that, for a set S to remove all distinct subpermutations of σ and π, it suffices that it removes all their relative inversions, that is, all of their distinct subpermutations of length two. Basedon this, we may define the edit distance between σ and π [n] 1 by d1 (σ, π) = n {i, j} ∈ 2 : σ(i) < σ(j) ⇔ τ (i) > τ (j) . This distance (2) is the permutation analogue of the edit distance (or 1 -distance) for graphs, and therefore inherits the name. (It is also known as Kendall’s tau distance). We may now declare a permutation property P to be testable through subpermutations, or testable for short, if there is a permutation property P satisfying the following. For any fixed ε > 0, there exists a positive integer k such that, if σ is a permutation on [n] with n ≥ k, then the two statements below hold for a random subpermutation sub(k, σ): (i) sub(k, σ) satisfies P with probability at least 1 − ε whenever σ satisfies P; (ii) sub(k, σ) does not satisfy P with probability at least 1−ε whenever d1 (σ, P) = min{d1 (σ, π) : |π| = n and π satisfies P} ≥ ε. A similar notion has been introduced in [4] with the name of weak testability. The difference between these two concepts lies in the definition of distance. Instead of using the edit distance d1 , the authors of [4] used a distance d based on the concept of discrepancy of a permutation, which plays a pivotal rˆole in the study of convergence of permutation sequences. The term ‘weak’ comes from the fact that proximity with respect to d1 implies proximity with respect to d in the following sense: for every ε > 0 there exists δ > 0 such that, if σ and π satisfy d1 (σ, π) < δ, then d (σ, π) < ε. However, the converse is not true in general, as one can verify that, when n grows, the distance between two random permutations on [n] chosen independently tends to zero with high probability with respect to d , but not with respect to d1 . In spite of this, we show that, as far as hereditary properties P are concerned, proximity to P with respect to d implies proximity to P with respect to d1 . As usual, a permutation property P is said to be hereditary if the fact that a permutation σ satisfies P implies that all its subpermutations also satisfy P.
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Main results
Our first result is as follows. Theorem 1.1 Let P be a hereditary permutation property. Then, for every ε > 0, there exists δ > 0 such that any permutation σ satisfying d (σ, P) < δ also satisfies d1 (σ, P) < ε. Theorem 1.1 had been conjectured in [4]. It contributes to unify the notion of permutation property testing with its graph-theoretical counterpart, as Theorem 1.1 is true when P is a graph property, σ is a graph and d and d1 are replaced by the rectangular and the edit distances for graphs, respectively. Moreover, one of the consequences of this result is the following permutation counterpart of the Alon-Shapira result [1] on the testability of hereditary graph properties, which we deem to be the main result in this work. Theorem 1.2 Every hereditary permutation property is testable. Theorem 1.2 is an improvement on Theorem 1.6 in [4], where the analogous conclusion was reached in terms of weak testability. A concept that is closely related to property testing is parameter testing. Here, the objective is to accurately estimate a numerical function, or parameter, associated with a permutation, for instance, the number of fixed points in the permutation. A permutation parameter f is testable through subpermutations if, for every ε > 0, there exists k > 0 such that, if σ is a permutation of length n > k, then we may compute an estimate f˜ of f (σ) based on a random subpermutation sub(k, σ) satisfying P |f (σ) − f˜| > ε ≤ ε. We may show the following. Theorem 1.3 If P is a hereditary permutation property, then the edit distance f (σ) = d1 (σ, P) of σ to P is a testable permutation parameter. In the remainder of this extended abstract, we introduce the main ingredients in the proof of Theorem 1.1, from which the proofs of Theorems 1.2 and 1.3 follow easily. This proof relies on the existence of the limit of a permutation sequence (see [5]), which can be related to the limit of an ordered graph sequence. It then follows the main steps of an alternative proof of the Alon-Shapira heredity result due to Lov´asz and Szegedy [6]. The main ingredient in the proof of Theorem 1.1 is to relate permutations, as well as limit permutations, to a class of ordered graphs and their limits. The proof will then follow the main steps of the short version of [6], adapting it so that ordered graphs play the rˆole of graphs. To this end, we let an ordered graph G be a labelled graph whose vertex set is [n] for some positive
