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18
Finding Hard Instances of the Satisfiability Problem: A Survey
, 1997
"... . Finding sets of hard instances of propositional satisfiability is of interest for understanding the complexity of SAT, and for experimentally evaluating SAT algorithms. In discussing this we consider the performance of the most popular SAT algorithms on random problems, the theory of average case ..."
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Cited by 114 (1 self)
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. Finding sets of hard instances of propositional satisfiability is of interest for understanding the complexity of SAT, and for experimentally evaluating SAT algorithms. In discussing this we consider the performance of the most popular SAT algorithms on random problems, the theory of average case complexity, the threshold phenomenon, known lower bounds for certain classes of algorithms, and the problem of generating hard instances with solutions.
Lookahead versus lookback for satisfiability problems
 THIRD INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING, LECTURE NOTES IN COMPUTER SCIENCE
, 1997
"... CNF propositional satis ability (SAT) is a special kind of the more general Constraint Satisfaction Problem (CSP). While lookback techniques appear to be of little use to solve hard random SAT problems, it is supposed that they are necessary to solve hard structured SAT problems. In this paper, we ..."
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Cited by 62 (1 self)
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CNF propositional satis ability (SAT) is a special kind of the more general Constraint Satisfaction Problem (CSP). While lookback techniques appear to be of little use to solve hard random SAT problems, it is supposed that they are necessary to solve hard structured SAT problems. In this paper, we propose a very simple DPL procedure called Satz which only employs some lookahead techniques: a variable ordering heuristic, a forward consistency checking (Unit Propagation) and a limited resolution before the search, where the heuristic is itself based on unit propagation. Satz is favorably compared on random 3SAT problems with three DPL procedures among the best in the literature for these problems. Furthermore on a great number of problems in 4 wellknown SAT benchmarks Satz reaches or outspeeds the performance of three other DPL procedures among the best in the literature for structured SAT problems. The comparative results suggest that a suitable exploitation of lookahead techniques, while very simple and efficient for random SAT problems, may allow to do without sophisticated lookback techniques in a DPL procedure.
Maintaining ArcConsistency within Dynamic Backtracking
 IN PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING (CP 2000), NUMBER 1894 IN LECTURE NOTES IN COMPUTER SCIENCE
, 2000
"... Most of complete search algorithms over Constraint Satisfaction Problems (csp) are based on Standard Backtracking. Two main ..."
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Cited by 59 (14 self)
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Most of complete search algorithms over Constraint Satisfaction Problems (csp) are based on Standard Backtracking. Two main
Frozen Development in Graph Coloring
 THEORETICAL COMPUTER SCIENCE
, 2000
"... We define the `frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are the same color in all legal colorings. This is analogous to studies of the development of a backbone or spine in SAT (the Satisability problem). We first describe in detail the algori ..."
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Cited by 34 (5 self)
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We define the `frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are the same color in all legal colorings. This is analogous to studies of the development of a backbone or spine in SAT (the Satisability problem). We first describe in detail the algorithmic techniques used to study frozen development. We present strong empirical evidence that freezing in 3coloring is sudden. A single edge typically causes the size of the graph to collapse in size by 28%. We also use the frozen development to calculate unbiased estimates of probability of colorability in random graphs, even where this probability is as low as 10^300. We investigate the links between frozen development and the solution cost of graph coloring. In SAT, a discontinuity in the order parameter has been correlated with the hardness of SAT instances, and our data for coloring is suggestive of an asymptotic discontinuity. The uncolorability threshold is known to give rise to har...
Approximating Minimal Unsatisfiable Subformulae by Means of Adaptive Core Search
 Discrete Applied Mathematics
, 2002
"... The paper is concerned with the relevant practical problem of selecting a small unsatisfiable subset of clauses inside an unsatisfiable CNF formula. Moreover, it deals with the algorithmic problem of improving an enumerative (DPLLstyle) approach to SAT, in order to overcome some structural defects ..."
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Cited by 20 (1 self)
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The paper is concerned with the relevant practical problem of selecting a small unsatisfiable subset of clauses inside an unsatisfiable CNF formula. Moreover, it deals with the algorithmic problem of improving an enumerative (DPLLstyle) approach to SAT, in order to overcome some structural defects of such approach. Within a complete solution framework, we are able to evaluate the di#culty of each clause, by analyzing the history of the search. Such clause hardness evaluation is used in order to rapidly select an unsatisfiable subformula (of the given CNF) which is a good approximation of a minimal unsatisfiable subformula (MUS). Unsatisfiability is proved by solving only such subformula. Very small unsatisfiable subformulae are detected inside famous Dimacs unsatisfiable problems and in real world problems. Comparison with the very e#cient solver SATO 3.2 used as a stateoftheart DPLL procedure (disabling learning of new clauses) shows the e#ectiveness of such enumeration guide.
Well out of reach: Why hard problems are hard
 APES RESEARCH GROUP
, 1999
"... We show that problems at the uncolorability phase transition are well out of reach of intelligent algorithms. Since there are not small and easily checkable subgraphs which can be used to confirm uncolorability quickly, we cannot hope to build more intelligent algorithms to avoid hard problems at t ..."
