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205
Property Testing and its connection to Learning and Approximation
"... We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the fun ..."
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Cited by 421 (57 self)
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We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the function on instances of its choice. First, we establish some connections between property testing and problems in learning theory. Next, we focus on testing graph properties, and devise algorithms to test whether a graph has properties such as being kcolorable or having a aeclique (clique of density ae w.r.t the vertex set). Our graph property testing algorithms are probabilistic and make assertions which are correct with high probability, utilizing only poly(1=ffl) edgequeries into the graph, where ffl is the distance parameter. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph which corre...
Probabilistic checking of proofs: a new characterization of NP
 Journal of the ACM
, 1998
"... Abstract. We give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from ..."
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Cited by 365 (28 self)
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Abstract. We give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof. We discuss implications of this characterization; specifically, we show that approximating Clique and Independent Set, even in a very weak sense, is NPhard.
SmallBias Probability Spaces: Efficient Constructions and Applications
 SIAM J. Comput
, 1993
"... We show how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with "almost" equal probability. They are called fflbiased random variables. The number of random bits needed to generate the random variables is ..."
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Cited by 258 (15 self)
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We show how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with "almost" equal probability. They are called fflbiased random variables. The number of random bits needed to generate the random variables is O(log n + log 1 ffl ). Thus, if ffl is polynomially small, then the size of the sample space is also polynomial. Random variables that are fflbiased can be used to construct "almost" kwise independent random variables where ffl is a function of k. These probability spaces have various applications: 1. Derandomization of algorithms: many randomized algorithms that require only k wise independence of their random bits (where k is bounded by O(log n)), can be derandomized by using fflbiased random variables. 2. Reducing the number of random bits required by certain randomized algorithms, e.g., verification of matrix multiplication. 3. Exhaustive testing of combinatorial circui...
Minwise Independent Permutations
 Journal of Computer and System Sciences
, 1998
"... We define and study the notion of minwise independent families of permutations. We say that F ⊆ Sn is minwise independent if for any set X ⊆ [n] and any x ∈ X, when π is chosen at random in F we have Pr(min{π(X)} = π(x)) = 1 X . In other words we require that all the elements of any fixed set ..."
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Cited by 191 (11 self)
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We define and study the notion of minwise independent families of permutations. We say that F ⊆ Sn is minwise independent if for any set X ⊆ [n] and any x ∈ X, when π is chosen at random in F we have Pr(min{π(X)} = π(x)) = 1 X . In other words we require that all the elements of any fixed set X have an equal chance to become the minimum element of the image of X under π. Our research was motivated by the fact that such a family (under some relaxations) is essential to the algorithm used in practice by the AltaVista web index software to detect and filter nearduplicate documents. However, in the course of our investigation we have discovered interesting and challenging theoretical questions related to this concept – we present the solutions to some of them and we list the rest as open problems.
Visual Cryptography
, 1995
"... In this paper we consider a new type of cryptographic scheme, which can decode concealed images without any cryptographic computations. The scheme is perfectly secure and very easy to implement. We extend it into a visual variant of the k out of n secret sharing problem, in which a dealer provides a ..."
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Cited by 175 (4 self)
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In this paper we consider a new type of cryptographic scheme, which can decode concealed images without any cryptographic computations. The scheme is perfectly secure and very easy to implement. We extend it into a visual variant of the k out of n secret sharing problem, in which a dealer provides a transparency to each one of the n users; any k of them can see the image by stacking their transparencies, but any k  1 of them gain no information about it.
Improved NonApproximability Results
, 1994
"... We indicate strong nonapproximability factors for central problems: N^{1/4} for Max Clique; N^{1/10} for Chromatic Number; and 66/65 for Max 3SAT. Underlying the Max Clique result is a proof system in... ..."
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Cited by 118 (14 self)
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We indicate strong nonapproximability factors for central problems: N^{1/4} for Max Clique; N^{1/10} for Chromatic Number; and 66/65 for Max 3SAT. Underlying the Max Clique result is a proof system in...
ChernoffHoeffding Bounds for Applications with Limited Independence
 SIAM J. Discrete Math
, 1993
"... ChernoffHoeffding bounds are fundamental tools used in bounding the tail probabilities of the sums of bounded and independent random variables. We present a simple technique which gives slightly better bounds than these, and which more importantly requires only limited independence among the rando ..."
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Cited by 104 (10 self)
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ChernoffHoeffding bounds are fundamental tools used in bounding the tail probabilities of the sums of bounded and independent random variables. We present a simple technique which gives slightly better bounds than these, and which more importantly requires only limited independence among the random variables, thereby importing a variety of standard results to the case of limited independence for free. Additional methods are also presented, and the aggregate results are sharp and provide a better understanding of the proof techniques behind these bounds. They also yield improved bounds for various tail probability distributions and enable improved approximation algorithms for jobshop scheduling. The "limited independence" result implies that a reduced amount of randomness and weaker sources of randomness are sufficient for randomized algorithms whose analyses use the ChernoffHoeffding bounds, e.g., the analysis of randomized algorithms for random sampling and oblivious packet routi...
Hardness Of Approximations
, 1996
"... This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems. ..."
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Cited by 101 (4 self)
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This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems.
On the Construction of PseudoRandom Permutations: LubyRackoff Revisited
 JOURNAL OF CRYPTOLOGY
, 1997
"... Luby and Rackoff [27] showed a method for constructing a pseudorandom permutation from a pseudorandom function. The method is based on composing four (or three for weakened security) so called Feistel permutations, each of which requires the evaluation of a pseudorandom function. We reduce somewh ..."
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Cited by 93 (8 self)
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Luby and Rackoff [27] showed a method for constructing a pseudorandom permutation from a pseudorandom function. The method is based on composing four (or three for weakened security) so called Feistel permutations, each of which requires the evaluation of a pseudorandom function. We reduce somewhat the complexity of the construction and simplify its proof of security by showing that two Feistel permutations are sufficient together with initial and final pairwise independent permutations. The revised construction and proof provide a framework in which similar constructions may be brought up and their security can be easily proved. We demonstrate this by presenting some additional adjustments of the construction that achieve the following:  Reduce the success probability of the adversary.  Provide a construction of pseudorandom permutations with large input size using pseudorandom functions with small input size.