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38
Which Problems Have Strongly Exponential Complexity?
 Journal of Computer and System Sciences
, 1998
"... For several NPcomplete problems, there have been a progression of better but still exponential algorithms. In this paper, we address the relative likelihood of subexponential algorithms for these problems. We introduce a generalized reduction which we call SubExponential Reduction Family (SERF) t ..."
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Cited by 128 (5 self)
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For several NPcomplete problems, there have been a progression of better but still exponential algorithms. In this paper, we address the relative likelihood of subexponential algorithms for these problems. We introduce a generalized reduction which we call SubExponential Reduction Family (SERF) that preserves subexponential complexity. We show that CircuitSAT is SERFcomplete for all NPsearch problems, and that for any fixed k, kSAT, kColorability, kSet Cover, Independent Set, Clique, Vertex Cover, are SERFcomplete for the class SNP of search problems expressible by second order existential formulas whose first order part is universal. In particular, subexponential complexity for any one of the above problems implies the same for all others. We also look at the issue of proving strongly exponential lower bounds for AC 0 ; that is, bounds of the form 2 \Omega\Gamma n) . This problem is even open for depth3 circuits. In fact, such a bound for depth3 circuits with even l...
The Complexity Of Propositional Proofs
 Bulletin of Symbolic Logic
, 1995
"... This paper of Tseitin is a landmark as the first to give nontrivial lower bounds for propositional proofs; although it predates the first papers on ..."
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Cited by 105 (2 self)
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This paper of Tseitin is a landmark as the first to give nontrivial lower bounds for propositional proofs; although it predates the first papers on
An Improved Exponentialtime Algorithm for kSAT
, 1998
"... We propose and analyze a simple new randomized algorithm, called ResolveSat, for finding satisfying assignments of Boolean formulas in conjunctive normal form. The algorithm consists of two stages: a preprocessing stage in which resolution is applied to enlarge the set of clauses of the formula, ..."
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Cited by 88 (5 self)
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We propose and analyze a simple new randomized algorithm, called ResolveSat, for finding satisfying assignments of Boolean formulas in conjunctive normal form. The algorithm consists of two stages: a preprocessing stage in which resolution is applied to enlarge the set of clauses of the formula, followed by a search stage that uses a simple randomized greedy procedure to look for a satisfying assignment. We show that, for each k, the running time of ResolveSat on a kCNF formula is significantly better than 2 n , even in the worst case. In particular, we show that the algorithm finds a satisfying assignment of a general satisfiable 3CNF in time O(2 :448n ) with high probability; where the best previous algorithm [13] has running time O(2 :562n ). We obtain a better upper bound of 2 (2 ln 2\Gamma1)n+o(n) = O(2 0:387n ) for 3CNF that have exactly one satisfying assignment (unique kSAT). For each k, the bounds for general kCNF are the best currently known for ...
Satisfiability Coding Lemma
 In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, FOCS’97
, 1997
"... We present and analyze two simple algorithms for finding satisfying assignments of kCNFs (Boolean formulae in conjunctive normal form with at most k literals per clause). The first is a randomized algorithm which, with probability approaching 1, finds a satisfying assignment of a satisfiable kCNF ..."
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Cited by 64 (6 self)
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We present and analyze two simple algorithms for finding satisfying assignments of kCNFs (Boolean formulae in conjunctive normal form with at most k literals per clause). The first is a randomized algorithm which, with probability approaching 1, finds a satisfying assignment of a satisfiable kCNF formula F in time O(n 2 jF j2 n\Gamman=k ). The second algorithm is deterministic, and its running time approaches 2 n\Gamman=2k for large n and k. The randomized algorithm is the best known algorithm for k ? 3; the deterministic algorithm is the best known deterministic algorithm for k ? 4. We also show an \Omega\Gamma n 1=4 2 p n ) lower bound on the size of depth 3 circuits of AND and OR gates computing the parity function. This bound is tight up to a constant factor. The key idea used in these upper and lower bounds is what we call the Satisfiability Coding Lemma. This basic lemma shows how to encode satisfying solutions of a kCNF succinctly. 1 Introduction The problem of ...
The History and Status of the P versus NP Question
, 1992
"... this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the re ..."
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Cited by 50 (0 self)
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this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the realization that certain problems are algorithmically unsolvable. At around this time, forerunners of the programmable computing machine were beginning to appear. As mathematicians contemplated the practical capabilities and limitations of such devices, computational complexity theory emerged from the theory of algorithmic unsolvability. Early on, a particular type of computational task became evident, where one is seeking an object which lies
The Permanent Requires Large Uniform Threshold Circuits
, 1999
"... We show that the permanent cannot be computed by uniform constantdepth threshold circuits of size T (n) for any function T such that for all k, T (k) (n) = o(2 n ). More generally, we show that any problem that is hard for the complexity class C=P requires circuits of this size (on the unif ..."
