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Disproof of the neighborhood conjecture with implications to SAT
 17th Annual European Symposium on Algorithms (ESA 2009), volume 5757 of Lecture Notes in Computer Science
, 2009
"... We study a Maker/Breaker game described by Beck. As a result we disprove a conjecture of Beck on positional games, establish a connection between this game and SAT and construct an unsatisfiable kCNF formula with few occurrences per variable, thereby improving a previous result by Hoory and Szeider ..."
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We study a Maker/Breaker game described by Beck. As a result we disprove a conjecture of Beck on positional games, establish a connection between this game and SAT and construct an unsatisfiable kCNF formula with few occurrences per variable, thereby improving a previous result by Hoory and Szeider and showing that the bound obtained from the Lovász Local Lemma is tight up to a constant factor. The Maker/Breaker game we study is as follows. Maker and Breaker take turns in choosing vertices from a given nuniform hypergraph F, with Maker going first. Maker’s goal is to completely occupy a hyperedge and Breaker tries to avoid this. Beck conjectures that if the maximum neighborhood size of F is at most 2n−1 then Breaker has a winning strategy. We disprove this conjecture by establishing an nuniform hypergraph with maximum neighborhood size 3 · 2n−3 where Maker has a winning strategy. Moreover, we show how to construct an nuniform hypergraph with maximum degree 2n−1 n where Maker has a winning strategy. In addition we show that each nuniform hypergraph with maximum degree at most 2n−2 en has a proper halving 2coloring, which solves another open problem posed by Beck related to the Neighborhood Conjecture. Finally, we establish a connection between SAT and the Maker/Breaker game we study. We can use this connection to derive new results in SAT. A (k, s)CNF formula is a boolean formula in conjunctive normal form where every clause contains exactly k literals and every variable occurs in at most s clauses. The (k, s)SAT problem is the satisfiability problem restricted to (k, s)CNF formulas. Kratochvíl, Savick´y and Tuza showed that for every k ≥ 3 there is an integer f(k) such that every (k, f(k))formula is satisfiable, but (k, f(k) + 1)SAT is already NPcomplete (it is not known whether f(k) is ( computable).) Kratochvíl, Savick´y and Tuza also k 2, which is a consequence of the Lovász Local gave the best known lower bound f(k) = Ω Lemma. We prove that, in fact, f(k) = Θ
UNSATISFIABLE LINEAR CNF FORMULAS ARE LARGE AND COMPLEX
, 2010
"... Abstract. We call a CNF formula linear if any two clauses have at most one variable in common. We show that there exist unsatisfiable linear kCNF formulas with at most 4k 2 4 k 4 k 8e 2 k 2 clauses is clauses, and on the other hand, any linear kCNF formula with at most satisfiable. The upper bound ..."
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Abstract. We call a CNF formula linear if any two clauses have at most one variable in common. We show that there exist unsatisfiable linear kCNF formulas with at most 4k 2 4 k 4 k 8e 2 k 2 clauses is clauses, and on the other hand, any linear kCNF formula with at most satisfiable. The upper bound uses probabilistic means, and we have no explicit construction coming even close to it. One reason for this is that unsatisfiable linear formulas exhibit a more complex structure than general (nonlinear) formulas: First, any treelike resolution refutation of any unsatisfiable linear kCNF formula has size at least 2 2 k 2 −1. This implies that small unsatisfiable linear kCNF formulas are hard instances for DavisPutnam style splitting algorithms. Second, if we require that the formula F have a strict resolution tree,..a a. i.e. every clause of F is used only once in the resolution tree, then we need at least a clauses, where a ≈ 2 and the height of this tower is roughly k. 1.
The Lovász Local Lemma and Satisfiability
, 2009
"... We consider boolean formulas in conjunctive normal form (CNF). If all clauses are large, it needs many clauses to obtain an unsatisfiable formula; moreover, these clauses have to interleave. We review quantitative results for the amount of interleaving required, many of which rely on the Lovász Loca ..."
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We consider boolean formulas in conjunctive normal form (CNF). If all clauses are large, it needs many clauses to obtain an unsatisfiable formula; moreover, these clauses have to interleave. We review quantitative results for the amount of interleaving required, many of which rely on the Lovász Local Lemma, a probabilistic lemma with many applications in combinatorics. In positive terms, we are interested in simple combinatorial conditions which guarantee for a CNF formula to be satisfiable. The criteria obtained are nontrivial in the sense that even though they are easy to check, it is by far not obvious how to compute a satisfying assignment efficiently in case the conditions are fulfilled; until recently, it was not known how to do so. It is also remarkable that while deciding satisfiability is trivial for formulas that satisfy the conditions, a slightest relaxation of the conditions leads us into the territory of NPcompleteness. Several open problems remain, some of which we mention in the concluding section.
The Local Lemma is Tight for SAT
"... We construct unsatisfiable kCNF formulas where every clause has k distinct literals and every variable appears in at most ( 2 e + o(1)) 2 k clauses. The lopsided Local Lemma k shows that our result is asymptotically best possible: every kCNF formula where every variable appears in at most 2 k+1 − ..."
