Testing Hereditary Properties of Ordered Graphs and Matrices

Alon, Noga; Ben‐Eliezer, Omri; Fischer, Eldar

doi:10.1109/focs.2017.83

Cited by 14 publications

(31 citation statements)

References 56 publications

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“…The problem of understanding whether the statement of Theorem 1.3 holds for any finite family F of binary matrices, was raised in [5] and is still open. Only recently in [2] it was shown that the statement holds if we ignore the polynomial dependence, as stated in Theorem 1.1. Problem 1.4.…”

Section: Background and Main Resultsmentioning

confidence: 91%

“…Here we do not try to optimize the dependence between the parameters, but rather to show that such a removal lemma exists. Note that in two dimensions this removal lemma follows from Theorem 1.1, but our results here suggest a direction to prove a weak high dimensional removal lemma without trying to generalize the heavy machinery used in [2] to the high dimensional setting. Our main result here states that this problem is equivalent in some sense to the problem of showing that if a hypermatrix M contains many pairwise-disjoint copies of a hypermatrix A, then it contains a "wide" copy of A; more details are given later.…”

Section: Multi-dimensional Matrices Over Arbitrary Alphabetsmentioning

confidence: 82%

“…Very recently, the authors and Fischer [2] generalized the (finite and infinite) induced graph removal lemma by obtaining an order-preserving version of it, and also showed that the same type of proof can be used to obtain a removal lemma for two-dimensional matrices (with row and column order) over a finite alphabet; this it Theorem 1.1 above.…”

Section: Related Workmentioning

confidence: 99%

“…While the investigation of graph removal lemmas has been quite extensive, as described in Section 2 below, the first known removal lemma for ordered graph-like two-dimensional structures, and specifically for (row and column ordered) matrices, was only obtained very recently by the authors and Fischer [2]. Theorem 1.1 [2]. Fix a finite alphabet Γ.…”

Section: Introductionmentioning

confidence: 99%

“…However, even when P is characterized by a single forbidden submatrix, the upper bound on f P (ǫ) guaranteed by the removal lemma in [2] is very large; in fact, it is at least as large as a wowzer (tower of towers) type function of ǫ. On the other hand, a lower bound of Fischer and Rozenberg [15] implies that one cannot hope for a polynomial dependence of f P (ǫ) in ǫ −1 in general (for the non-binary case), even when P is characterized by a single forbidden submatrix.…”

Section: Introductionmentioning

confidence: 99%

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Efficient Removal Lemmas for Matrices

Alon

Ben‐Eliezer

2019

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The authors and Fischer recently proved that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing the following (ordered) matrix removal lemma: For any finite alphabet Σ, any hereditary property P of matrices over Σ, and any ǫ > 0, there exists f P (ǫ) such that for any matrix M over Σ that is ǫ-far from satisfying P, most of the f P (ǫ) × f P (ǫ) submatrices of M do not satisfy P. Here being ǫ-far from P means that one needs to modify at least an ǫ-fraction of the entries of M to make it satisfy P.However, in the above general removal lemma, f P (ǫ) grows very fast as a function of ǫ −1 , even when P is characterized by a single forbidden submatrix. In this work we establish much more efficient removal lemmas for several special cases of the above problem. In particular, we show the following: For any fixed s × t binary matrix A and any ǫ > 0 there exists δ > 0 polynomial in ǫ, such that for any binary matrix M in which less than a δ-fraction of the s × t submatrices are equal to A, there exists a set of less than an ǫ-fraction of the entries of M that intersects every A-copy in M .We generalize the work of Alon, Fischer and Newman [SICOMP'07] and make progress towards proving one of their conjectures. The proofs combine their efficient conditional regularity lemma for matrices with additional combinatorial and probabilistic ideas.

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Section: Background and Main Resultsmentioning

confidence: 91%

Section: Multi-dimensional Matrices Over Arbitrary Alphabetsmentioning

confidence: 82%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Efficient Removal Lemmas for Matrices

Alon

Ben‐Eliezer

2019

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Testing convexity of figures under the uniform distribution

Berman

Murzabulatov

Raskhodnikova

2018

Random Struct Algorithms

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We consider the following basic geometric problem: Given ϵ∈(0,1/2), a 2‐dimensional black‐and‐white figure is ∊‐ far from convex if it differs in at least an ∊ fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is ∊‐ far from convex are needed to detect a violation of convexity with constant probability? This question arises in the context of designing property testers for convexity. We show that Θ(ϵ−4/3) uniform samples (and the same running time) are necessary and sufficient for detecting a violation of convexity in an ∊‐far figure and, equivalently, for testing convexity of figures with 1‐sided error. Our algorithm beats the Ω(ϵ−3/2) lower bound by Schmeltz [32] on the number of samples required for learning convex figures under the uniform distribution. It demonstrates that, with uniform samples, we can check if a set is approximately convex much faster than we can find an approximate representation of a convex set.

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