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15
A PolynomialTime Approximation Algorithm for the Permanent of a Matrix with NonNegative Entries
 Journal of the ACM
, 2004
"... Abstract. We present a polynomialtime randomized algorithm for estimating the permanent of an arbitrary n ×n matrix with nonnegative entries. This algorithm—technically a “fullypolynomial randomized approximation scheme”—computes an approximation that is, with high probability, within arbitrarily ..."
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Cited by 324 (25 self)
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Abstract. We present a polynomialtime randomized algorithm for estimating the permanent of an arbitrary n ×n matrix with nonnegative entries. This algorithm—technically a “fullypolynomial randomized approximation scheme”—computes an approximation that is, with high probability, within arbitrarily small specified relative error of the true value of the permanent. Categories and Subject Descriptors: F.2.2 [Analysis of algorithms and problem complexity]: Nonnumerical
Limits on the Provable Consequences of Oneway Permutations
, 1989
"... We present strong evidence that the implication, "if oneway permutations exist, then secure secret key agreement is possible" is not provable by standard techniques. Since both sides of this implication are widely believed true in real life, to show that the implication is false requires a new m ..."
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Cited by 162 (0 self)
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We present strong evidence that the implication, "if oneway permutations exist, then secure secret key agreement is possible" is not provable by standard techniques. Since both sides of this implication are widely believed true in real life, to show that the implication is false requires a new model. We consider a world where dl parties have access to a black box or a randomly selected permutation. Being totally random, this permutation will be strongly oneway in provable, informationthevretic way. We show that, if P = NP, no protocol for secret key agreement is secure in such setting. Thus, to prove that a secret key greement protocol which uses a oneway permutation as a black box is secure is as hrd as proving F NP. We also obtain, as corollary, that there is an oracle relative to which the implication is false, i.e., there is a oneway permutation, yet secretexchange is impossible. Thus, no technique which relativizes can prove that secret exchange can be based on any oneway permutation. Our results present a general framework for proving statements of the form, "Cryptographic application X is not likely possible based solely on complexity assumption Y." 1
Oracles and Queries that are Sufficient for Exact Learning
 Journal of Computer and System Sciences
, 1996
"... We show that the class of all circuits is exactly learnable in randomized expected polynomial time using weak subset and weak superset queries. This is a consequence of the following result which we consider to be of independent interest: circuits are exactly learnable in randomized expected poly ..."
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Cited by 83 (5 self)
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We show that the class of all circuits is exactly learnable in randomized expected polynomial time using weak subset and weak superset queries. This is a consequence of the following result which we consider to be of independent interest: circuits are exactly learnable in randomized expected polynomial time with equivalence queries and the aid of an NPoracle. We also show that circuits are exactly learnable in deterministic polynomial time with equivalence queries and a \Sigma 3 oracle. The hypothesis class for the above learning algorithms is the class of circuits of largerbut polynomially relatedsize. Also, the algorithms can be adapted to learn the class of DNF formulas with hypothesis class consisting of depth3  formulas (by the work of Angluin [A90], this is optimal in the sense that the hypothesis class cannot be reduced to DNF formulas, i.e. depth2  formulas).
On Counting Independent Sets in Sparse Graphs
, 1998
"... We prove two results concerning approximate counting of independent sets in graphs with constant maximum degree \Delta. The first implies that the Monte Carlo Markov chain technique is likely to fail if \Delta 6. The second shows that no fully polynomial randomized approximation scheme can exist if ..."
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Cited by 62 (12 self)
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We prove two results concerning approximate counting of independent sets in graphs with constant maximum degree \Delta. The first implies that the Monte Carlo Markov chain technique is likely to fail if \Delta 6. The second shows that no fully polynomial randomized approximation scheme can exist if \Delta 25, unless RP = NP. 1 Introduction Counting independent sets in graphs is one of several combinatorial counting problems which have received recent attention. The problem is known to be #Pcomplete, even for low degree graphs [3]. On the other hand, it has been shown that, for graphs of maximum degree \Delta = 4, randomized approximate counting is possible [7, 3]. This success has been achieved using the Monte Carlo Markov chain method to construct a fully polynomial randomized approximation scheme (fpras). This has led to a natural question as to how far this success might extend. Here we consider in more detail this question of counting independent sets in graphs with constant m...
OneWay Functions are Essential for NonTrivial ZeroKnowledge(Extended Abstract)
 IN PROC. 2ND ISRAEL SYMP. ON THEORY OF COMPUTING AND SYSTEMS (ISTCS93), IEEE COMPUTER
, 1993
"... It was known that if oneway functions exist, then there are zeroknowledge proofs for every language in PSPACE. We prove that unless very weak oneway functions exist, ZeroKnowledge proofs can be given only for languages in BPP. For averagecase definitions of BPP we prove an analogous result und ..."
