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Elimination Of Infrequent Variables Improves Average Case Performance Of Satisfiability Algorithms
 SIAM J. Comput
, 1991
"... . We consider preprocessing a random instance I of CNF Satisfiability in order to remove infrequent variables (those which appear once or twice in an instance) from I. The model used to generate random instances is the popular randomclausesize model with parametersn, the number of clauses, r, the ..."
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Cited by 16 (5 self)
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. We consider preprocessing a random instance I of CNF Satisfiability in order to remove infrequent variables (those which appear once or twice in an instance) from I. The model used to generate random instances is the popular randomclausesize model with parametersn, the number of clauses, r, the number of Boolean variables from which clauses are composed, and p, the probability that a variable appears in a clause as a positive (or negative) literal. It is shown that exhaustive search over such preprocessed instances runs in polynomial average time over a significantly larger parameter space than has been shown for any other algorithm under the randomclausesize model when n = r ffl , ffl ! 1, and pr ! p fflr ln(r). Specifically, the results are that random instances of Satisfiability are "easy" in the average case if n = r ffl , 2=3 ? ffl ? 0, and pr ! (ln(n)=4) 1=3 r 2=3\Gammaffl ; or n = r ffl , 1 ? ffl 2=3, pr ! (1 \Gamma ffl \Gamma ffi) ln(n)=ffl for any ffi ? 0...
Average Case Results for Satisfiability Algorithms Under the Random Clause Width Model
 Annals of Mathematics and Artificial Intelligence
, 1995
"... In the probabilistic analysis of algorithms for the Satisfiability problem, the randomclausewidth model is one of the most popular for generating random instances. This model is parameterized and it is not difficult to show that virtually the entire parameter space is covered by a collection of ..."
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Cited by 9 (1 self)
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In the probabilistic analysis of algorithms for the Satisfiability problem, the randomclausewidth model is one of the most popular for generating random instances. This model is parameterized and it is not difficult to show that virtually the entire parameter space is covered by a collection of polynomial time algorithms that find solutions to random instances with probability tending to 1 as instance size increases. But finding a collection of polynomial average time algorithms that cover the parameter space has proved much harder and such results have spanned approximately ten years. However, it can now be said that virtually the entire parameter space is covered by polynomial average time algorithms. This paper relates dominant, exploitable properties of random formulas over the parameter space to mechanisms of polynomial average time algorithms. The probabilistic discussion of such properties is new; main averagecase results over the last ten years are reviewed. 1 Intr...
On the Occurrence of Null Clauses in Random Instances of Satisfiability
 Discrete Applied Mathematics
, 1989
"... We analyze a popular probabilistic model for generating instances of Satisfiability. According to this model, each literal of a set L = fv 1 ; v 1 ; v 2 ; v 2 ; :::; v r ; v r g of literals appears independently in each of n clauses with probability p. This model allows null clauses and the frequ ..."
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We analyze a popular probabilistic model for generating instances of Satisfiability. According to this model, each literal of a set L = fv 1 ; v 1 ; v 2 ; v 2 ; :::; v r ; v r g of literals appears independently in each of n clauses with probability p. This model allows null clauses and the frequency of occurrence of such clauses depends on the relationship between the parameters n, r, and p. If an instance contains a null clause it is trivially unsatisfiable. Several papers present polynomial average time results under this model when null clauses are numerous (e.g. [4,5]) but, until now, not all such cases have been covered by averagecase efficient algorithms. In fact, a recent paper [2] shows that the average complexity of the pure literal rule is superpolynomial even when most random instances contain a null clause. We show here that a simple strategy based on locating null clauses in a given random input has polynomial average complexity if either n r :5 , and pr ! ln(n)=2...