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We construct unsatisfiable kCNF formulas where every clause has k distinct literals and every variable appears in at most ( 2 e + o(1)) 2 k clauses. The lopsided Local Lemma k shows that our result is asymptotically best possible: every kCNF formula where every variable appears in at most 2 k+1 − 1 clauses is satisfiable. The determination of this e(k+1) extremal function is particularly important as it represents the value where the kSAT problem exhibits its complexity hardness jump: from having every instance being a YESinstance it becomes NPhard just by allowing each variable to occur in one more clause. The asymptotics of other related extremal functions are also determined. Let l(k) denote the maximum number, such that every kCNF formula with each clause containing k distinct literals and each clause having a common variable with at most l(k) other clauses, is satisfiable. We establish that the bound on l(k) obtained from the Local Lemma is asymptotically optimal, i.e., l(k) = ( 1 e + o(1)) 2 k. The constructed formulas are all in the class MU(1) of minimal unsatisfiable formulas having one more clause than variables and thus they resolve these asymptotic questions within that class as well. The SATformulas are constructed via the binary trees of [10]. In order to construct the trees a continuous setting of the problem is defined, giving rise to a differential equation. The solution of the equation diverges at 0, which in turn implies that the binary tree obtained from the discretization of this solution has the required properties. 1
The Local Lemma is asymptotically tight for SAT
"... The Local Lemma is a fundamental tool of probabilistic combinatorics and theoretical computer science, yet there are hardly any natural problems known where it provides an asymptotically tight answer. The main theme of our paper is to identify several of these problems, among them a couple of widely ..."
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The Local Lemma is a fundamental tool of probabilistic combinatorics and theoretical computer science, yet there are hardly any natural problems known where it provides an asymptotically tight answer. The main theme of our paper is to identify several of these problems, among them a couple of widely studied extremal functions related to certain restricted versions of the kSAT problem, where the Local Lemma does give essentially optimal answers. As our main contribution, we construct unsatisfiable kCNF formulas where every clause has k distinct literals and every variable appears in at most(
Durham Research Online Deposited in DRO: Citation for published item: Use policy The Complexity of Resolution with Generalized Symmetry Rules *
"... Use policy The fulltext may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or notforprot purposes provided that: • a full bibliographic reference is made to the original source • a li ..."
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Use policy The fulltext may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or notforprot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the fulltext is not changed in any way The fulltext must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Abstract We generalize Krishnamurthy's wellstudied symmetry rule for resolution systems by considering homomorphisms instead of symmetries; symmetries are injective maps of literals which preserve complements and clauses; homomorphisms arise from symmetries by releasing the constraint of being injective. We prove that the use of homomorphisms yields a strictly more powerful system than the use of symmetries by exhibiting an infinite sequence of sets of clauses for which the consideration of global homomorphisms allows exponentially shorter proofs than the consideration of local symmetries. It is known that local symmetries give rise to a strictly more powerful system than global symmetries; we prove a similar result for local and global homomorphisms. Finally, we obtain an exponential lower bound for the resolution system enhanced by the local homomorphism rule.
Without Loss of Generality  Symmetric Reasoning for Resolution Systems
"... Abstract Krishnamurthy [1985] introduced symmetry rules that make the informal "without loss of generality" reasoning available for resolutionbased systems. The homomorphism rules of Szeider [2005] are more powerful variants of Krishnamurthy's rules that can save in certain cases a ..."
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Abstract Krishnamurthy [1985] introduced symmetry rules that make the informal "without loss of generality" reasoning available for resolutionbased systems. The homomorphism rules of Szeider [2005] are more powerful variants of Krishnamurthy's rules that can save in certain cases an exponential number of inference steps over symmetry rules. In this talk we will review the concepts of symmetry and homomorphism rules and discuss various questions and results that arise in that context.
A random formula lower bound for ordered DLL extended with local symmetry recognition
, 2004
"... ..."
(Extended Abstract for SymCon’07 Invited Talk)
"... Krishnamurthy [1985] introduced symmetry rules that make the informal “without loss of generality ” reasoning available for resolutionbased systems. The homomorphism rules of Szeider [2005] are more powerful variants of Krishnamurthy’s rules and can save in certain cases an exponential number of inf ..."
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Krishnamurthy [1985] introduced symmetry rules that make the informal “without loss of generality ” reasoning available for resolutionbased systems. The homomorphism rules of Szeider [2005] are more powerful variants of Krishnamurthy’s rules and can save in certain cases an exponential number of inference steps over symmetry rules. In this talk we will review the concepts of symmetry and homomorphism rules for resolutionbased systems and discuss various questions and results that arise in that context. 1
Efficient Autarkies
"... Abstract. Autarkies are partial truth assignments that satisfy all clauses having literals in the assigned variables. Autarkies provide important information in the analysis of unsatisfiable formulas. Indeed, clauses satisfied by autarkies cannot be included in minimal explanations or in minimal co ..."
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Abstract. Autarkies are partial truth assignments that satisfy all clauses having literals in the assigned variables. Autarkies provide important information in the analysis of unsatisfiable formulas. Indeed, clauses satisfied by autarkies cannot be included in minimal explanations or in minimal corrections of unsatisfiability. Computing the maximum autarky allows identifying all such clauses. In recent years, a number of alternative approaches have been proposed for computing a maximum autarky. This paper develops new models for representing autarkies, and proposes new algorithms for computing the maximum autarky. Experimental results, obtained on a large number of problem instances, show orders of magnitude performance improvements over existing approaches, and solving instances that could not otherwise be solved. 1