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159
A New Kind of Science
, 2002
"... “Somebody says, ‘You know, you people always say that space is continuous. How do you know when you get to a small enough dimension that there really are enough points in between, that it isn’t just a lot of dots separated by little distances? ’ Or they say, ‘You know those quantum mechanical amplit ..."
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Cited by 502 (0 self)
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“Somebody says, ‘You know, you people always say that space is continuous. How do you know when you get to a small enough dimension that there really are enough points in between, that it isn’t just a lot of dots separated by little distances? ’ Or they say, ‘You know those quantum mechanical amplitudes you told me about, they’re so complicated and absurd, what makes you think those are right? Maybe they aren’t right. ’ Such remarks are obvious and are perfectly clear to anybody who is working on this problem. It does not do any good to point this out.” —Richard Feynman [1, p.161]
Finding Hard Instances of the Satisfiability Problem: A Survey
, 1997
"... . Finding sets of hard instances of propositional satisfiability is of interest for understanding the complexity of SAT, and for experimentally evaluating SAT algorithms. In discussing this we consider the performance of the most popular SAT algorithms on random problems, the theory of average case ..."
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Cited by 113 (1 self)
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. Finding sets of hard instances of propositional satisfiability is of interest for understanding the complexity of SAT, and for experimentally evaluating SAT algorithms. In discussing this we consider the performance of the most popular SAT algorithms on random problems, the theory of average case complexity, the threshold phenomenon, known lower bounds for certain classes of algorithms, and the problem of generating hard instances with solutions.
Relations Between Average Case Complexity and Approximation Complexity (Extended Abstract)
 In Proceedings of the 34th Annual ACM Symposium on Theory of Computing
, 2002
"... We investigate relations between average case complexity and the complexity of approximation. Our preliminary findings indicate that this is a research direction that leads to interesting insights. Under the assumption that refuting 3SAT is hard on average on a natural distribution, we derive hardne ..."
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Cited by 87 (9 self)
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We investigate relations between average case complexity and the complexity of approximation. Our preliminary findings indicate that this is a research direction that leads to interesting insights. Under the assumption that refuting 3SAT is hard on average on a natural distribution, we derive hardness of approximation results for min bisection, dense ksubgraph, max bipartite clique and the 2catalog segmentation problem. No NPhardness of approximation results are currently known for these problems.
Randomness vs. Time: Derandomization under a uniform assumption
 Journal of Computer and System Sciences
, 1998
"... We prove that if BPP 6= EXP, then every problem in BPP can be solved deterministically in subexponential time on almost every input ( on every samplable ensemble for infinitely many input sizes). This is the first derandomization result for BPP based on uniform, noncryptographic hardness assumptions ..."
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Cited by 77 (11 self)
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We prove that if BPP 6= EXP, then every problem in BPP can be solved deterministically in subexponential time on almost every input ( on every samplable ensemble for infinitely many input sizes). This is the first derandomization result for BPP based on uniform, noncryptographic hardness assumptions. It implies the following gap in the averageinstance complexities of problems in BPP : either these complexities are always subexponential or they contain arbitrarily large exponential functions. We use a construction of a small "pseudorandom " set of strings from a "hard function" in EXP which is identical to that used in the analogous nonuniform results of [21, 3]. However, previous proofs of correctness assume the "hard function" is not in P=poly. They give a nonconstructive argument that a circuit distinguishing the pseudorandom strings from truly random strings implies that a similarlysized circuit exists computing the "hard function". Our main technical contribution is to show ...
On the spheredecoding algorithm I. Expected complexity
 IEEE Trans. Sig. Proc
, 2005
"... Abstract—The problem of finding the leastsquares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The ..."
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Cited by 76 (5 self)
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Abstract—The problem of finding the leastsquares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The problem is equivalent to finding the closest lattice point to a given point and is known to be NPhard. In communications applications, however, the given vector is not arbitrary but rather is an unknown lattice point that has been perturbed by an additive noise vector whose statistical properties are known. Therefore, in this paper, rather than dwell on the worstcase complexity of the integer leastsquares problem, we study its expected complexity, averaged over the noise and over the lattice. For the “sphere decoding” algorithm of Fincke and Pohst, we find a closedform expression for the expected complexity, both for the infinite and finite lattice.
