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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 ...
On the Complexity of kSAT
, 2001
"... The kSAT problem is to determine if a given kCNF has a satisfying assignment. It is a celebrated open question as to whether it requires exponential time to solve kSAT for k 3. Here exponential time means 2 $n for some $>0. In this paper, assuming that, for k 3, kSAT requires exponential time ..."
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Cited by 110 (8 self)
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The kSAT problem is to determine if a given kCNF has a satisfying assignment. It is a celebrated open question as to whether it requires exponential time to solve kSAT for k 3. Here exponential time means 2 $n for some $>0. In this paper, assuming that, for k 3, kSAT requires exponential time complexity, we show that the complexity of kSAT increases as k increases. More precisely, for k 3, define s k=inf[$: there exists 2 $n algorithm for solving kSAT]. Define ETH (ExponentialTime Hypothesis) for kSAT as follows: for k 3, s k>0. In this paper, we show that s k is increasing infinitely often assuming ETH for kSAT. Let s be the limit of s k. We will in fact show that s k (1&d k) s for some constant d>0. We prove this result by bringing together the ideas of critical clauses and the Sparsification Lemma to reduce the satisfiability of a kCNF to the satisfiability of a disjunction of 2 =n k$CNFs in fewer variables for some k $ k and arbitrarily small =>0. We also show that such a disjunction can be computed in time 2 =n for arbitrarily small =>0.
A Fast Deterministic Algorithm for Formulas That Have . . .
, 1998
"... How can we find any satisfying assignment for a Boolean formula that has many satisfying assignments ? There exists an obvious randomized algorithm for solving this problem: one can just pick an assignment at random and check the truth value of the formula for this assignment, this is iterated until ..."
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Cited by 7 (0 self)
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How can we find any satisfying assignment for a Boolean formula that has many satisfying assignments ? There exists an obvious randomized algorithm for solving this problem: one can just pick an assignment at random and check the truth value of the formula for this assignment, this is iterated until a satisfying assignment occurs. Does there exist a polynomialtime deterministic algorithm that solves the same problem? This paper presents such an algorithm and shows that its worstcase running time is linear when input formulas are in kCNF and a fraction of satisfying assignments (among all possible assignments) is greater than a constant. This algorithm is almost the same as the algorithm proposed by Monien and Speckenmeyer (and independently by Dantsin) in the early 1980s for less than 2 steps 3SAT decision. Another
Local Search Algorithms for SAT: WorstCase Analysis
 In: Proceedings of the 6th Scandinavian Workshop on Algorithm Theory, LNCS 1432
, 1998
"... Recent experiments demonstrated that local search algorithms (e.g. GSAT) are able to find satisfying assignments for many "hard" Boolean formulas. However, no nontrivial worstcase upper bounds were proved, although many such bounds of the form 2 ffn (ff ! 1 is a constant) are known for ..."
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Recent experiments demonstrated that local search algorithms (e.g. GSAT) are able to find satisfying assignments for many "hard" Boolean formulas. However, no nontrivial worstcase upper bounds were proved, although many such bounds of the form 2 ffn (ff ! 1 is a constant) are known for other SAT algorithms, e.g. resolutionlike algorithms. In the present paper we prove such a bound for a local search algorithm, namely for CSAT. The class of formulas we consider covers most of DIMACS benchmarks, the satisfiability problem for this class of formulas is NPcomplete. 1 Introduction Recently there has been an increased interest to local search algorithms for the Boolean satisfiability problem. Though this problem is NPcomplete (see e.g. [GaJo]), B. Selman, H. Levesque and D. Mitchell have shown in [SeLeMi] that an algorithm that uses local search can easily handle some of "hard" instances of SAT. They proposed a randomized greedy local search procedure GSAT (see Figure 1) for the Boo...
Separating signs in the propositional satisfiability problem. Preprint; http://logic.pdmi.ras.ru/ hirsch/index.html
, 1998
"... ..."
Satisfiability  algorithms and logic (Extended Abstract)
"... We present some recent results on algorithms for satisfiability of kCNF formulas: fastest probabilistic algorithms. We mention some results in proof complexity that can be used to derive lower bounds on classes of algorithms for satisfiability. ..."
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We present some recent results on algorithms for satisfiability of kCNF formulas: fastest probabilistic algorithms. We mention some results in proof complexity that can be used to derive lower bounds on classes of algorithms for satisfiability.
and
"... but still exponential algorithms. In this paper, we address the relative likelihoodofsubexponentialalgorithmsfortheseproblems.Weintroducea generalizedreductionthatwecallSubexponentialReductionFamily(SERF) that preserves subexponential complexity. We show that CircuitSAT is SERFcompleteforall NP ..."
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but still exponential algorithms. In this paper, we address the relative likelihoodofsubexponentialalgorithmsfortheseproblems.Weintroducea generalizedreductionthatwecallSubexponentialReductionFamily(SERF) that preserves subexponential complexity. We show that CircuitSAT is SERFcompleteforall NPsearchproblems,andthatforanyfixed k \ 3, kSAT, kColorability, kSetCover,IndependentSet,Clique,andVertexCover,are SERFcompletefortheclass SNPofsearchproblemsexpressiblebysecondorderexistentialformulaswhosefirstorderpartisuniversal.Inparticular, subexponentialcomplexityforanyoneoftheaboveproblemsimpliesthe sameforallothers. Wealsolookattheissueofprovingstronglyexponentiallowerboundsfor AC0,thatis,boundsoftheform 2W(n).Thisproblemisevenopenfordepth3 circuits.Infact,suchaboundfordepth3circuitswithevenlimited(atmost ne)faninforbottomlevelgateswouldimplyanonlinearsizelowerbound for logarithmic depth circuits. We show that with high probability even randomdegree2GF(2)polynomialsrequirestronglyexponentialsizefor S k 3 circuitsfor k=o(loglog n).Wethusexhibitamuchsmallerspaceof 2O(n2) functions such that almost every function in this class requires strongly exponential size S k 3 circuits. As a corollary, we derive a pseudorandom generator (requiring O(n 2) bits of advice) that maps n bits into a larger numberofbitssothatcomputingparityontherangeishardfor S k
Fundamental Study New methods for 3SAT decision and worstcase analysis 1
, 1995
"... Kullmann, O. (1999). New methods for 3SAT decision and worstcase analysis. Theoretical Computer Science, 223 ..."
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Kullmann, O. (1999). New methods for 3SAT decision and worstcase analysis. Theoretical Computer Science, 223