scholarly journals An Introduction to Satisfiability Algorithms

2003 ◽  
Vol 7 (20) ◽  
Author(s):  
Carlos Ansotegui ◽  
Felip Manyà
Keyword(s):  
Satisfiability Algorithms

Chapter 16. Worst-Case Upper Bounds

2021 ◽  
Author(s):  
Evgeny Dantsin ◽  
Edward A. Hirsch
Keyword(s):  
Structural Complexity ◽  
Upper Bounds ◽  
Worst Case ◽  
Exponential Time ◽  
Running Time ◽  
Dichotomy Theorem ◽  
Exponential Time Hypothesis ◽  
Satisfiability Algorithms

The chapter is a survey of ideas and techniques behind satisfiability algorithms with the currently best asymptotic upper bounds on the worst-case running time. The survey also includes related structural-complexity topics such as Schaefer’s dichotomy theorem, reductions between various restricted cases of SAT, the exponential time hypothesis, etc.

Chapter 10. Random Satisfiabiliy

2021 ◽  
Author(s):  
Dimitris Achlioptas
Keyword(s):  
Statistical Physics ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Generative Models ◽  
Research Activity ◽  
Mathematical Properties ◽  
Satisfiability Algorithms ◽  
Resolution Proof ◽  
Intense Research ◽  
Cnf Formulas

In the last twenty years a significant amount of effort has been devoted to the study of randomly generated satisfiability instances. While a number of generative models have been proposed, uniformly random k-CNF formulas are by now the dominant and most studied model. One reason for this is that such formulas enjoy a number of intriguing mathematical properties, including the following: for each k≥3, there is a critical value, rk, of the clauses-to-variables ratio, r, such that for r<rk a random k-CNF formula is satisfiable with probability that tends to 1 as n→∞, while for r>rk it is unsatisfiable with probability that tends to 1 as n→∞. Algorithmically, even at densities much below rk, no polynomial-time algorithm is known that can find any solution even with constant probability, while for all densities greater than rk, the length of every resolution proof of unsatisfiability is exponential (and, thus, so is the running time of every DPLL-type algorithm). By now, the study of random k-CNF formulas has also attracted attention in areas such as mathematics and statistical physics and is at the center of an area of intense research activity. At the same time, random k-SAT instances are a popular benchmark for testing and tuning satisfiability algorithms. Indeed, some of the better practical ideas in use today come from insights gained by studying the performance of algorithms on them. We review old and recent mathematical results about random k-CNF formulas, demonstrating that the connection between computational complexity and phase transitions is both deep and highly nuanced.

Chapter 28. Pseudo-Boolean and Cardinality Constraints

2021 ◽  
Author(s):  
Olivier Roussel ◽  
Vasco Manquinho
Keyword(s):  
Optimization Problems ◽  
Expressive Power ◽  
Natural Generalization ◽  
Satisfiability Problem ◽  
Weighted Sum ◽  
Inference Rules ◽  
Cardinality Constraint ◽  
Cardinality Constraints ◽  
Satisfiability Algorithms

Pseudo-Boolean and cardinality constraints are a natural generalization of clauses. While a clause expresses that at least one literal must be true, a cardinality constraint expresses that at least n literals must be true and a pseudo-Boolean constraint states that a weighted sum of literals must be greater than a constant. These contraints have a high expressive power, have been intensively studied in 0/1 programming and are close enough to the satisfiability problem to benefit from the recents advances in this field. Besides, optimization problems are naturally expressed in the pseudo-Boolean context. This chapter presents the inference rules on pseudo-Boolean constraints and demonstrates their increased inference power in comparison with resolution. It also shows how the modern satisfiability algorithms can be extended to deal with pseudo-Boolean constraints.

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