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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 107 (10 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 66 (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 ...
Stochastic Boolean Satisfiability
 Journal of Automated Reasoning
, 2000
"... . Satisfiability problems and probabilistic models are core topics of artificial intelligence and computer science. This paper looks at the rich intersection between these two areas, opening the door for the use of satisfiability approaches in probabilistic domains. The paper examines a generic stoc ..."
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Cited by 61 (9 self)
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. Satisfiability problems and probabilistic models are core topics of artificial intelligence and computer science. This paper looks at the rich intersection between these two areas, opening the door for the use of satisfiability approaches in probabilistic domains. The paper examines a generic stochastic satisfiability problem, SSat, which can function for probabilistic domains as Sat does for deterministic domains. It shows the connection between SSat and well studied problems in belief network inference and planning under uncertainty, and defines algorithms, both systematic and stochastic, for solving SSat instances. These algorithms are validated on random SSat formulae generated under the fixedclause model. In spite of the large complexity gap between SSat (PSPACE) and Sat (NP), the paper suggests that much of what we've learned about Sat transfers to the probabilistic domain. 1. Introduction There has been a recent focus in artificial intelligence (AI) on solving problems exh...
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...
Rank Bounds and Integrality Gaps for Cutting Planes Procedures
"... We present a new method for proving rank lower bounds for Cutting Planes (CP) and several procedures based on lifting due to Lovász and Schrijver (LS), when viewed as proof systems for unsatisfiability. We apply this method to obtain the following new results: First, we prove nearoptimal rank bound ..."
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Cited by 42 (8 self)
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We present a new method for proving rank lower bounds for Cutting Planes (CP) and several procedures based on lifting due to Lovász and Schrijver (LS), when viewed as proof systems for unsatisfiability. We apply this method to obtain the following new results: First, we prove nearoptimal rank bounds for Cutting Planes and LovászSchrijver proofs for several prominent unsatisfiable CNF examples, including random kCNF formulas and the Tseitin graph formulas. It follows from these lower bounds that a linear number of rounds of CP or LS procedures when applied to relaxations of integer linear programs is not sufficient for reducing the integrality gap. Secondly, we give unsatisfiable examples that have constant rank CP and LS proofs but that require linear rank Resolution proofs. Thirdly, we give examples where the CP rank is O ¡ logn¢ but the LS rank is linear. Finally, we address the question of size versus rank: we show that, for both proof systems, rank does not accurately reflect proof size. Specifically, there are examples with polynomialsize CP/LS proofs, but requiring linear rank.
Satisfiability Solvers
, 2008
"... The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and h ..."
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Cited by 37 (0 self)
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The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worstcase exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and hardware verification [29–31, 228], automatic test pattern generation [138, 221], planning [129, 197], scheduling [103], and even challenging problems from algebra [238]. Annual SAT competitions have led to the development of dozens of clever implementations of such solvers [e.g. 13,
The Resolution Complexity of Random Graph kColorability
 In preparation
, 2003
"... We consider the resolution proof complexity of propositional formulas which encode random instances of graph kcolorability. We obtain a tradeoff between the graph density and the resolution proof complexity. For random graphs with linearly many edges we obtain linearexponential lower bounds on ..."
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Cited by 16 (6 self)
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We consider the resolution proof complexity of propositional formulas which encode random instances of graph kcolorability. We obtain a tradeoff between the graph density and the resolution proof complexity. For random graphs with linearly many edges we obtain linearexponential lower bounds on the length of resolution refutations. For any # > 0, we obtain subexponential lower bounds of the form for some # > 0 for nonkcolorability proofs of graphs with n vertices and O(n # ) edges. We obtain sharper lower bounds for DavisPutnamDPLL proofs and for proofs in a system considered by McDiarmid.
The Complexity of Properly Learning Simple Concept Classes
, 2007
"... We consider the complexity of properly learning concept classes, i.e. when the learner must output a hypothesis of the same form as the unknown concept. We present the following new upper and lower bounds on wellknown concept classes: • We show that unless NP = RP, there is no polynomialtime PAC l ..."
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Cited by 12 (1 self)
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We consider the complexity of properly learning concept classes, i.e. when the learner must output a hypothesis of the same form as the unknown concept. We present the following new upper and lower bounds on wellknown concept classes: • We show that unless NP = RP, there is no polynomialtime PAC learning algorithm for DNF formulas where the hypothesis is an ORofthresholds. Note that as special cases, we show that neither DNF nor ORofthresholds are properly learnable unless NP = RP. Previous hardness results have required strong restrictions on the size of the output DNF formula. We also prove that it is NPhard to learn the intersection of ℓ ≥ 2 halfspaces by the intersection of k halfspaces for any constant k ≥ 0. Previous work held for the case when k = ℓ. • Assuming that NP � ⊆ DTIME(2nɛ) for a certain constant ɛ < 1 we show that it is not possible to learn size s decision trees by size sk decision trees for any k ≥ 0. Previous hardness results for learning decision trees held for k ≤ 2. • We present the first nontrivial upper bounds on properly learning DNF formulas. More specifically, we show how to learn size s DNF by DNF in time 2 Õ( √ n log s). The hardness results for DNF formulas and intersections of halfspaces are obtained via specialized
Exponential lower bounds and Integrality Gaps for Treelike LovászSchrijver Procedures
, 2007
"... The matrix cuts of Lovász and Schrijver are methods for tightening linear relaxations of zeroone programs by the addition of new linear inequalities. We address the question of how many new inequalities are necessary to approximate certain combinatorial problems with strong guarantees, and to solve ..."
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Cited by 10 (1 self)
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The matrix cuts of Lovász and Schrijver are methods for tightening linear relaxations of zeroone programs by the addition of new linear inequalities. We address the question of how many new inequalities are necessary to approximate certain combinatorial problems with strong guarantees, and to solve certain instances of Boolean satisfiability. We show that relaxations of linear programs, obtained by tightening via any subexponentialsize semidefinite LovászSchrijver derivation tree, cannot approximate maxkSAT to a factor better than 1+ 1 2k−1, maxkXOR to a factor better than 2 − ε, nor vertex cover to a factor better than 7/6. We prove exponential size lower bounds for treelike LovászSchrijver proofs of unsatisfiability for several prominent unsatisfiable CNFs, including random 3CNF formulas, random systems of linear equations, and the Tseitin graph formulas. Furthermore, we prove that treelike LS+ cannot polynomially simulate treelike cutting planes, and that treelike LS+ cannot polynomially simulate unrestricted resolution. All of our size lower bounds for derivation trees are based upon connections between the size and height of the derivation tree (its rank). The primary method is a treesize/rank tradeoff for LovászSchrijver refutations: Small tree size implies small rank. Surprisingly, this does not hold for derivations of arbitrary linear inequalities. We show that for LS0 and LS, there are examples with polynomialsize treelike derivations, but requiring linear rank.