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Cryptographic Limitations on Learning Boolean Formulae and Finite Automata
 PROCEEDINGS OF THE TWENTYFIRST ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1989
"... In this paper we prove the intractability of learning several classes of Boolean functions in the distributionfree model (also called the Probably Approximately Correct or PAC model) of learning from examples. These results are representation independent, in that they hold regardless of the syntact ..."
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Cited by 311 (16 self)
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In this paper we prove the intractability of learning several classes of Boolean functions in the distributionfree model (also called the Probably Approximately Correct or PAC model) of learning from examples. These results are representation independent, in that they hold regardless of the syntactic form in which the learner chooses to represent its hypotheses. Our methods reduce the problems of cracking a number of wellknown publickey cryptosystems to the learning problems. We prove that a polynomialtime learning algorithm for Boolean formulae, deterministic finite automata or constantdepth threshold circuits would have dramatic consequences for cryptography and number theory: in particular, such an algorithm could be used to break the RSA cryptosystem, factor Blum integers (composite numbers equivalent to 3 modulo 4), and detect quadratic residues. The results hold even if the learning algorithm is only required to obtain a slight advantage in prediction over random guessing. The techniques used demonstrate an interesting duality between learning and cryptography. We also apply our results to obtain strong intractability results for approximating a generalization of graph coloring.
Approximating Maximum Independent Sets by Excluding Subgraphs
 BIT
, 1992
"... An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known to O(n/(log n)²). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strong simultaneous performance guarantee ..."
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Cited by 137 (10 self)
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An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known to O(n/(log n)²). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strong simultaneous performance guarantee for the clique and coloring problems. The framework
On the Hardness of Approximating the Chromatic Number
, 1993
"... We study the hardness of approximating the chromatic number when the input graph is kcolorable for some fixed k 3. Our main result is that it is NPhard to find a 4coloring of a 3chromatic graph. As an immediate corollary we obtain that it is NPhard to color a kchromatic graph with at most ..."
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Cited by 72 (6 self)
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We study the hardness of approximating the chromatic number when the input graph is kcolorable for some fixed k 3. Our main result is that it is NPhard to find a 4coloring of a 3chromatic graph. As an immediate corollary we obtain that it is NPhard to color a kchromatic graph with at most k + 2bk=3c 1 colors. We also give simple proofs of two results of Lund and Yannakakis [20]. The first result shows that it is NPhard to approximate the chromatic number to within n for some fixed > 0. We point
Learning binary relations and total orders
 In Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science
, 1989
"... Abstract. We study the problem of designing polynomial prediction algorithms for learning binary relations. We study these problems under an online model in which the instances are drawn by the learner, by a helpful teacher, by an adversary or according to a probability distribution on the instance ..."
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Cited by 36 (6 self)
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Abstract. We study the problem of designing polynomial prediction algorithms for learning binary relations. We study these problems under an online model in which the instances are drawn by the learner, by a helpful teacher, by an adversary or according to a probability distribution on the instance space. We represent the relation as an n x m binary matrix, and present results for when the matrix is restricted to have at most k distinct row types, and when it is constrained by requiring that the predicate form a total order. 1
2003) "Financial Communities
"... ABSTRACT. Whereas many studies in finance have examined and established a strong link between stock returns and information, the physical mechanics of this link have been relatively unexplored. With the advent of stock message boards, it has become feasible to look more closely at the group process ..."
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Cited by 3 (0 self)
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ABSTRACT. Whereas many studies in finance have examined and established a strong link between stock returns and information, the physical mechanics of this link have been relatively unexplored. With the advent of stock message boards, it has become feasible to look more closely at the group process by which information impacts prices and vice versa. This paper utilizes a large universe of messages posted to stock market discussion forums to understand how opinions are linked across tickers during small investor discussion. We define a collective information unit, the financial community. These are clusters of tickers sharing and accessing the same information generators. Graph theoretic techniques are used to detect financial communities and to summarize their properties. Community stocks display connectedness, and we find that the greater the connectedness in a financial community, the greater the covariance of returns within the community as opposed to that amongst stocks that are not part of a major financial community. Highly connected stocks, on average, have lower return variance and higher mean returns. Using eigenvector techniques, we detect stocks that are hubs for information flow, using a measure known as centrality. We find that stocks with high centrality scores tend to have greater average covariance with other stocks than those with low scores. Our analysis of connectedness and centrality establishes a link between one arena of the information generation process and stock return correlations. 1.
A Structural View Of Approximation
, 1996
"... The discovery of efficient probabilistically checkable proofs for NP, and their surprising connection to hardness of approximation, has resulted in a quantum leap in our understanding of approximability of many important optimization problems. Yet much of this research has stayed focused on problems ..."
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The discovery of efficient probabilistically checkable proofs for NP, and their surprising connection to hardness of approximation, has resulted in a quantum leap in our understanding of approximability of many important optimization problems. Yet much of this research has stayed focused on problemspecific results. Building on many such results, this dissertation develops frameworks which unify a variety of seemingly different results and thereby highlights the intrinsic structural properties which govern the approximability of optimization problems. Broadly speaking, our work comprises of three distinct parts. In the first part, we develop a structural basis for computationallydefined approximation classes such as APX (constantfactor approximable problems) and polyAPX (polynomialfactor approximable problems). We show that there exist canonical transformations whereby every problem in an approximation class can be "expressed" as a problem in a syntacticallydefined optimization cla...