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Cited by 27 (8 self)
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We show that the permanent cannot be computed by uniform constantdepth threshold circuits of size T (n) for any function T such that for all k, T (k) (n) = o(2 n ). More generally, we show that any problem that is hard for the complexity class C=P requires circuits of this size (on the uniform constantdepth threshold circuit model). In particular, this lower bound applies to any problem that is hard for the complexity classes PP or #P.
SizeDepth Tradeoffs for Threshold Circuits (Extended Abstract)
 SIAM J. Comput
, 1997
"... Russell Impagliazzo , Ramamohan Paturi , Michael E. Saks y Abstract The following sizedepth tradeoff for threshold circuits is obtained: any threshold circuit of depth d that computes the parity function on n variables must have at least n 1+c` \Gammad edges, where constants c ? 0 and ..."
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Cited by 21 (1 self)
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Russell Impagliazzo , Ramamohan Paturi , Michael E. Saks y Abstract The following sizedepth tradeoff for threshold circuits is obtained: any threshold circuit of depth d that computes the parity function on n variables must have at least n 1+c` \Gammad edges, where constants c ? 0 and ` 3 are constants independent of n and d. Previously known constructions show that up to the choice of c and ` this bound is best possible. In particular, the lower bound implies an affirmative answer to the conjecture of Paturi and Saks that a bounded depth threshold circuit that computes parity requires a superlinear number of edges. This is the first super linear lower bound for an explicit function that holds for any fixed depth, and the first that applies to threshold circuits with unrestricted weights. The tradeoff is obtained as a consequence of a general restriction theorem for threshold circuits with a small number of edges: For any threshold circuit with n inputs, de...
Counting Hierarchies: Polynomial Time And Constant Depth Circuits
, 1990
"... In the spring of 1989, Seinosuke Toda of the University of ElectroCommunications in Tokyo, Japan, proved that the polynomial hierarchy is contained in P PP [To89]. In this Structural Complexity Column, we will briefly review Toda's result, and explore how it relates to other topics of interest i ..."
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Cited by 19 (4 self)
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In the spring of 1989, Seinosuke Toda of the University of ElectroCommunications in Tokyo, Japan, proved that the polynomial hierarchy is contained in P PP [To89]. In this Structural Complexity Column, we will briefly review Toda's result, and explore how it relates to other topics of interest in computer science. In particular, we will introduce the reader to The Counting Hierarchy: a hierarchy of complexity classes contained in PSPACE and containing the Polynomial Hierarchy. Threshold Circuits: circuits constructed of MAJORITY gates; this notion of circuit is being studied not only by complexity theoreticians, but also by researchers in an active subfield of AI studying "neural networks". Along the way, we'll review the important notion of an operator on a complexity class. 1. The Counting Hierarchy, and Operators on Complexity Classes The counting hierarchy was defined in [Wa86] and independently by Parberry and Schnitger in [PS88]. (The motivation for [Wa86] was the desir...
On the Correlation of Symmetric Functions
 MATH. SYSTEMS THEORY
, 1996
"... The correlation between two Boolean functions of n inputs is defined as the number of times the functions agree minus the number of times they disagree, all divided by 2 n . In this paper, we compute, in closed form, the correlation between any two symmetric Boolean functions. As a consequence of ..."
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Cited by 17 (6 self)
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The correlation between two Boolean functions of n inputs is defined as the number of times the functions agree minus the number of times they disagree, all divided by 2 n . In this paper, we compute, in closed form, the correlation between any two symmetric Boolean functions. As a consequence of our main result, we get that every symmetric Boolean function having an odd period has an exponentially small correlation (in n) with the parity function. This improves a result of Smolensky [12] restricted to symmetric Boolean functions: the correlation between parity and any circuit consisting of a Mod q gate over ANDgates of small fanin, where q is odd and the function computed by the sum of the ANDgates is symmetric, is bounded by 2 \Gamma\Omega\Gamma n) . In addition, we find that for a large class of symmetric functions the correlation with parity is identically zero for infinitely many n. We characterize exactly those symmetric Boolean functions having this property.
Analog versus Discrete Neural Networks
 Neural Computation
, 1996
"... We show that neural networks with threetimes continuously differentiable activation functions are capable of computing a certain family of nbit Boolean functions with two gates, whereas networks composed of binary threshold functions require at least \Omega\Gammaast n) gates. Thus, for a large cla ..."
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Cited by 17 (2 self)
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We show that neural networks with threetimes continuously differentiable activation functions are capable of computing a certain family of nbit Boolean functions with two gates, whereas networks composed of binary threshold functions require at least \Omega\Gammaast n) gates. Thus, for a large class of activation functions, analog neural networks can be more powerful than discrete neural networks, even when computing Boolean functions. 1 Introduction. Artificial neural networks have become a popular model for machine learning and many results have been obtained regarding their application to practical problems. Typically, the network is trained to encode complex associations between inputs and outputs during supervised training cycles, where the associations are encoded by the weights of the network. Once trained, the network will compute an input/output mapping which (hopefully) is a good approximation of the original mapping. 1 Partially supported by NSF Grant CCR9114545 In thi...