Results 1  10
of
158
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 ..."
Abstract

Cited by 242 (11 self)
 Add to MetaCart
(Show Context)
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...
OneDimensional Quantum Walks
 STOC'01
, 2001
"... We define and analyze quantum computational variants of random walks on onedimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the Hadamard walk. Several striking differences between the quantum and classical cases are observed. For example, ..."
Abstract

Cited by 134 (10 self)
 Add to MetaCart
We define and analyze quantum computational variants of random walks on onedimensional lattices. In particular, we analyze a quantum analog of the symmetric random walk, which we call the Hadamard walk. Several striking differences between the quantum and classical cases are observed. For example, when unrestricted in either direction, the Hadamard walk has position that is nearly uniformly distributed in the range [\Gamma t= p
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, ..."
Abstract

Cited by 111 (7 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 104 (8 self)
 Add to MetaCart
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.
Contingent Planning Under Uncertainty via Stochastic Satisfiability
 Artificial Intelligence
, 1999
"... We describe two new probabilistic planning techniques cmaxplan and zanderthat generate contingent plans in probabilistic propositional domains. Both operate by transforming the planning problem into a stochastic satisfiability problem and solving that problem instead. cmaxplan encodes t ..."
Abstract

Cited by 71 (11 self)
 Add to MetaCart
(Show Context)
We describe two new probabilistic planning techniques cmaxplan and zanderthat generate contingent plans in probabilistic propositional domains. Both operate by transforming the planning problem into a stochastic satisfiability problem and solving that problem instead. cmaxplan encodes the problem as an EMajsat instance, while zander encodes the problem as an SSat instance. Although SSat problems are in a higher complexity class than EMajsat problems, the problem encodings produced by zander are substantially more compact and appear to be easier to solve than the corresponding EMajsat encodings. Preliminary results for zander indicate that it is competitive with existing planners on a variety of problems. Introduction When planning under uncertainty, any information about the state of the world is precious. A contingent plan is one that can make action choices contingent on such information. In this paper, we present an implemented framework for contingent pl...
UnitWalk: A new SAT solver that uses local search guided by unit clause elimination
, 2002
"... In this paper we present a new randomized algorithm for SAT, i.e., the satisfiability problem for Boolean formulas in conjunctive normal form. Despite its simplicity, this algorithm performs well on many common benchmarks ranging from graph coloring problems to microprocessor verification. ..."
Abstract

Cited by 69 (1 self)
 Add to MetaCart
(Show Context)
In this paper we present a new randomized algorithm for SAT, i.e., the satisfiability problem for Boolean formulas in conjunctive normal form. Despite its simplicity, this algorithm performs well on many common benchmarks ranging from graph coloring problems to microprocessor verification.
Measure and Conquer: Domination  A case study
"... DavisPutnamstyle exponentialtime backtracking algorithms are the most common algorithms used for finding exact solutions of NPhard problems. The analysis of such recursive algorithms is based on the bounded search tree technique: a measure of the size of the subproblems is defined; this measure ..."
Abstract

Cited by 57 (22 self)
 Add to MetaCart
DavisPutnamstyle exponentialtime backtracking algorithms are the most common algorithms used for finding exact solutions of NPhard problems. The analysis of such recursive algorithms is based on the bounded search tree technique: a measure of the size of the subproblems is defined; this measure is used to lower bound the progress made by the algorithm at each branching step. For the last 30 years the research on exact algorithms has been mainly focused on the design of more and more sophisticated algorithms. However, measures used in the analysis of backtracking algorithms are usually very simple. In this paper we stress that a more careful choice of the measure can lead to significantly better the worst case time analysis. As an example, we consider the minimum dominating set problem. The currently fastest algorithm for this problem has running time O(2
A measure & conquer approach for the analysis of exact algorithms
, 2007
"... For more than 40 years Branch & Reduce exponentialtime backtracking algorithms have been among the most common tools used for finding exact solutions of NPhard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worstcase running time bounds. ..."
Abstract

Cited by 49 (11 self)
 Add to MetaCart
For more than 40 years Branch & Reduce exponentialtime backtracking algorithms have been among the most common tools used for finding exact solutions of NPhard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worstcase running time bounds. Motivated by this we use an approach, that we call “Measure & Conquer”, as an attempt to step beyond such limitations. The approach is based on the careful design of a nonstandard measure of the subproblem size; this measure is then used to lower bound the progress made by the algorithm at each branching step. The idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit, and hence lead to a significantly better worstcase time analysis. In order to show the potentialities of Measure & Conquer, we consider two wellstudied NPhard problems: minimum dominating set and maximum independent set. For the first problem, we consider the current best algorithm, and prove (thanks to a better measure) a much tighter running time bound for it. For the second problem, we describe a new, simple algorithm, and show that its running time is competitive with the current best time bounds, achieved with far more complicated algorithms (and standard analysis). Our examples
Improved upper bounds for 3sat
 In 15th ACMSIAM Symposium on Discrete Algorithms (SODA 2004). ACM and SIAM
"... The CNF Satisfiability problem is to determine, given a CNF formula F, whether or not there exists a satisfying assignment for F. If each clause of F contains at most k literals, then F is called a kCNF formula and the problem is called kSAT. For small k’s, especially for k = 3, there exists a lot ..."
Abstract

Cited by 48 (3 self)
 Add to MetaCart
(Show Context)
The CNF Satisfiability problem is to determine, given a CNF formula F, whether or not there exists a satisfying assignment for F. If each clause of F contains at most k literals, then F is called a kCNF formula and the problem is called kSAT. For small k’s, especially for k = 3, there exists a lot of algorithms which run significantly faster than the trivial 2n bound. The following list summarizes those algorithms where a constant c means that the algorithm runs in time O(cn). Roughly speaking most algorithms are based on DavisPutnam. [Sch99] is the first local search algorithm which gives a guaranteed performance for general instances and [DGH+02], [HSSW02], [BS03] and [Rol03] follow up this Schöning’s approach. 3SAT 4SAT 5SAT 6SAT type ref. 1.782 1.835 1.867 1.888 det. [PPZ97]
Improved Algorithms for 3Coloring, 3EdgeColoring, and Constraint Satisfaction
, 2001
"... We consider worst case time bounds for NPcomplete problems including 3SAT, 3coloring, 3edgecoloring, and 3list coloring. Our algorithms are based on a constraint satisfaction (CSP) formulation of these problems; 3SAT is equivalent to (2, 3)CSP while the other problems above are special cases ..."
Abstract

Cited by 46 (3 self)
 Add to MetaCart
We consider worst case time bounds for NPcomplete problems including 3SAT, 3coloring, 3edgecoloring, and 3list coloring. Our algorithms are based on a constraint satisfaction (CSP) formulation of these problems; 3SAT is equivalent to (2, 3)CSP while the other problems above are special cases of (3, 2)CSP. We give a fast algorithm for (3, 2) CSP and use it to improve the time bounds for solving the other problems listed above. Our techniques involve a mixture of DavisPutnamstyle backtracking with more sophisticated matching and network flow based ideas.