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Simplified and Improved Resolution Lower Bounds
 IN PROCEEDINGS OF THE 37TH IEEE FOCS
, 1996
"... We give simple new lower bounds on the lengths of Resolution proofs for the pigeonhole principle and for randomly generated formulas. For random formulas, our bounds significantly extend the range of formula sizes for which nontrivial lower bounds are known. For example, we show that with probabili ..."
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Cited by 103 (8 self)
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We give simple new lower bounds on the lengths of Resolution proofs for the pigeonhole principle and for randomly generated formulas. For random formulas, our bounds significantly extend the range of formula sizes for which nontrivial lower bounds are known. For example, we show that with probability approaching 1, any Resolution refutation of a randomly chosen 3CNF formula with at most n 6=5\Gammaffl clauses requires exponential size. Previous bounds applied only when the number of clauses was at most linear in the number of variables. For the pigeonhole principle our bound is a small improvement over previous bounds. Our proofs are more elementary than previous arguments, and establish a connection between Resolution proof size and maximum clause size.
The efficiency of resolution and DavisPutnam procedures
 SIAM Journal on Computing
, 1999
"... We consider several problems related to the use of resolutionbased methods for determining whether a given boolean formula in conjunctive normal form is satisfiable. First, building on work of Clegg, Edmonds and Impagliazzo, we give an algorithm for satisfiability that when given an unsatisfiabl ..."
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Cited by 64 (1 self)
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We consider several problems related to the use of resolutionbased methods for determining whether a given boolean formula in conjunctive normal form is satisfiable. First, building on work of Clegg, Edmonds and Impagliazzo, we give an algorithm for satisfiability that when given an unsatisfiable formula of F finds a resolution proof of F , and the runtime of our algorithm is nontrivial as a function of the size of the shortest resolution proof of F . Next we investigate a class of backtrack search algorithms, commonly known as DavisPutnam procedures and provide the first averagecase complexity analysis for their behavior on random formulas. In particular, for a simple algorithm in this class, called ordered DLL we prove that the running time of the algorithm on a randomly generated kCNF formula with n variables and m clauses is 2 Q(n(n/m) 1/(k2) ) with probability 1  o(1). Finally, we give new lower bounds on res(F), the size of the smallest resolution refutation ...
On the Complexity of Unsatisfiability Proofs for Random kCNF Formulas
 In 30th Annual ACM Symposium on the Theory of Computing
, 1997
"... We study the complexity of proving unsatisfiability for random kCNF formulas with clause density D = m=n where m is number of clauses and n is the number of variables. We prove the first nontrivial general upper bound, giving algorithms that, in particular, for k = 3 produce refutations almost cer ..."
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Cited by 51 (1 self)
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We study the complexity of proving unsatisfiability for random kCNF formulas with clause density D = m=n where m is number of clauses and n is the number of variables. We prove the first nontrivial general upper bound, giving algorithms that, in particular, for k = 3 produce refutations almost certainly in time 2 O(n=D) . This is polynomial when m n 2 =logn. We show that our upper bounds are tight for certain natural classes of DavisPutnam algorithms. We show further that random 3CNF formulas of clause density D almost certainly have no resolution refutation of size smaller than 2 W(n=D 4+e ) , which implies the same lower bound on any DavisPutnam algorithm. We also give a much simpler argument based on a novel form of selfreduction that yields a slightly weaker 2 W(n=D 5+e ) lower bound. 1 Introduction The random kCNF model has been widely studied for several good reasons. First, it is an intrinsically natural model, analogous to the random graph model, that shed...
A Perspective on Certain Polynomial Time Solvable Classes of Satisfiability
 Discrete Applied Mathematics
, 1998
"... The scope of certain wellstudied polynomialtime solvable classes of Satisfiability is investigated relative to a polynomialtime solvable class consisting of what we call matched formulas. The class of matched formulas has not been studied in the literature, probably because it seems not to contai ..."