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integer n. More generally, given a graph H whose vertices v1 < · · · < vn are ordered, the ordered graph of H is the graph G with vertex set [n] such that i ∼ j if and only if vi ∼ vj . An ordered (induced) subgraph of a graph G is the ordered graph of an (induced) subgraph of G. Given a permutation σ on [n], we may associate the ordered graph Gσ such that, for i < j ∈ [n], the pair ij is an edge of Gσ if and only if σ(i) < σ(j). With this, it is clear that τ is a subpermutation of σ if and only if Gτ is an ordered induced subgraph of Gσ . We now relate limit permutations to limits of graph sequences (for details, see [5] and [2]). A limit permutation Z is a pair Z = (X, Y ) of uniform random variables X and Y in [0, 1] (not necessarily independent), given by their joint distribution function. A limit permutation Z = (X, Y ) can be identified with the regular conditional distribution function of Y given X, given by the Lebesgue measurable function Z : [0, 1]2 → [0, 1] such that Z(x, y) = P(Y ≤ y | X = x). It is clear that, given x ∈ [0, 1], the function Z(x, ·) : [0, 1] → [0, 1] is a cumulative distribution function (cdf). Let YZ (x) be the r.v. with cdf Z(x, ·). A limit graph, or graphon, is a symmetric Lebesgue measurable function W : [0, 1]2 → [0, 1]. With a limit permutation Z = (X, Y ), we associate the graphon WZ (x1 , x2 ) = P (YZ (x1 ) < YZ (x2 )) for every x1 < x2 ∈ [0, 1]. By definition, we see that, for permutations σ and σ of the same length, we have d1 (σ, σ ) = d1 (Gσ , Gσ ), which relates the edit distance between permutations. The key technical tool is the following observation, which translates convergence with respect to the rectangular distance of limit permutations into convergence with respect to the rectangular distance of graphons, and thus allows us to prove our main results. Formally, we have d (Z1 , Z2 ) = supS,T |P ((X1 , Y1 ) ∈ S × T ) − P ((X2 , Y2 ) ∈ S × T )|, where Z1 = (X1 , Y1 ) and Z2 = (X2 , Y2 ) are limit permutations, and where S and T range over all intervals of [0, 1]. Now, if W1 and W2 are graphons, the rectangu lar distance is given by d (W1 , W2 ) = supS,T S×T W1 (x, y) − W2 (x, y) dx dy , where S and T range over all Borel subsets of [0, 1]. Moreover, it is a fact that a permutation σ may be identified with a limit permutation Zσ , while a graph G may be identified with a graphon WG , so that these two distances are also distances between two permutations (or two graphs), or between a permutation and a limit permutation (between a graph and a graphon, respectively). Lemma 1.4 Let (σn ) be a permutation sequence and let Z = (X, Y ) be a limit permutation. Then limn→∞ d (σn , Z) = 0 ⇒ limn→∞ d (Gσn , WZ ) = 0.
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We mention that Schacht [4,7] has obtained a general removal lemma for permutations. Theorem 1.2 would follow from this lemma if one could show that, (*) for every ε > 0 and every hereditary property P, there exists δ > 0 such that, for any permutation σ on [n], if d1 (σ, P) ≥ ε, then one needs to remove ≥ δn2 pairs {i, j} ⊂ [n] to destroy all subpermutations τ of σ that do not satisfy P. However, to the best of our knowledge, the validity of (*) is still open. We should mention that, in contrast to the graph case (where it is trivial), the combinatorial structure obtained by removing pairs in a permutation is not a permutation. As a consequence, it might well be that, for some property P, although all subpermutations of some permutation σ satisfy P after the removal of o(n2 ) pairs, there is no permutation satisfying P that is 1 -close to σ.
References [1] N. Alon and A. Shapira, A characterization of the (natural) graph properties testable with one-sided error, Proc. of the 46th IEEE FOCS (2005), 429–438. [2] C. Borgs, J. T. Chayes, L. Lov´asz, V. T. S´ os, B. Szegedy and K. Vesztergombi: Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing, Advances in Mathematics 219 (6) (2008), 1801–1851. [3] O. Goldreich, S. Goldwasser and D. Ron, Property testing and its connection to learning an approximation, Proc. of the 37th IEEE FOCS (1996), 339–348. [4] C. Hoppen, Y. Kohayakawa, C. G. Moreira and R. M. Sampaio, Testing permutation properties through subpermutations, to appear in Theoretical Computer Science (2011). [5] C. Hoppen, Y. Kohayakawa, C. G. Moreira, B. R´ ath and R. M. Sampaio, Limits of permutation sequences, preprint (2011). [6] L. Lov´asz and B. Szegedy, Testing properties of graphs and functions, to appear in Israel Journal of Mathematics. Short version: Graph limits and testing hereditary graph properties, manuscript (2005). [7] M. Schacht, personal communication.
