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Proof verification and hardness of approximation problems
 IN PROC. 33RD ANN. IEEE SYMP. ON FOUND. OF COMP. SCI
, 1992
"... We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probabilit ..."
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Cited by 797 (39 self)
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We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof " with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [6] whose verifiers examine a nonconstant number of bits in the proof (though this number is a very slowly growing function of the input length). As a consequence we prove that no MAX SNPhard problem has a polynomial time approximation scheme, unless NP=P. The class MAX SNP was defined by Papadimitriou and Yannakakis [82] and hard problems for this class include vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige, Goldwasser, Lovász, Safra and Szegedy [42], and Arora and Safra [6] and shows that there exists a positive ɛ such that approximating the maximum clique size in an Nvertex graph to within a factor of N ɛ is NPhard.
Hybrid Dynamic Data Race Detection
, 2003
"... We present a new method for dynamically detecting potential data races in multithreaded programs. Our method improves on the state of the art in accuracy, in usability, and in overhead. We improve accuracy by combining two previously known race detection techniques — locksetbased detection and happ ..."
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Cited by 165 (0 self)
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We present a new method for dynamically detecting potential data races in multithreaded programs. Our method improves on the state of the art in accuracy, in usability, and in overhead. We improve accuracy by combining two previously known race detection techniques — locksetbased detection and happensbeforebased detection — to obtain fewer false positives than locksetbased detection alone. We enhance usability by reporting more information about detected races than any previous dynamic detector. We reduce overhead compared to previous detectors — particularly for large applications such as Web application servers — by not relying on happensbefore detection alone, by introducing a new optimization to discard redundant information, and by using a “two phase” approach to identify errorprone program points and then focus instrumentation on those points. We justify our claims by presenting the results of applying our tool to a range of Java programs, including the widelyused Web application servers Resin and Apache Tomcat. Our paper also presents a formalization of locksetbased and happensbeforebased approaches in a common framework, allowing us to prove a “folk theorem” that happensbefore detection reports fewer false positives than locksetbased detection (but can report more false negatives), and to prove that two key optimizations are correct.
Efficient probabilistically checkable proofs and applications to approximation
 IN PROCEEDINGS OF STOC93
, 1993
"... ..."
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 117 (5 self)
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This chapter is a selfcontained survey of recent results about the hardness of approximating NPhard optimization problems.
Two Formal Analyses of Attack Graphs
 IN PROCEEDINGS OF THE 15TH COMPUTER SECURITY FOUNDATION WORKSHOP
, 2002
"... An attack graph is a succinct representation of all paths through a system that end in a state where an intruder has successfully achieved his goal. Today Red Teams determine the vulnerability of networked systems by drawing gigantic attack graphs by hand. Constructing attack graphs by hand is tedio ..."
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Cited by 90 (2 self)
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An attack graph is a succinct representation of all paths through a system that end in a state where an intruder has successfully achieved his goal. Today Red Teams determine the vulnerability of networked systems by drawing gigantic attack graphs by hand. Constructing attack graphs by hand is tedious, errorprone, and impractical for large systems. By viewing an attack as a violation of a safety property, we can use offtheshelf model checking technology to produce attack graphs automatically: a successful path from the intruder's viewpoint is a counterexample produced by the model checker. In this paper we present an algorithm for generating attack graphs using model checking as a subroutine. Security analysts use attack graphs for detection, defense and forensics. In this paper we present a minimization analysis technique that allows analysts to decide which minimal set of security measures would guarantee the safety of the system. We provide a formal characterization of this problem: we prove that it is polynomially equivalent to the minimum hitting set problem and we present a greedy algorithm with provable bounds. We also present a reliability analysis technique that allows analysts to perform a simple costbenefit tradeoff depending on the likelihoods of attacks. By interpreting attack graphs as Markov Decision Processes we can use the value iteration algorithm to compute the probabilities of intruder success for each attack the graph.
Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems
, 1992
"... The definition of the class NP [Coo71, Lev73] highlights the problem of verification of proofs as one of central interest to theoretical computer science. Recent efforts have shown that the efficiency of the verification can be greatly improved by allowing the verifier access to random bits and acce ..."
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Cited by 65 (8 self)
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The definition of the class NP [Coo71, Lev73] highlights the problem of verification of proofs as one of central interest to theoretical computer science. Recent efforts have shown that the efficiency of the verification can be greatly improved by allowing the verifier access to random bits and accepting probabilistic guarantees from the verifier [BFL91, BFLS91, FGL + 91, AS92]. We improve upon the efficiency of the proof systems developed above and obtain proofs which can be verified probabilistically by examining only a constant number of (randomly chosen) bits of the proof. The efficiently verifiable proofs constructed here rely on the structural properties of lowdegree polynomials. We explore the properties of these functions by examining some simple and basic questions about them. We consider questions of the form: • (testing) Given an oracle for a function f, is f close to a lowdegree polynomial? • (correcting) Let f be close to a lowdegree polynomial g, is it possible to efficiently reconstruct the value of g on any given input using an oracle for f? 2 The questions described above have been raised before in the context of coding theory as the problems of errordetecting and errorcorrecting of codes. More recently
The complexity of approximating a nonlinear program
 IBM Research Report RC 17831
, 1992
"... We consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope. We show that there is no polynomial time approximation algorithm for this problem, even one with a very poor guarantee, unless P = NP. We show that even when the polynomial is quadratic (i.e. quadra ..."
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Cited by 58 (3 self)
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We consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope. We show that there is no polynomial time approximation algorithm for this problem, even one with a very poor guarantee, unless P = NP. We show that even when the polynomial is quadratic (i.e. quadratic programming) there is no polynomial time approximation unless NP is contained in quasipolynomial time. Our results rely on recent advances in the theory of interactive proof systems. They exemplify an interesting interplay of discrete and continuous mathematics—using a combinatorial argument to get a hardness result for a continuous optimization problem.
Logical Definability of NP Optimization Problems
 Information and Computation
, 1994
"... : We investigate here NP optimization problems from a logical definability standpoint. We show that the class of optimization problems whose optimum is definable using firstorder formulae coincides with the class of polynomially bounded NP optimization problems on finite structures. After this, we ..."
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Cited by 40 (2 self)
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: We investigate here NP optimization problems from a logical definability standpoint. We show that the class of optimization problems whose optimum is definable using firstorder formulae coincides with the class of polynomially bounded NP optimization problems on finite structures. After this, we analyze the relative expressive power of various classes of optimization problems that arise in this framework. Some of our results show that logical definability has different implications for NP maximization problems than it has for NP minimization problems, in terms of both expressive power and approximation properties. To appear in Information and Computation. Research partially supported by NSF Grants CCR8905038 and CCR9108631. y email addresses: kolaitis@cse.ucsc.edu, thakur@cse.ucsc.edu z supersedes Technical report UCSCCRL9048 1 Introduction and Summary of Results It is well known that optimization problems had a major influence on the development of the theory of NPco...
On approximation preserving reductions: Complete problems and robust measures
, 1987
"... We investigate the wellknown anomalous differences in the approximability properties of NPcomplete optimization problems. We define a notion of polynomial time reduction between optimization problems, and introduce conditions guaranteeing that such reductions preserve various types of approximate ..."
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Cited by 39 (0 self)
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We investigate the wellknown anomalous differences in the approximability properties of NPcomplete optimization problems. We define a notion of polynomial time reduction between optimization problems, and introduce conditions guaranteeing that such reductions preserve various types of approximate solutions. We then prove that a weighted version of the satisfiability problem, the traveling salesperson problem, and the zeroone integer programming problem are in a strong sense approximation complete for the class of NP minimization problems. Finally, we discuss the reasons that cause the standard relative error approximation quality measure to break down in computationally simple problem transformations, and give a general construction for producing quality measures that are more robust with respect to an arbitrary given class of invertible transformations. 1