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22
HeavyTailed Phenomena in Satisfiability and Constraint Satisfaction Problems
 J. of Autom. Reasoning
, 2000
"... Abstract. We study the runtime distributions of backtrack procedures for propositional satisfiability and constraint satisfaction. Such procedures often exhibit a large variability in performance. Our study reveals some intriguing properties of such distributions: They are often characterized by ver ..."
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Cited by 164 (27 self)
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Abstract. We study the runtime distributions of backtrack procedures for propositional satisfiability and constraint satisfaction. Such procedures often exhibit a large variability in performance. Our study reveals some intriguing properties of such distributions: They are often characterized by very long tails or “heavy tails”. We will show that these distributions are best characterized by a general class of distributions that can have infinite moments (i.e., an infinite mean, variance, etc.). Such nonstandard distributions have recently been observed in areas as diverse as economics, statistical physics, and geophysics. They are closely related to fractal phenomena, whose study was introduced by Mandelbrot. We also show how random restarts can effectively eliminate heavytailed behavior. Furthermore, for harder problem instances, we observe long tails on the lefthand side of the distribution, which is indicative of a nonnegligible fraction of relatively short, successful runs. A rapid restart strategy eliminates heavytailed behavior and takes advantage of short runs, significantly reducing expected solution time. We demonstrate speedups of up to two orders of magnitude on SAT and CSP encodings of hard problems in planning, scheduling, and circuit synthesis. Key words: satisfiability, constraint satisfaction, heavy tails, backtracking 1.
The hardest constraint problems: A double phase transition
 Artif. Intell
, 1994
"... The distribution of hard graph coloring problems as a function of graph connectivity is shown to have two distinct transition behaviors. The first, previously recognized, is a peak in the median search cost near the connectivity at which half the graphs have solutions. This region contains a high pr ..."
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Cited by 95 (2 self)
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The distribution of hard graph coloring problems as a function of graph connectivity is shown to have two distinct transition behaviors. The first, previously recognized, is a peak in the median search cost near the connectivity at which half the graphs have solutions. This region contains a high proportion of relatively hard problem instances. However, the hardest instances are in fact concentrated at a second, lower, transition point. Near this point, most problems are quite easy, but there are also a few very hard cases. This region of exceptionally hard problems corresponds to the transition between polynomial and exponential scaling of the average search cost, whose location we also estimate theoretically. These behaviors also appear to arise in other constraint problems. This work also shows the limitations of simple measures of the cost distribution, such as mean or median, for identifying outlying cases. 1
Towards a characterisation of the behaviour of stochastic local search algorithms for SAT
 ARTIFICIAL INTELLIGENCE
, 1999
"... Stochastic local search (SLS) algorithms have been successfully applied to hard combinatorial problems from different domains. Due to their inherent randomness, the runtime behaviour of these algorithms is characterised by a random variable. The detailed knowledge of the runtime distribution provi ..."
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Cited by 56 (16 self)
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Stochastic local search (SLS) algorithms have been successfully applied to hard combinatorial problems from different domains. Due to their inherent randomness, the runtime behaviour of these algorithms is characterised by a random variable. The detailed knowledge of the runtime distribution provides important information about the behaviour of SLS algorithms. In this paper we investigate the empirical runtime distributions for Walksat, one of the most powerful SLS algorithms for the Propositional Satisfiability Problem (SAT). Using statistical analysis techniques, we show that on hard Random3SAT problems, Walksat's runtime behaviour can be characterised by exponential distributions. This characterisation can be generalised to various SLS algorithms for SAT and to encoded problems from other domains. This result also has a number of consequences which are of theoretical as well as practical interest. One of these is the fact that these algorithms can be easily parallelised such that optimal speedup is achieved for hard problem instances.
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 48 (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,
Systematic versus stochastic constraint satisfaction
 Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI’95
, 1995
"... ..."
Exploiting a Theory of Phase Transitions in ThreeSatisfiability Problems
 In Proc. AAAI '96
, 1996
"... In the past few years there have been several empirical discoveries of phase transitions in constraint satisfaction problems (CSPs), and a growth of interest in the area among the artificial intelligence community. This paper extends a simple analytical theory of phase transitions in threesatisfiab ..."