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Cited by 37 (10 self)
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It was known that if oneway functions exist, then there are zeroknowledge proofs for every language in PSPACE. We prove that unless very weak oneway functions exist, ZeroKnowledge proofs can be given only for languages in BPP. For averagecase definitions of BPP we prove an analogous result under the assumption that uniform oneway functions do not exist. Thus, very loosely speaking, zeroknowledge is either useless (exists only for "easy" languages), or universal (exists for every provable language).
Limits on the Provable Consequences of Oneway Functions
, 1989
"... This technical point will prevent the reader from suspecting any measuretheoretic fallacy. ..."
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Cited by 32 (1 self)
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This technical point will prevent the reader from suspecting any measuretheoretic fallacy.
Expected Solution Quality
 In Proceedings of the 14th International Joint Conference on Artificial Intelligence
, 1995
"... This paper presents the Expected Solution Quality (esq) method for statistically characterizing scheduling problems and the performance of schedulers. The esq method is demonstrated by applying it to a practical telescope scheduling problem. The method addresses the important and difficult issue of ..."
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Cited by 8 (2 self)
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This paper presents the Expected Solution Quality (esq) method for statistically characterizing scheduling problems and the performance of schedulers. The esq method is demonstrated by applying it to a practical telescope scheduling problem. The method addresses the important and difficult issue of how to meaningfully evaluate the performance of a scheduler on a constrained optimization problem for which an optimal solution is not known. At the heart of esq is a Monte Carlo algorithm that estimates a problem's probability density function with respect to solution quality. This "quality density function" provides a useful characterization of a scheduling problem, and it also provides a background against which scheduler performance can be meaningfully evaluated. esq provides a unitless measure that combines both schedule quality and the amount of time to generate a schedule. 1 Introduction This paper presents a method for statistically characterizing both scheduling problems and the p...
The Computational Complexity of Counting
 Proceedings of the International Congress of Mathematicians
, 1994
"... The complexity theory of counting contrasts intriguingly with that of existence or optimisation. 1 Counting versus existence The branch of theoretical computer science known as computational complexity is concerned with quantifying the computational resources required to achieve specified computa ..."
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Cited by 6 (1 self)
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The complexity theory of counting contrasts intriguingly with that of existence or optimisation. 1 Counting versus existence The branch of theoretical computer science known as computational complexity is concerned with quantifying the computational resources required to achieve specified computational goals. Classically, the goal is often to decide the existence of a certain combinatorial structure, for example, whether a given graph G contains a Hamilton cycle. Alternatively, the goal might be to find an occurrence of the structure that is optimal with respect to a certain measure; in the context of the structure "Hamilton cycle," the notorious Travelling Salesman Problem may be cited as an example. Less well studied, and somewhat less well understood, are counting problems, such as determining how many Hamilton cycles a graph G contains. In some areas, such as statistical physics, counting problems arise directly; in many others they appear in the guise of discrete approximations...
A SubExponential Time Algorithm for Approximating the Number of Solutions to a Multidimensional Knapsack Problem
, 1993
"... We describe an O(2 O(r p n(log n) 5=2 ) ffl \Gamma2 ) time randomized algorithm which estimates the number of feasible solutions of a multidimensional knapsack problem within 1 \Sigma ffl of the exact number. (Here r is the number of constraints and n is the number of integer variables.) The ..."
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Cited by 2 (0 self)
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We describe an O(2 O(r p n(log n) 5=2 ) ffl \Gamma2 ) time randomized algorithm which estimates the number of feasible solutions of a multidimensional knapsack problem within 1 \Sigma ffl of the exact number. (Here r is the number of constraints and n is the number of integer variables.) The 1 algorithm uses a Markov chain to generate an almost uniform random solution to the problem. 1 Introduction An instance of the multidimensional knapsack problem in n variables is defined by a nonnegative r \Theta n integer matrix A, a vector d 2 ZZ n , and a vector b 2 ZZ r . We assume, to avoid trivialities, that all components of b and d are positive. The set of feasible solutions, K, is then K = fx 2 ZZ n j Ax b; 0 x dg: (1) We will assume, without loss that d j min i b i =a ij . Let d max = max j d j . In particular, if d max = 1, we have the zeroone multidimensional knapsack problem, and if r = 1 we have the knapsack problem. The optimization problem consists of fi...
The Complexity of Estimating MinEntropy
, 2012
"... Goldreich, Sahai, and Vadhan (CRYPTO 1999) proved that the promise problem for estimating the Shannon entropy of a distribution sampled by a given circuit is NISZKcomplete. We consider the analogous problem for estimating the minentropy and prove that it is SBPcomplete, even when restricted to 3l ..."
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Cited by 2 (1 self)
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Goldreich, Sahai, and Vadhan (CRYPTO 1999) proved that the promise problem for estimating the Shannon entropy of a distribution sampled by a given circuit is NISZKcomplete. We consider the analogous problem for estimating the minentropy and prove that it is SBPcomplete, even when restricted to 3local samplers. For logarithmicspace samplers, we observe that this problem is NPcomplete by a result of Lyngsø and Pedersen on hidden Markov models (JCSS 2002). 1