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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 249 (9 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...
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 116 (7 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 IEEE Conference on Foundations of Computer Science
, 1997
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Exponential lower bounds for depth 3 boolean circuits
 Preliminary version in 29th annual ACM Symposium on Theory of Computing, 96–91
, 2000
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An improved exponentialtime algorithm for kSAT
 PPZ99] Ramamohan Paturi, Pavel Pudlák, and Francis
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Abstract Satisfiability Coding Lemma
"... We present and analyze two simple algorithms for finding satisfying assignments ¢ ofCNFs (Boolean formulae in conjunctive normal form with at ¢ most literals per clause). The first is a randomized algorithm which, with probability approaching 1, finds a satisfying assignment of a ¢ satisfiableCNF ..."
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We present and analyze two simple algorithms for finding satisfying assignments ¢ ofCNFs (Boolean formulae in conjunctive normal form with at ¢ most literals per clause). The first is a randomized algorithm which, with probability approaching 1, finds a satisfying assignment of a ¢ satisfiableCNF £ formula in time. The second algorithm is deterministic, and its running time approaches ����������©�� for large § and. The randomized algorithm is the best known al¢��� � gorithm for; the deterministic algorithm is the best known deterministic ¢��� � algorithm for. We also show an 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 aCNF succinctly. 1
An Improved Exponentialtime Algorithm for SAT
"... We propose and analyze a simple new algorithm for finding satisfying assignments of Boolean formulae in conjunctive normal form. The algorithm, ResolveSat, is a randomized variant of the DLL (Davis, Longeman and Loveland) [2] or DavisPutnam procedure. Rather than applying the DLL procedure to the i ..."
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We propose and analyze a simple new algorithm for finding satisfying assignments of Boolean formulae in conjunctive normal form. The algorithm, ResolveSat, is a randomized variant of the DLL (Davis, Longeman and Loveland) [2] or DavisPutnam procedure. Rather than applying the DLL procedure to the input ¥ formula, however, ResolveSat ¥ enlarges by adding additional clauses using limited resolution before performing DLL. The basic idea behind our analysis is the same as in [6]: a critical clause for a variable at a satisfying assignment gives rise to a unit clause in the DLL procedure with sufficiently high probability, thus increasing the probability of finding a satisfying assignment. In the current paper, we analyze the effect of multiple critical clauses (obtained through resolution)