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Cited by 9 (1 self)
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In the past few years there have been several empirical discoveries of phase transitions in constraint satisfaction problems (CSPs), and a growth of interest in the area among the artificial intelligence community. This paper extends a simple analytical theory of phase transitions in threesatisfiability (3SAT) problems in two directions. First, a more accurate, problemdependent calculation leads to a new polynomial time probabilistic estimate of the satisfiability of 3SAT problems called PESAT (Probabilistic Estimate SATisfiability algorithm). PESAT empirically classifies 3SAT problems with about 70% accuracy at the hardest region (the socalled crossover point or 50% satisfiable region) of random 3SAT space. Furthermore, the estimate has a meaningful magnitude such that extreme estimates are much more likely to be correct. Second, the same estimate is used to improve the running time of a backtracking search for a solution to 3SAT by pruning unlikely branches of the search. T...
Backdoors to Combinatorial Optimization: Feasibility and Optimality
 CPAIOR
, 2009
"... Abstract. There has been considerable interest in the identification of structural properties of combinatorial problems that lead, directly or indirectly, to the development of efficient algorithms for solving them. One such concept is that of a backdoor set—a set of variables such that once they ar ..."
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Cited by 7 (4 self)
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Abstract. There has been considerable interest in the identification of structural properties of combinatorial problems that lead, directly or indirectly, to the development of efficient algorithms for solving them. One such concept is that of a backdoor set—a set of variables such that once they are instantiated, the remaining problem simplifies to a tractable form. While backdoor sets were originally defined to capture structure in decision problems with discrete variables, here we introduce a notion of backdoors that captures structure in optimization problems, which often have both discrete and continuous variables. We show that finding a feasible solution and proving optimality are characterized by backdoors of different kinds and size. Surprisingly, in certain mixed integer programming problems, proving optimality involves a smaller backdoor set than finding the optimal solution. We also show extensive results on the number of backdoors of various sizes in optimization problems. Overall, this work demonstrates that backdoors, appropriately generalized, are also effective in capturing problem structure in optimization problems.
The GSAT/SAFamiliy  Relating greedy satisifability testing to simulated annealing
, 1994
"... In this paper, we investigate and relate various variants of the greedy satisfiability tester GSAT. We present these algorithms as members of a whole family of algorithms for finding a model for satisfiable propositional logic formulas. In particular, all algorithms can be formulated as instances of ..."
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Cited by 6 (6 self)
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In this paper, we investigate and relate various variants of the greedy satisfiability tester GSAT. We present these algorithms as members of a whole family of algorithms for finding a model for satisfiable propositional logic formulas. In particular, all algorithms can be formulated as instances of the same generic frame GenSAT. Comparing the algorithms, we do not only concentrate on their overall performance, but are also interested in how properties like locality or different kinds of randomness influence the performance. To the end we define a new, theoretically complete instance of GenSAT. This variant can be viewed as a reformulation of simulated annealing (SA) within the GSAT family and thus, defines a link between GSAT and SA. For most of the algorithms experiments have been performed using very hard, randomly generated propositional logic formulas. The results of these experiments are also reported.
PORTFOLIOS WITH DEADLINES FOR BACKTRACKING SEARCH
 INTERNATIONAL JOURNAL ON ARTIFICIAL INTELLIGENCE TOOLS
, 2008
"... Backtracking search is often the method of choice for solving constraint satisfaction and propositional satisfiability problems. Previous studies have shown that portfolios of backtracking algorithms—a selection of one or more algorithms plus a schedule for executing the algorithms—can dramatically ..."
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Cited by 4 (0 self)
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Backtracking search is often the method of choice for solving constraint satisfaction and propositional satisfiability problems. Previous studies have shown that portfolios of backtracking algorithms—a selection of one or more algorithms plus a schedule for executing the algorithms—can dramatically improve performance on some instances. In this paper, we consider a setting that often arises in practice where the instances to be solved arise over time, the instances all belong to some class of problem instances, and a limit or deadline is placed on the computational resources that can be consumed in solving any instance. For such a scenario, we present a simple scheme for learning a good portfolio of backtracking algorithms from a small sample of instances. We demonstrate the effectiveness of our approach through an extensive empirical evaluation using two testbeds: realworld instruction scheduling problems and the widely used quasigroup completion problems.