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A Threshold of ln n for Approximating Set Cover
 JOURNAL OF THE ACM
, 1998
"... Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhar ..."
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Cited by 619 (6 self)
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Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhard. We prove that (1 \Gamma o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low order terms) between the ratio of approximation achievable by the greedy algorithm (which is (1 \Gamma o(1)) ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log 2 n)=2 ' 0:72 lnn. For max kcover we show an approximation threshold of (1 \Gamma 1=e) (up to low order terms), under the assumption that P != NP .
On Defining Proofs of Knowledge
, 1998
"... The notion of a "proof of knowledge," suggested by Gold wasset, Micali and Rackoff, has been used in many works as a tool for the construction of cryptographic protocols and other schemes. Yet the commonly cited formalizations of this notion are unsatisfactory and in particular inadequate for s ..."
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Cited by 139 (23 self)
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The notion of a "proof of knowledge," suggested by Gold wasset, Micali and Rackoff, has been used in many works as a tool for the construction of cryptographic protocols and other schemes. Yet the commonly cited formalizations of this notion are unsatisfactory and in particular inadequate for some of the applications in which they are used. Consequently,
Making Games Short (Extended Abstract)
"... We study the complexity of refereed games, in which two computationally unlimited players play against each other, and a polynomial time referee monitors the game and announces the winner. The players may exchange messages with the referee in private, resulting in a game of perfect recall but incomp ..."
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Cited by 8 (0 self)
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We study the complexity of refereed games, in which two computationally unlimited players play against each other, and a polynomial time referee monitors the game and announces the winner. The players may exchange messages with the referee in private, resulting in a game of perfect recall but incomplete information. We show that any EXPTIME statement can be efficiently transformed into a refereed game in which if the statement is true, the first player wins with overwhelming probability, and if the statement is false, the second player wins with overwhelming probability. We also prove matching PSPACE upper and lower bounds on the complexity of statements that have refereed games that take one round of communication.
A Relationship between OneWayness and Correlation Intractability
 Proceedings of PKC'99
, 1999
"... The notion of correlation intractability was introduced in an attempt to capture the "unpredictability " property of random oracles: It is assumed that if R is a random oracle then it is infeasible to find an input x such that the inputoutput pair (x, R(x)) has some desired property. It is desirabl ..."
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Cited by 7 (0 self)
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The notion of correlation intractability was introduced in an attempt to capture the "unpredictability " property of random oracles: It is assumed that if R is a random oracle then it is infeasible to find an input x such that the inputoutput pair (x, R(x)) has some desired property. It is desirable that a plausible construction of correlation intractable function ensembles will be provided since the unpredictability property is often useful to design many cryptographic applications in the random oracle model. However, no plausibility result has been proposed. In this paper, we show that proving the implication, "if uniform oneway functions exist then uniform correlation intractable function ensembles exist", is as hard as proving a claim regarding the triviality of 3round auxiliaryinput zeroknowledge ArthurMerlin proofs without making any assumptions. We believe that it is unlikely that one can prove it unconditionally. Therefore, we conclude that it will be di#cult to construct...
Interactive Verification of coNP Statements
"... n tosses of V ). We shall describe the LFKN interactive proof for every language in coNP. The original version of the protocol involved special properties of the permanent function (see [10] regarding the definition and the complexity of this important function), and we present here a later version ..."
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n tosses of V ). We shall describe the LFKN interactive proof for every language in coNP. The original version of the protocol involved special properties of the permanent function (see [10] regarding the definition and the complexity of this important function), and we present here a later version of the protocol that more easily extends to later PCPs that we will present. We remark that Shamir [8] extended the results of LFKN and designed interactive proofs for every language in PSPACE. However, we choose not to present the Shamir protocol as its additional complication are not needed for our later PCP constructions. An informative and entertaining (though possibly biased) presentation of the exciting research activities in the end of 1989, beginning of 1990, appears in [2]. 1.2 Soundness not bounded away from 1 It suffices to design an interactive proof system for a coNPcomplete language, and then by polynomial time reductions we have interactive proofs for all coNP language