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44
Improvements To Propositional Satisfiability Search Algorithms
, 1995
"... ... quickly across a wide range of hard SAT problems than any other SAT tester in the literature on comparable platforms. On a Sun SPARCStation 10 running SunOS 4.1.3 U1, POSIT can solve hard random 400variable 3SAT problems in about 2 hours on the average. In general, it can solve hard nvariable ..."
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Cited by 161 (0 self)
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... quickly across a wide range of hard SAT problems than any other SAT tester in the literature on comparable platforms. On a Sun SPARCStation 10 running SunOS 4.1.3 U1, POSIT can solve hard random 400variable 3SAT problems in about 2 hours on the average. In general, it can solve hard nvariable random 3SAT problems with search trees of size O(2 n=18:7 ). In addition to justifying these claims, this dissertation describes the most significant achievements of other researchers in this area, and discusses all of the widely known general techniques for speeding up SAT search algorithms. It should be useful to anyone interested in NPcomplete problems or combinatorial optimization in general, and it should be particularly useful to researchers in either Artificial Intelligence or Operations Research.
Typical random 3SAT formulae and the satisfiability threshold
 in Proceedings of the Eleventh ACMSIAM Symposium on Discrete Algorithms
, 2000
"... Abstract: We present a new structural (or syntactic) approach for estimating the satisfiability threshold of random 3SAT formulae. We show its efficiency in obtaining a jump from the previous upper bounds, lowering them to 4.506. The method combines well with other techniques, and also applies to o ..."
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Cited by 87 (2 self)
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Abstract: We present a new structural (or syntactic) approach for estimating the satisfiability threshold of random 3SAT formulae. We show its efficiency in obtaining a jump from the previous upper bounds, lowering them to 4.506. The method combines well with other techniques, and also applies to other problems, such as the 3colourability of random graphs. 1
The Probabilistic Analysis of a Greedy Satisfiability Algorithm
, 2002
"... Consider the following simple, greedy DavisPutnam algorithm applied to a random 3CNF formula of fixed density (clauses to variables ratio): Arbitrarily select and set to True a literal that appears in as many clauses as possible, irrespective of their size (and irrespective of the number of occu ..."
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Cited by 72 (5 self)
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Consider the following simple, greedy DavisPutnam algorithm applied to a random 3CNF formula of fixed density (clauses to variables ratio): Arbitrarily select and set to True a literal that appears in as many clauses as possible, irrespective of their size (and irrespective of the number of occurrences of the negation of the literal). Delete these clauses from the formula, and also delete the negation of this literal from any clauses it appears. Repeat. If however unit clauses ever appear, then first repeatedly and in any order set the literals in them to True and delete and shrink clauses accordingly, until no unit clause remains. Also if at any step an empty clause appears, then do not backtrack, but just terminate the algorithm and report failure. A slight modification of this algorithm is probabilistically analyzed in this paper (rigorously). It is proved that for random formulas of n variables and density up to 3.42, it succeeds in producing a satisfying truth assignment with bounded away from zero probability, as n approaches infinity. Therefore the satisfiability threshold is at least 3.42.
A backbonesearch heuristic for efficient solving of hard 3SAT formulae
, 2001
"... Of late, new insight into the study of random kSAT formulae has been gained from the introduction of a concept inspired by models of physics, the `backbone ' of a SAT formula which corresponds to the variables having a fixed truth value in all assignments satisfying the maximum number of claus ..."
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Cited by 62 (1 self)
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Of late, new insight into the study of random kSAT formulae has been gained from the introduction of a concept inspired by models of physics, the `backbone ' of a SAT formula which corresponds to the variables having a fixed truth value in all assignments satisfying the maximum number of clauses.
The Satisfiability Constraint Gap
 Artificial Intelligence
, 1996
"... We describe an experimental investigation of the satisfiability phase transition for several different classes of randomly generated problems. We show that the "conventional" picture of easyhardeasy problem difficulty is inadequate. In particular, there is a region of very variable problem difficu ..."
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Cited by 33 (10 self)
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We describe an experimental investigation of the satisfiability phase transition for several different classes of randomly generated problems. We show that the "conventional" picture of easyhardeasy problem difficulty is inadequate. In particular, there is a region of very variable problem difficulty where problems are typically underconstrained and satisfiable. Within this region, problems can be orders of magnitude harder than problems in the middle of the satisfiability phase transition. These extraordinary hard problems appear to be associated with a constraint gap, a minimum in the amount of constraint propagation compared to the amount of search. We show that the position and shape of this constraint gap are very consistent with problem size. Unlike hard problems in the middle of satisfiability phase transition, hard problems in the variable region are not critically constrained between satisfiability and unsatisfiability. Indeed, hard problems in the variable region often cont...
