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257
Fast Monte-Carlo algorithms for finding low-rank approximations
- IN PROCEEDINGS OF THE 39TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 1998
"... We consider the problem of approximating a given m * n matrix A by another matrix of specified rank k, which is smaller than m and n. The Singular Value Decomposition (SVD) can be used to find the "best " such approximation. However, it takes time polynomial in m, n which is prohib ..."
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Cited by 237 (16 self)
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We consider the problem of approximating a given m * n matrix A by another matrix of specified rank k, which is smaller than m and n. The Singular Value Decomposition (SVD) can be used to find the &quot;best &quot; such approximation. However, it takes time polynomial in m, n which is prohibitive for some modern applications. In this paper, we develop an algorithm which is qualitatively faster, provided we may sample the entries of the matrix according to a natural probability distribution. In many applications such sampling can be done efficiently. Our main result is a randomized algorithm to find the description of a matrix D * of rank at most k so that ||A- D*||2F < = min D,rank(D)<=k ||A- D||
Efficient Testing of Large Graphs
- Combinatorica
"... Let P be a property of graphs. An -test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it h ..."
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Cited by 176 (47 self)
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Let P be a property of graphs. An -test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than n 2 edges to make it satisfy P . The property P is called testable, if for every there exists an -test for P whose total number of queries is independent of the size of the input graph. Goldreich, Goldwasser and Ron [8] showed that certain graph properties admit an -test. In this paper we make a first step towards a logical characterization of all testable graph properties, and show that properties describable by a very general type of coloring problem are testable. We use this theorem to prove that first order graph properties not containing a quantifier alternation of type "89" are always testable, while we show that some properties containing this alternation are not. Our results are proven using a combinatorial lemma, a special case of which, that may be of independent interest, is the following. A graph H is called -unavoidable in G if all graphs that differ from G in no more than jGj 2 places contain an induced copy of H . A graph H is called -abundant in G if G contains at least jGj jHj induced copies of H. If H is -unavoidable in G then it is also ( ; jHj)-abundant.
The art of uninformed decisions -- A primer to property testing
- BULLETIN OF THE EATCS
, 2001
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Quick Approximation to Matrices and Applications
, 1998
"... We give algorithms to find the following simply described approximation to a given matrix. Given an m \Theta n matrix A with entries between say-1 and 1, and an error parameter ffl between 0 and 1, we find a matrix D (implicitly) which is the sum of O(1=ffl 2 ) simple rank 1 matrices so that the ..."
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Cited by 145 (7 self)
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We give algorithms to find the following simply described approximation to a given matrix. Given an m \Theta n matrix A with entries between say-1 and 1, and an error parameter ffl between 0 and 1, we find a matrix D (implicitly) which is the sum of O(1=ffl 2 ) simple rank 1 matrices so that the sum of entries of any submatrix (among the 2 m+n ) of (A \Gamma D) is at most fflmn in absolute value. Our algorithm takes time dependent only on ffl and the allowed probability of failure (not on m;n). We draw on two lines of research to develop the algorithms: one is built around the fundamental Regularity Lemma of Szemerédi in Graph Theory and the constructive version of Alon, Duke, Leffman, Rödl and Yuster. The second one is from the papers of Arora, Karger and Karpinski, Fernandez de la Vega and most directly Goldwasser, Goldreich and Ron who develop approximation algorithms for a set of graph problems, typical of which is the maximum cut problem. ?From our matrix approximation, the...
A characterization of the (natural) graph properties testable with one-sided error
- Proc. of FOCS 2005
, 2005
"... 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 decis ..."
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Cited by 107 (18 self)
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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
The counting lemma for regular k-uniform hypergraphs
, 2004
"... Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓ-partite graph with V (G) = V1 ∪ · · · ∪ Vℓ and |Vi | = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are ε-regular of ..."
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Cited by 105 (14 self)
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Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an ℓ-partite graph with V (G) = V1 ∪ · · · ∪ Vℓ and |Vi | = n for all i ∈ [ℓ], and all pairs (Vi, Vj) are ε-regular of density d for ℓ 1 ≤ i < j ≤ ℓ, then G contains (1 ± fℓ(ε))d
Regularity lemma for k-uniform hypergraphs, Random Structures and Algorithms
, 2005
"... Szemerédi’s Regularity Lemma proved to be a very powerful tool in extremal graph theory with a large number of applications. Chung [Regularity lemmas for hypergraphs and quasi-randomness, Random ..."
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Cited by 90 (8 self)
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Szemerédi’s Regularity Lemma proved to be a very powerful tool in extremal graph theory with a large number of applications. Chung [Regularity lemmas for hypergraphs and quasi-randomness, Random
A combinatorial characterization of the testable graph properties: it’s all about regularity
- Proc. of STOC 2006
, 2006
"... A common thread in all the recent results concerning testing dense graphs is the use of Szemerédi’s regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemerédi-partition is testable with a constant ..."
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Cited by 83 (15 self)
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A common thread in all the recent results concerning testing dense graphs is the use of Szemerédi’s regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemerédi-partition is testable with a constant number of queries. Our second and main result is a purely combinatorial characterization of the graph properties that are testable with a constant number of queries. This characterization (roughly) says that a graph property P can be tested with a constant number of queries if and only if testing P can be reduced to testing the property of satisfying one of finitely many Szemerédi-partitions. This means that in some sense, testing for Szemerédi-partitions is as hard as testing any testable graph property. We thus resolve one of the main open problems in the area of property-testing, which was first raised in the 1996 paper of Goldreich, Goldwasser and Ron [24] that initiated the study of graph property-testing. This characterization also gives an intuitive explanation as to what makes a graph property testable.
A variant of the hypergraph removal lemma
, 2006
"... Abstract. Recent work of Gowers [10] and Nagle, Rödl, Schacht, and Skokan [15], [19], [20] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [26] and Furstenberg-Katznelson [7] concerning one-dimensional and multi-dimensional arithmetic progressions respecti ..."
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Cited by 75 (7 self)
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Abstract. Recent work of Gowers [10] and Nagle, Rödl, Schacht, and Skokan [15], [19], [20] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [26] and Furstenberg-Katznelson [7] concerning one-dimensional and multi-dimensional arithmetic progressions respectively. In this paper we shall give a self-contained proof of this hypergraph removal lemma. In fact we prove a slight strengthening of the result, which we will use in a subsequent paper [29] to establish (among other things) infinitely many constellations of a prescribed shape in the Gaussian primes. 1.