Average Case Completeness
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1991
"... We explain and advance Levin's theory of average case completeness. In particular, we exhibit examples of problems complete in the average case and prove a limitation on the power of deterministic reductions. ..."
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Cited by 71 (2 self)
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We explain and advance Levin's theory of average case completeness. In particular, we exhibit examples of problems complete in the average case and prove a limitation on the power of deterministic reductions.
Parameterized Complexity: A Framework for Systematically Confronting Computational Intractability
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1997
"... In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixedparameter tractability techniques can deliver practical algorithms in two different ways: (1) by ..."
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Cited by 67 (15 self)
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In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixedparameter tractability techniques can deliver practical algorithms in two different ways: (1) by providing useful exact algorithms for small parameter ranges, and (2) by providing guidance in the design of heuristic algorithms. In particular, we describe an improved FPT kernelization algorithm for Vertex Cover, a practical FPT algorithm for the Maximum Agreement Subtree (MAST) problem parameterized by the number of species to be deleted, and new general heuristics for these problems based on FPT techniques. In the course of making this overview, we also investigate some structural and hardness issues. We prove that an important naturally parameterized problem in artificial intelligence, STRIPS Planning (where the parameter is the size of the plan) is complete for W [1]. As a corollary, this implies that kStep Reachability for Petri Nets is complete for W [1]. We describe how the concept of treewidth can be applied to STRIPS Planning and other problems of logic to obtain FPT results. We describe a surprising structural result concerning the top end of the parameterized complexity hierarchy: the naturally parameterized Graph kColoring problem cannot be resolved with respect to XP either by showing membership in XP, or by showing hardness for XP without settling the P = NP question one way or the other.
On Yao’s XOR lemma
 Electronic Colloquium on Computational Complexity
, 1995
"... Abstract. A fundamental lemma of Yao states that computational weakunpredictability of Boolean predicates is amplified when the results of several independent instances are XOR together. We survey two known proofs of Yao’s Lemma and present a third alternative proof. The third proof proceeds by firs ..."
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Cited by 59 (7 self)
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Abstract. A fundamental lemma of Yao states that computational weakunpredictability of Boolean predicates is amplified when the results of several independent instances are XOR together. We survey two known proofs of Yao’s Lemma and present a third alternative proof. The third proof proceeds by first proving that a function constructed by concatenating the values of the original function on several independent instances is much more unpredictable, with respect to specified complexity bounds, than the original function. This statement turns out to be easier to prove than the XORLemma. Using a result of Goldreich and Levin (1989) and some elementary observation, we derive the XORLemma.
Extracting randomness from samplable distributions
 In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... The standard notion of a randomness extractor is a procedure which converts any weak source of randomness into an almost uniform distribution. The conversion necessarily uses a small amount of pure randomness, which can be eliminated by complete enumeration in some, but not all, applications. Here, ..."
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Cited by 58 (8 self)
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The standard notion of a randomness extractor is a procedure which converts any weak source of randomness into an almost uniform distribution. The conversion necessarily uses a small amount of pure randomness, which can be eliminated by complete enumeration in some, but not all, applications. Here, we consider the problem of deterministically converting a weak source of randomness into an almost uniform distribution. Previously, deterministic extraction procedures were known only for sources satisfying strong independence requirements. In this paper, we look at sources which are samplable, i.e. can be generated by an efficient sampling algorithm. We seek an efficient deterministic procedure that, given a sample from any samplable distribution of sufficiently large minentropy, gives an almost uniformly distributed output. We explore the conditions under which such deterministic extractors exist. We observe that no deterministic extractor exists if the sampler is allowed to use more computational resources than the extractor. On the other hand, if the extractor is allowed (polynomially) more resources than the sampler, we show that deterministic extraction becomes possible. This is true unconditionally in the nonuniform setting (i.e., when the extractor can be computed by a small circuit), and (necessarily) relies on complexity assumptions in the uniform setting. One of our uniform constructions is as follows: assuming that there are problems in���ÌÁÅ�ÇÒthat are not solvable by subexponentialsize circuits with¦� gates, there is an efficient extractor that transforms any samplable distribution of lengthÒand minentropy Ò into an output distribution of length ÇÒ, whereis any sufficiently small constant. The running time of the extractor is polynomial inÒand the circuit complexity of the sampler. These extractors are based on a connection be