Random 3SAT: The Plot Thickens
 IN PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING
, 2000
"... This paper presents an experimental investigation of the following questions: how does the averagecase complexity of random 3SAT, understood as a function of the order (number of variables) for xed density (ratio of number of clauses to order) instances, depend on the density? Is there a phase tra ..."
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Cited by 28 (2 self)
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This paper presents an experimental investigation of the following questions: how does the averagecase complexity of random 3SAT, understood as a function of the order (number of variables) for xed density (ratio of number of clauses to order) instances, depend on the density? Is there a phase transition in which the complexity shifts from polynomial to exponential in the order? Is the transition dependent or independent of the solver? Our experiment design uses three complete SAT solvers embodying dierent algorithms: GRASP, CPLEX, and CUDD. We observe new phase transitions for all three solvers, where the median running time shifts from polynomial in the order to exponential. The location of the phase transition appears to be solverdependent. While GRASP and CUDD shift from polynomial to exponential complexity at a density of about 3.8, CUDD exhibits this transition between densities of 0.1 and 0.5. This experimental result underscores the dependence between the solver and the complexity phase transition, and challenges the widely held belief that random 3SAT exhibits a phase transition in computational complexity very close to the crossover point.
Scaling Effects in the CSP Phase Transition
, 1995
"... Phase transitions in constraint satisfaction problems (CSP's) are the subject of intense study. We identify an order parameter for random binary CSP's. There is a rapid transition in the probability of a CSP having a solution at a critical value of this parameter. The order parameter allows differen ..."
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Cited by 27 (16 self)
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Phase transitions in constraint satisfaction problems (CSP's) are the subject of intense study. We identify an order parameter for random binary CSP's. There is a rapid transition in the probability of a CSP having a solution at a critical value of this parameter. The order parameter allows different phase transition behaviour to be compared in an uniform manner, for example CSP's generated under different regimes. We then show that within classes, the scaling of behaviour can be modelled by a tehnique called "finite size scaling". This applies not only to probability of solubility, as has been observed before in other NPproblems, but also to search cost, the first time this has been observed. Furthermore, the technique applies with equal validity to several different methods of varying problem size. As well as contributing to the understanding of phase transitions, we contribute by allowing much finer grained comparison of algorithms, and for accurate empirical extrapolations of beha...
Some Pitfalls for Experimenters with Random SAT
 Artificial Intelligence
, 1996
"... We consider the use of random CNF formulas in evaluating the performance of SAT testing algorithms, and in particular the role that the phase transition phenomenon plays in this use. Examples from the literature illustrate the importance of understanding the properties of formula distributions prior ..."
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Cited by 19 (3 self)
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We consider the use of random CNF formulas in evaluating the performance of SAT testing algorithms, and in particular the role that the phase transition phenomenon plays in this use. Examples from the literature illustrate the importance of understanding the properties of formula distributions prior to designing an experiment. We expect this to be of increasing importance in the field. 1 Introduction Satisfiability testing lies at the core of many computational problems and because of its close relationship to various reasoning tasks, this is especially so in Artificial Intelligence. Randomly generated CNF formulas are a popular class of test problems for evaluating the performance of SAT testing programs. Not surprisingly, the choice of formula distribution is crucial to the validity of any investigation using random formulas. In [23], we argued that some families of distributions were more useful sources of test material than others, and suggested choosing formulas from the "hard reg...
Protecting data privacy through hardtoreverse negative databases
 In Springer LNCS, editor, In proceedings of the 9th Information Security Conference (ISC’06
, 2006
"... Abstract. The paper extends the idea of negative representations of information for enhancing privacy. Simply put, a set DB of data elements can be represented in terms of its complement set. That is, all the elements not in DB are depicted and DB itself is not explicitly stored. We review the negat ..."
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Cited by 16 (6 self)
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Abstract. The paper extends the idea of negative representations of information for enhancing privacy. Simply put, a set DB of data elements can be represented in terms of its complement set. That is, all the elements not in DB are depicted and DB itself is not explicitly stored. We review the negative database (NDB) representation scheme for storing a negative image compactly and propose a design for depicting a multiple record DB using a collection of NDBs—in contrast to the single NDB approach of previous work. Finally, we present a method for creating negative databases that are hard to reverse in practice, i.e., from which it is hard to obtain DB, by adapting a technique for generating 3SAT formulas. 1