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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 779 (5 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 inadeq ..."
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Cited by 171 (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 13 (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 i ..."
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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...
A BlackBox Query Complexity of polynomial summation
, 2007
"... For any given Boolean formula φ(x1,..., xn), one can efficiently construct (using arithmetization) a lowdegree polynomial p(x1,..., xn) that agrees with φ over all points in the Boolean cube {0, 1} n; the constructed polynomial p can be interpreted as a polynomial over an arbitrary field F. The pro ..."
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For any given Boolean formula φ(x1,..., xn), one can efficiently construct (using arithmetization) a lowdegree polynomial p(x1,..., xn) that agrees with φ over all points in the Boolean cube {0, 1} n; the constructed polynomial p can be interpreted as a polynomial over an arbitrary field F. The problem #SAT (of counting the number of satisfying assignments of φ) thus reduces to the polynomial summation � x∈{0,1} n p(x). Motivated by this connection, we study the query complexity of the polynomial summation problem: Given (oracle access to) a polynomial p(x1,..., xn), compute � x∈{0,1} n p(x). Obviously, querying p at all 2n points in {0, 1} n suffices. Is there a field F such that, for every polynomial p ∈ F[x1,..., xn], the sum � x∈{0,1} n p(x) can be computed using fewer than 2n queries from Fn? We show that the simple upper bound 2n is in fact tight for any field F in the blackbox model where one has only oracle access to the polynomial p. We prove these lower bounds for the adaptive query model where the next query can depend on the values of p at previously queried points. Our lower bounds hold even for polynomials that have degree at most 2 in each variable. In contrast, for polynomials that have degree at most 1 in each variable (i.e., multilinear polynomials), we observe that a single query is sufficient over any field of characteristic other than 2. We also give query lower bounds for certain extensions of the polynomial summation problem.