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Cited by 16 (5 self)
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We show that problems at the uncolorability phase transition are well out of reach of intelligent algorithms. Since there are not small and easily checkable subgraphs which can be used to confirm uncolorability quickly, we cannot hope to build more intelligent algorithms to avoid hard problems at the phase transition. Also, our results suggest that a conjectured double phase transition in graph coloring occurs only in small graphs. Similar results are likely in other NPcomplete problems where instances from phase transitions are hard for all known algorithms, and will help to explain the phenomenon. Furthermore, our results help to elucidate the distinction between polynomial and nonpolynomial search behavior.
Using Configurable Computing to Accelerate Boolean Satisfiability
, 1999
"... The issues of software compute time and complexity are very important in current CAD tools. As FPGA speeds and densities increase, the opportunity for effective hardware accelerators built from FPGA technology has opened up. This paper describes and evaluates a formulaspecific method for implementi ..."
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Cited by 13 (0 self)
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The issues of software compute time and complexity are very important in current CAD tools. As FPGA speeds and densities increase, the opportunity for effective hardware accelerators built from FPGA technology has opened up. This paper describes and evaluates a formulaspecific method for implementing Boolean satisfiability solver circuits in configurable hardware. That is, using a template generator, we create circuits specific to the problem instance to be solved. This approach yields impressive runtime speedups of up to several hundred times compared to the software approaches. The high performance comes from realizing finegrained parallelism inherent in the clause evaluation and implication and from direct mapping of Boolean relations into logic gates. Our implementation uses a commerciallyavailable hardware system for proof of concept. This system yields more than 100 times runtime speedup on many problems, even though the clock rate of the hardware is 100 times slower than tha...
Backtracking Search Algorithms
, 2006
"... There are three main algorithmic techniques for solving constraint satisfaction problems: backtracking search, local search, and dynamic programming. In this chapter, I survey backtracking search algorithms. Algorithms based on dynamic programming [15]— sometimes referred to in the literature as var ..."
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Cited by 10 (2 self)
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There are three main algorithmic techniques for solving constraint satisfaction problems: backtracking search, local search, and dynamic programming. In this chapter, I survey backtracking search algorithms. Algorithms based on dynamic programming [15]— sometimes referred to in the literature as variable elimination, synthesis, or inference algorithms—are the topic of Chapter 7. Local or stochastic search algorithms are the topic of Chapter 5. An algorithm for solving a constraint satisfaction problem (CSP) can be either complete or incomplete. Complete, or systematic algorithms, come with a guarantee that a solution will be found if one exists, and can be used to show that a CSP does not have a solution and to find a provably optimal solution. Backtracking search algorithms and dynamic programming algorithms are, in general, examples of complete algorithms. Incomplete, or nonsystematic algorithms, cannot be used to show a CSP does not have a solution or to find a provably optimal solution. However, such algorithms are often effective at finding a solution if one exists and can be used to find an approximation to an optimal solution. Local or stochastic search algorithms are examples of incomplete algorithms. Of the two
Search algorithms in type theory
, 2000
"... In this paper, we take an abstract view of search by describing search procedures via particular kinds of proofs in type theory. We rely on the proofsasprograms interpretation to extract programs from our proofs. Using these techniques we explore, in depth, a large family of search problems by par ..."
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Cited by 8 (2 self)
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In this paper, we take an abstract view of search by describing search procedures via particular kinds of proofs in type theory. We rely on the proofsasprograms interpretation to extract programs from our proofs. Using these techniques we explore, in depth, a large family of search problems by parameterizing the speci cation of the problem. A constructive proof is presented which has as its computational content a correct search procedure for these problems. We show how a classical extension to an otherwise constructive system can be used to describe a typical use of the nonlocal control operator call/cc. Using the classical typing of nonlocal control we extend our purely constructive proof to incorporate a sophisticated backtracking technique known as ‘con ictdirected backjumping’ (CBJ). A variant of this proof is formalized in Nuprl yielding a correctbyconstruction implementation of CBJ. The extracted program has been translated into Scheme and serves as the basis for an implementation of a new solution to the Hamiltonian circuit problem. This paper demonstrates a nontrivial application of the proofsasprograms paradigm by applying the technique to the derivation of a sophisticated search algorithm; also, it shows the generality of the resulting implementation by demonstrating its application in a new problem
The Logic of Search Algorithms: Theory and Applications
 In Principles and Practice of Constraint Programming { CP97
, 1997
"... . Many search algorithms have been introduced without correctness proofs, or proved only with respect to an informal semantics of the algorithm. We address this problem by taking advantage of the correspondence between programs and proofs. We give a single proof of the correctness of a very general ..."
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Cited by 5 (2 self)
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. Many search algorithms have been introduced without correctness proofs, or proved only with respect to an informal semantics of the algorithm. We address this problem by taking advantage of the correspondence between programs and proofs. We give a single proof of the correctness of a very general search algorithm, for which we provide Scheme code. It is straightforward to implement service functions to implement algorithms such as DavisPutnam for satisfiability or forward checking (FC) for constraint satisfaction, and to incorporate conflictdirected backjumping (CBJ) and heuristics for variable and value ordering. By separating the search algorithm from problem features, our work should enable the much speedier implementation of sophisticated search methods such as FCCBJ in new domains, and we illustrate this by sketching an implementation for the Hamiltonian Circuit problem. 1 Introduction The constraint satisfaction community has an excellent record of introducing intelligent se...