Electronic Notes in Discrete Mathematics 38 (2011) 123–128 www.elsevier.com/locate/endm
Every hereditary permutation property is testable Antˆonio J. O. Bastos a,2 Carlos Hoppen b,2 Yoshiharu Kohayakawa c,2 Rudini M. Sampaio a,2 a
b
Universidade Federal do Cear´ a, Fortaleza, Brazil Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil c Universidade de S˜ ao Paulo, S˜ ao Paulo, Brazil
Abstract In 2010 Moreira and three of the current authors introduced a notion of property testing for permutations based on a distance that measures the randomness of a permutation. They proved that every hereditary permutation property is testable in this sense, or weakly testable in their terminology. In this paper, we show that hereditary permutation properties are testable in a stronger sense, namely when testability is defined by means of the usual 1 -distance for permutations. This is the permutation analogue of the well-known Alon-Shapira result on the testability of hereditary graph properties. Keywords: Property testing, permutations, hereditary properties.
Given a combinatorial structure S and a property P, there has been great interest in determining whether one can decide, with high accuracy, if an object s ∈ S satisfies P, or if it is ‘far’ from satisfying it, based on a randomly chosen substructure of s with sufficiently large but constant size. A property P whose occurrence can be estimated in this way is called a testable S-property. 1
The second author acknowledges the support by FAPERGS (Proc. 10/0388-2) and CNPq (Proc. 484154/2010-9). The third author was partially supported by CNPq (Proc. 308509/2007-2, 484154/2010-9). The fourth author was partially supported by Funcap (Proc. 07.013.00/09) and CNPq (Proc. 484154/2010-9). 2 Email: {ajob,rudini}@lia.ufc.br, [email protected], [email protected] 1571-0653/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2011.09.021
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A rich theory has been developed in the case where S is the set of finite dense graphs. Following the seminal paper of Goldreich, Goldwasser and Ron [3], a graph property P is said to be testable if, for every ε > 0, there exist a positive integer k and a randomized algorithm T , called a tester, which has the ability to query whether a desired pair of vertices in the input graph are adjacent or not. Moreover, the following properties are satisfied for any input graph G with at least k vertices: (i) the number of queries made by T is bounded above by a function that depends only on the error bound ε, and not on the size of the input; (ii) if G satisfies P, then the tester identifies this with probability at least 1 − ε; (iii) if G is “ε-far” from satisfying P, then, with probability at least 1 − ε, the tester confirms that G does not satisfy P, where, by being ε-far, we mean that no graph obtained from G by the addition or removal of at most εn2 edges satisfies P. There is a long list of surprisingly general results characterizing testable graph properties. For instance, it is known that, for graphs, the class of testable properties remains unchanged if we consider only testers of a particular form, namely those that randomly choose a k-subset X of vertices in G and then verify whether the induced subgraph G[X] satisfies some related property P . Moreover, using a variant of the Szemer´edi Regularity Lemma, Alon and Shapira [1] proved that every hereditary graph property is testable, that is, every graph property that is inherited by induced subgraphs, is testable. More recently, three of the current authors considered property testing for permutations [4]. Here, a permutation σ of length |σ| = n is a bijective function on [n] = {1, 2, . . . , n}, and it is represented by σ = (σ(1), . . . , σ(n)). In parallel with the graph case, the testability for permutation parameters has been defined in terms of subpermutations: given permutations τ on [m] and σ on [n], we say that a strictly increasing m-tuple (x1 , . . . , xm ) ∈ [n]m induces the subpermutation τ in σ if τ (i) < τ (j) if and only if σ(xi ) < σ(xj ) for every (i, j) ∈ [m]2 . The set S = {x1 , . . . , xm } is the index set of this subpermutation, which is called the subpermutation of σ induced by S and is denoted by σ[S]. As an illustration, if τ = (3, 1, 4, 2) and σ = (5, 6, 2, 4, 7, 1, 3), then τ = σ[{1, 3, 5, 7}]. With this, if k < n are integers and σ is a permutation on [n], then the random subpermutation sub(k, σ) of σ is the induced subpermutation of σ of size k whose index set S = {s1 < · · · < sk } is chosen uniformly at random in [n]. Following the graph case, in order to define property testing for permutations, it remains to introduce a metric with respect to
