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22
Regularity lemmas and combinatorial algorithms
 In Proc. FOCS
"... Abstract — We present new combinatorial algorithms for Boolean matrix multiplication (BMM) and preprocessing a graph to answer independent set queries. We give the first asymptotic improvements on combinatorial algorithms for dense BMM in many years, improving on the “Four Russians ” O(n 3 /(w log n ..."
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Abstract — We present new combinatorial algorithms for Boolean matrix multiplication (BMM) and preprocessing a graph to answer independent set queries. We give the first asymptotic improvements on combinatorial algorithms for dense BMM in many years, improving on the “Four Russians ” O(n 3 /(w log n)) bound for machine models with wordsize w. (For a pointer machine, we can set w = log n.) The algorithms utilize notions from Regularity Lemmas for graphs in a novel way. • We give two randomized combinatorial algorithms for BMM. The first algorithm is essentially a reduction from BMM to the Triangle Removal Lemma. The best known bounds for the Triangle Removal Lemma only imply an O ` (n 3 log β)/(βw log n) ´ time algorithm for BMM where β = (log ∗ n) δ for some δ> 0, but improvements on the Triangle Removal Lemma would yield corresponding runtime improvements. The second algorithm applies the Weak Regularity Lemma of Frieze and Kannan along with “ several information compression ideas, running in O n 3 (log log n) 2 /(log n) 9/4 ”) time with probability exponentially “ close to 1. When w ≥ log n, it can be implemented in O n 3 (log log n) 2 /(w log n) 7/6 ”) time. Our results immediately imply improved combinatorial methods for CFG parsing, detecting trianglefreeness, and transitive closure. Using Weak Regularity, we also give an algorithm for answering queries of the form is S ⊆ V an independent set? in a graph. Improving on prior work, we show how to randomly preprocess a graph in O(n 2+ε) time (for all ε> 0) so that with high probability, all subsequent batches of log n independent “ set queries can be answered deterministically in O n 2 (log log n) 2 /((log n) 5/4 ”) time. When w ≥ log n, w queries can be answered in O n 2 (log log n) 2 /((log n) 7/6 ” time. In addition to its nice applications, this problem is interesting in that it is not known how to do better than O(n 2) using “algebraic ” methods. 1.
A proof of Green’s conjecture regarding the removal properties of sets of linear equations
"... A system of ℓ linear equations in p unknowns Mx = b is said to have the removal property if every set S ⊆ {1,..., n} which contains o(n p−ℓ) solutions of Mx = b can be turned into a set S ′ containing no solution of Mx = b, by the removal of o(n) elements. Green [GAFA 2005] proved that a single homo ..."
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Cited by 17 (1 self)
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A system of ℓ linear equations in p unknowns Mx = b is said to have the removal property if every set S ⊆ {1,..., n} which contains o(n p−ℓ) solutions of Mx = b can be turned into a set S ′ containing no solution of Mx = b, by the removal of o(n) elements. Green [GAFA 2005] proved that a single homogenous linear equation always has the removal property, and conjectured that every set of homogenous linear equations has the removal property. In this paper we confirm Green’s conjecture by showing that every set of linear equations (even nonhomogenous) has the removal property. We also discuss some applications of our result in theoretical computer science, and in particular, use it to resolve a conjecture of Bhattacharyya, Chen, Sudan and Xie [7] related to algorithms for testing properties of boolean functions.
A unified framework for testing linearinvariant properties
 In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science
, 2010
"... In the history of property testing, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, ReedMuller codes, and F ..."
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Cited by 16 (6 self)
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In the history of property testing, a particularly important role has been played by linearinvariant properties, i.e., properties of Boolean functions on the hypercube which are closed under linear transformations of the domain. Examples of such properties include linearity, ReedMuller codes, and Fourier sparsity. In this work, we describe a framework that can lead to a unified analysis of the testability of all linearinvariant properties, drawing on techniques from additive combinatorics and from graph theory. Our main contributions here are the following: 1. We introduce a simple combinatorial condition, which we call subspaceheredity, and conjecture that any property of Boolean functions satisfying it can be efficiently tested. Verifying this conjecture will unify many individual results in this area. 2. We show that if our conjecture holds, then one can obtain a simple combinatorial characterization of properties of Boolean functions that can be efficiently tested with onesided error, thus addressing a challenge posed by Sudan recently. 3. We introduce a new technique for proving the testability of Boolean functions. Using it, we verify a special case of the conjecture. Our approach here is motivated by techniques that proved to be very successful previously in studying the testability of graph properties.
Invariance in property testing
 Electronic Colloquium on Computational Complexity (ECCC
"... Property testing considers the task of testing rapidly (in particular, with very few samples into the data), if some massive data satisfies some given property, or is far from satisfying the property. For “global properties”, i.e., properties that really depend somewhat on every piece of the data, o ..."