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Cited by 21 (2 self)
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The scope of certain wellstudied polynomialtime solvable classes of Satisfiability is investigated relative to a polynomialtime solvable class consisting of what we call matched formulas. The class of matched formulas has not been studied in the literature, probably because it seems not to contain many challenging formulas. Yet, we find that, in some sense, the matched formulas are more numerous than Horn, extended Horn, renamable Horn, qHorn, CCbalanced, or Single Lookahead Unit Resolution (SLUR) formulas. In addition, we find that relatively few unsatisfiable formulas are in any of the above classes. However, there are many unsatisfiable formulas that can be solved in polynomial time by restricting resolution so as not to generate resolvents of size greater than the number of literals in a maximum length clause. We use the wellstudied random kSAT model M(n;m;k) for generating CNF formulas with m clauses, each with k distinct literals, from n variables. We show, for all m=n ? 2...
Recognizing more unsatisfiable random kSAT instances efficiently
, 2001
"... It is known that random kSAT instances with at least cn clauses where c = ck is a suitable constant are unsatisfiable (with high probability). We consider the problem to certify efficiently the unsatisfiability of such formulas. A backtracking based algorithm of Beame et al. shows that kSAT instan ..."
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Cited by 13 (0 self)
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It is known that random kSAT instances with at least cn clauses where c = ck is a suitable constant are unsatisfiable (with high probability). We consider the problem to certify efficiently the unsatisfiability of such formulas. A backtracking based algorithm of Beame et al. shows that kSAT instances with at least n clauses can be certified unsatisfiable in polynomial time. We employ spectral methods to improve on this bound: For even k 4 we present a polynomial time algorithm which certifies random kSAT instances with at least clauses as unsatisfiable (with high probability). For odd k we focus on 3SAT instances and obtain an ecient algorithm for formulas with at least n clauses, where " > 0 is an arbitrary constant.
A Dichotomy Theorem for the Resolution Complexity of Random Constraint Satisfaction Problems
"... We consider random instances of constraint satisfaction problems where each variable has domain size O(1), each constraint is on O(1) variables and the constraints are chosen from a specified distribution. The number of constraints is cn where c is a constant. We prove that for every possible distri ..."
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Cited by 2 (2 self)
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We consider random instances of constraint satisfaction problems where each variable has domain size O(1), each constraint is on O(1) variables and the constraints are chosen from a specified distribution. The number of constraints is cn where c is a constant. We prove that for every possible distribution, either the resolution complexity is almost surely polylogarithmic for sufficiently large c, or it is almost surely exponential for every c> 0. We characterize the distributions of each type. To do so, we introduce a closure operation on a set of constraints which yields the set of all constraints that, in some sense, appear implicitly in the random CSP. 1
St˚almarck’s Method versus Resolution: A Comparative Theoretical Study
, 2001
"... This Master’s thesis presents a comparative analysis of St˚almarck’s proof method and resolution from a theoretical perspective. We give (to our knowledge) the first complete explicit formal description of the dilemma proof system underlying St˚almarck’s method. Based on this description we prove a ..."
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Cited by 1 (0 self)
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This Master’s thesis presents a comparative analysis of St˚almarck’s proof method and resolution from a theoretical perspective. We give (to our knowledge) the first complete explicit formal description of the dilemma proof system underlying St˚almarck’s method. Based on this description we prove a number of simulation and separation results between different subsystems of dilemma (defined by restrictions on possible branching assumptions and rules for merging the results derived in distinct branches). The key result of the thesis is that dilemma depth translates into resolution width. More precisely, a dilemma refutation in depth d and length L of a kCNF formula F can be transformed to a resolution refutation of F in width O (kd) and length � Lk d � O(1). From this depthwidth relation it follows that for kCNF formulas with k fixed, resolution psimulates dilemma restricted to minimumdepth proofs. Furthermore, the running time of the minimumwidth proof search algorithm
Probabilistic Analysis of Satisfiability Algorithms
, 2008
"... Probabilistic and averagecase analysis can give useful insight into the question of what algorithms for testing satisfiability might be effective and why. Under certain circumstances, one or more structural properties shared by each of a family or class of expressions may be exploited to solve such ..."
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Probabilistic and averagecase analysis can give useful insight into the question of what algorithms for testing satisfiability might be effective and why. Under certain circumstances, one or more structural properties shared by each of a family or class of expressions may be exploited to solve such expressions efficiently;