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which the ‘proximity’ to satisfying some given property is measured. Tothis end, let σ and π be permutations on [n]. We say that a set S ⊆ [n] re2 moves all distinct subpermutations of σ and π if, whenever a set of indices {a1 < · · · < ak } ⊂ [n] induces different subpermutations of σ and π, there are indices i < j ∈ [k] with {ai , aj } ∈ S. It is easy to see that, for a set S to remove all distinct subpermutations of σ and π, it suffices that it removes all their relative inversions, that is, all of their distinct subpermutations of length two. Basedon this, we may define the edit distance between σ and π [n] 1 by d1 (σ, π) = n {i, j} ∈ 2 : σ(i) < σ(j) ⇔ τ (i) > τ (j) . This distance (2) is the permutation analogue of the edit distance (or 1 -distance) for graphs, and therefore inherits the name. (It is also known as Kendall’s tau distance). We may now declare a permutation property P to be testable through subpermutations, or testable for short, if there is a permutation property P satisfying the following. For any fixed ε > 0, there exists a positive integer k such that, if σ is a permutation on [n] with n ≥ k, then the two statements below hold for a random subpermutation sub(k, σ): (i) sub(k, σ) satisfies P with probability at least 1 − ε whenever σ satisfies P; (ii) sub(k, σ) does not satisfy P with probability at least 1−ε whenever d1 (σ, P) = min{d1 (σ, π) : |π| = n and π satisfies P} ≥ ε. A similar notion has been introduced in [4] with the name of weak testability. The difference between these two concepts lies in the definition of distance. Instead of using the edit distance d1 , the authors of [4] used a distance d based on the concept of discrepancy of a permutation, which plays a pivotal rˆole in the study of convergence of permutation sequences. The term ‘weak’ comes from the fact that proximity with respect to d1 implies proximity with respect to d in the following sense: for every ε > 0 there exists δ > 0 such that, if σ and π satisfy d1 (σ, π) < δ, then d (σ, π) < ε. However, the converse is not true in general, as one can verify that, when n grows, the distance between two random permutations on [n] chosen independently tends to zero with high probability with respect to d , but not with respect to d1 . In spite of this, we show that, as far as hereditary properties P are concerned, proximity to P with respect to d implies proximity to P with respect to d1 . As usual, a permutation property P is said to be hereditary if the fact that a permutation σ satisfies P implies that all its subpermutations also satisfy P.
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Main results
Our first result is as follows. Theorem 1.1 Let P be a hereditary permutation property. Then, for every ε > 0, there exists δ > 0 such that any permutation σ satisfying d (σ, P) < δ also satisfies d1 (σ, P) < ε. Theorem 1.1 had been conjectured in [4]. It contributes to unify the notion of permutation property testing with its graph-theoretical counterpart, as Theorem 1.1 is true when P is a graph property, σ is a graph and d and d1 are replaced by the rectangular and the edit distances for graphs, respectively. Moreover, one of the consequences of this result is the following permutation counterpart of the Alon-Shapira result [1] on the testability of hereditary graph properties, which we deem to be the main result in this work. Theorem 1.2 Every hereditary permutation property is testable. Theorem 1.2 is an improvement on Theorem 1.6 in [4], where the analogous conclusion was reached in terms of weak testability. A concept that is closely related to property testing is parameter testing. Here, the objective is to accurately estimate a numerical function, or parameter, associated with a permutation, for instance, the number of fixed points in the permutation. A permutation parameter f is testable through subpermutations if, for every ε > 0, there exists k > 0 such that, if σ is a permutation of length n > k, then we may compute an estimate f˜ of f (σ) based on a random subpermutation sub(k, σ) satisfying P |f (σ) − f˜| > ε ≤ ε. We may show the following. Theorem 1.3 If P is a hereditary permutation property, then the edit distance f (σ) = d1 (σ, P) of σ to P is a testable permutation parameter. In the remainder of this extended abstract, we introduce the main ingredients in the proof of Theorem 1.1, from which the proofs of Theorems 1.2 and 1.3 follow easily. This proof relies on the existence of the limit of a permutation sequence (see [5]), which can be related to the limit of an ordered graph sequence. It then follows the main steps of an alternative proof of the Alon-Shapira heredity result due to Lov´asz and Szegedy [6]. The main ingredient in the proof of Theorem 1.1 is to relate permutations, as well as limit permutations, to a class of ordered graphs and their limits. The proof will then follow the main steps of the short version of [6], adapting it so that ordered graphs play the rˆole of graphs. To this end, we let an ordered graph G be a labelled graph whose vertex set is [n] for some positive