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Cited by 13 (2 self)
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Property testing considers the task of testing rapidly (in particular, with very few samples into the data), if some massive data satisfies some given property, or is far from satisfying the property. For “global properties”, i.e., properties that really depend somewhat on every piece of the data, one could ask how it can be tested by so few samples? We suggest that for “natural ” properties, this should happen because the property is invariant under “nice ” set of “relabellings ” of the data. We refer to this set of relabellings as the “invariance class ” of the property and advocate explicit identification of the invariance class of locally testable properties. Our hope is the explicit knowledge of the invariance class may lead to more general, broader, results. After pointing out the invariance classes associated with some the basic classes of testable properties, we focus on “algebraic properties ” which seem to be characterized by the fact that the properties are themselves vector spaces, while their domains are also vector spaces and the properties are invariant under affine transformations of the domain. We survey recent results (obtained with Tali Kaufman, Elena Grigorescu and Eli BenSasson) that give broad conditions that are sufficient for local testability among this class of properties, and some structural theorems that attempt to describe which properties exhibit the sufficient conditions. 1
Graph removal lemmas
 SURVEYS IN COMBINATORICS
, 2013
"... The graph removal lemma states that any graph on n vertices with o(nv(H)) copies of a fixed graph H may be made Hfree by removing o(n²) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and com ..."
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Cited by 9 (3 self)
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The graph removal lemma states that any graph on n vertices with o(nv(H)) copies of a fixed graph H may be made Hfree by removing o(n²) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.
Testing low complexity affineinvariant properties
, 2013
"... Abstract Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affineinvariant property of multivariate functions over finite fields is testable with a ..."
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Cited by 8 (3 self)
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Abstract Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affineinvariant property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, for instance, that the ReedMuller code over F p of degree d < p is testable, with an argument that uses no detailed algebraic information about polynomials, except that low degree is preserved by composition with affine maps. The complexity of an affineinvariant property P refers to the maximum complexity, as defined by Green and Tao (Ann. Math. 2008), of the sets of linear forms used to characterize P. A more precise statement of our main result is that for any fixed prime p ≥ 2 and fixed integer R ≥ 2, any affineinvariant property P of functions f : F n p → [R] is testable, assuming the complexity of the property is less than p. Our proof involves developing analogs of graphtheoretic techniques in an algebraic setting, using tools from higherorder Fourier analysis.
Every locally characterized affineinvariant property is testable
, 2013
"... Let F = Fp for any fixed prime p> 2. An affineinvariant property is a property of functions on Fn that is closed under taking affine transformations of the domain. We prove that all affineinvariant properties that have local characterizations are testable. In fact, we give a proximityoblivious ..."
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Cited by 6 (3 self)
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Let F = Fp for any fixed prime p> 2. An affineinvariant property is a property of functions on Fn that is closed under taking affine transformations of the domain. We prove that all affineinvariant properties that have local characterizations are testable. In fact, we give a proximityoblivious test for any such property P, meaning that given an input function f, we make a constant number of queries to f, always accept if f satisfies P, and otherwise reject with probability larger than a positive number that depends only on the distance between f and P. More generally, we show that any affineinvariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable. We also prove that any property that can be described as the property of being decomposable into a known structure of lowdegree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degreed polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degreestructural properties and are therefore locally characterized. Our results use a new Gowers inverse theorem by Tao and Ziegler for low characteristic fields that decomposes any polynomial with large Gowers norm into a function of a small number of lowdegree nonclassical polynomials. We establish a new equidistribution result for high rank nonclassical polynomials that drives the proofs of both the testability results and the local characterization of degreestructural properties.
Lower Bounds for Testing Trianglefreeness in Boolean Functions
, 2009
"... Let f1, f2, f3: F n 2 → {0, 1} be three Boolean functions. We ..."
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Let f1, f2, f3: F n 2 → {0, 1} be three Boolean functions. We
Improved lower bounds for testing trianglefreeness in boolean functions via fast matrix multiplication
 In RANDOM
, 2014
"... Abstract Understanding the query complexity for testing linearinvariant properties has been a central open problem in the study of algebraic property testing. Trianglefreeness in Boolean functions is a simple property whose testing complexity is unknown. ..."
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Abstract Understanding the query complexity for testing linearinvariant properties has been a central open problem in the study of algebraic property testing. Trianglefreeness in Boolean functions is a simple property whose testing complexity is unknown.
Tight Lower Bounds for Testing Linear Isomorphism
"... We study lower bounds for testing membership in families of linear/affineinvariant Boolean functions over the hypercube. A family of functions P ⊆ {{0, 1} n → {0, 1}} is linear/affine invariant if for any f ∈ P, it is the case that f ◦ L ∈ P for any linear/affine transformation L of the domain. Mot ..."
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We study lower bounds for testing membership in families of linear/affineinvariant Boolean functions over the hypercube. A family of functions P ⊆ {{0, 1} n → {0, 1}} is linear/affine invariant if for any f ∈ P, it is the case that f ◦ L ∈ P for any linear/affine transformation L of the domain. Motivated by the recent resurgence of attention to the permutation isomorphism problem, we first focus on families that are linearly/affinely isomorphic to some fixed function. A function f: {0, 1} n → {0, 1} is called linear isomorphic to a fixed Boolean function g if f = g ◦ A for some nonsingular transformation A. Our main result is a tight adaptive, twosided Ω(n2) lower bound for testing linear isomorphism to the innerproduct function. This is the first lower bound for testing linear isomorphism to a specific function that matches the trivial upper bound. Our proof exploits the elegant connection between testing and communication complexity discovered by Blais et al. (Computational Complexity, 2012.) Our results are also the first instance of this connection that gives better than Ω(n) lower bound for any property of Boolean functions. These results extend to testing linear isomorphism to any fixed function in the larger class of socalled MaioranaMcFarland bent functions. Our second result shows an Ω(2n/4) query lower bound for any adaptive, twosided tester for membership in the MaioranaMcFarland class of bent functions. This class of Boolean functions is also affineinvariant and its rich structure and pseudorandom properties have been wellstudied in mathematics, coding theory and cryptography.