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integer n. More generally, given a graph H whose vertices v1 < · · · < vn are ordered, the ordered graph of H is the graph G with vertex set [n] such that i ∼ j if and only if vi ∼ vj . An ordered (induced) subgraph of a graph G is the ordered graph of an (induced) subgraph of G. Given a permutation σ on [n], we may associate the ordered graph Gσ such that, for i < j ∈ [n], the pair ij is an edge of Gσ if and only if σ(i) < σ(j). With this, it is clear that τ is a subpermutation of σ if and only if Gτ is an ordered induced subgraph of Gσ . We now relate limit permutations to limits of graph sequences (for details, see [5] and [2]). A limit permutation Z is a pair Z = (X, Y ) of uniform random variables X and Y in [0, 1] (not necessarily independent), given by their joint distribution function. A limit permutation Z = (X, Y ) can be identified with the regular conditional distribution function of Y given X, given by the Lebesgue measurable function Z : [0, 1]2 → [0, 1] such that Z(x, y) = P(Y ≤ y | X = x). It is clear that, given x ∈ [0, 1], the function Z(x, ·) : [0, 1] → [0, 1] is a cumulative distribution function (cdf). Let YZ (x) be the r.v. with cdf Z(x, ·). A limit graph, or graphon, is a symmetric Lebesgue measurable function W : [0, 1]2 → [0, 1]. With a limit permutation Z = (X, Y ), we associate the graphon WZ (x1 , x2 ) = P (YZ (x1 ) < YZ (x2 )) for every x1 < x2 ∈ [0, 1]. By definition, we see that, for permutations σ and σ of the same length, we have d1 (σ, σ ) = d1 (Gσ , Gσ ), which relates the edit distance between permutations. The key technical tool is the following observation, which translates convergence with respect to the rectangular distance of limit permutations into convergence with respect to the rectangular distance of graphons, and thus allows us to prove our main results. Formally, we have d (Z1 , Z2 ) = supS,T |P ((X1 , Y1 ) ∈ S × T ) − P ((X2 , Y2 ) ∈ S × T )|, where Z1 = (X1 , Y1 ) and Z2 = (X2 , Y2 ) are limit permutations, and where S and T range over all intervals of [0, 1]. Now, if W1 and W2 are graphons, the rectangu lar distance is given by d (W1 , W2 ) = supS,T S×T W1 (x, y) − W2 (x, y) dx dy , where S and T range over all Borel subsets of [0, 1]. Moreover, it is a fact that a permutation σ may be identified with a limit permutation Zσ , while a graph G may be identified with a graphon WG , so that these two distances are also distances between two permutations (or two graphs), or between a permutation and a limit permutation (between a graph and a graphon, respectively). Lemma 1.4 Let (σn ) be a permutation sequence and let Z = (X, Y ) be a limit permutation. Then limn→∞ d (σn , Z) = 0 ⇒ limn→∞ d (Gσn , WZ ) = 0.
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We mention that Schacht [4,7] has obtained a general removal lemma for permutations. Theorem 1.2 would follow from this lemma if one could show that, (*) for every ε > 0 and every hereditary property P, there exists δ > 0 such that, for any permutation σ on [n], if d1 (σ, P) ≥ ε, then one needs to remove ≥ δn2 pairs {i, j} ⊂ [n] to destroy all subpermutations τ of σ that do not satisfy P. However, to the best of our knowledge, the validity of (*) is still open. We should mention that, in contrast to the graph case (where it is trivial), the combinatorial structure obtained by removing pairs in a permutation is not a permutation. As a consequence, it might well be that, for some property P, although all subpermutations of some permutation σ satisfy P after the removal of o(n2 ) pairs, there is no permutation satisfying P that is 1 -close to σ.
References [1] N. Alon and A. Shapira, A characterization of the (natural) graph properties testable with one-sided error, Proc. of the 46th IEEE FOCS (2005), 429–438. [2] C. Borgs, J. T. Chayes, L. Lov´asz, V. T. S´ os, B. Szegedy and K. Vesztergombi: Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing, Advances in Mathematics 219 (6) (2008), 1801–1851. [3] O. Goldreich, S. Goldwasser and D. Ron, Property testing and its connection to learning an approximation, Proc. of the 37th IEEE FOCS (1996), 339–348. [4] C. Hoppen, Y. Kohayakawa, C. G. Moreira and R. M. Sampaio, Testing permutation properties through subpermutations, to appear in Theoretical Computer Science (2011). [5] C. Hoppen, Y. Kohayakawa, C. G. Moreira, B. R´ ath and R. M. Sampaio, Limits of permutation sequences, preprint (2011). [6] L. Lov´asz and B. Szegedy, Testing properties of graphs and functions, to appear in Israel Journal of Mathematics. Short version: Graph limits and testing hereditary graph properties, manuscript (2005). [7] M. Schacht, personal communication.