749 research outputs found
The Satisfiability Threshold for a Seemingly Intractable Random Constraint Satisfaction Problem
We determine the exact threshold of satisfiability for random instances of a
particular NP-complete constraint satisfaction problem (CSP). This is the first
random CSP model for which we have determined a precise linear satisfiability
threshold, and for which random instances with density near that threshold
appear to be computationally difficult. More formally, it is the first random
CSP model for which the satisfiability threshold is known and which shares the
following characteristics with random k-SAT for k >= 3. The problem is
NP-complete, the satisfiability threshold occurs when there is a linear number
of clauses, and a uniformly random instance with a linear number of clauses
asymptotically almost surely has exponential resolution complexity.Comment: This is the long version of a paper that will be published in the
SIAM Journal on Discrete Mathematics. This long version includes an appendix
and a computer program. The contents of the paper are unchanged in the latest
version. The format of the arxiv submission was changed so that the computer
program will appear as an ancillary file. Some comments in the computer
program were update
Scale-Free Random SAT Instances
We focus on the random generation of SAT instances that have properties
similar to real-world instances. It is known that many industrial instances,
even with a great number of variables, can be solved by a clever solver in a
reasonable amount of time. This is not possible, in general, with classical
randomly generated instances. We provide a different generation model of SAT
instances, called \emph{scale-free random SAT instances}. It is based on the
use of a non-uniform probability distribution to select
variable , where is a parameter of the model. This results into
formulas where the number of occurrences of variables follows a power-law
distribution where . This property
has been observed in most real-world SAT instances. For , our model
extends classical random SAT instances.
We prove the existence of a SAT-UNSAT phase transition phenomenon for
scale-free random 2-SAT instances with when the clause/variable
ratio is . We also prove that scale-free
random k-SAT instances are unsatisfiable with high probability when the number
of clauses exceeds . %This implies that the SAT/UNSAT
phase transition phenomena vanishes when , and formulas are
unsatisfiable due to a small core of clauses. The proof of this result suggests
that, when , the unsatisfiability of most formulas may be due to
small cores of clauses. Finally, we show how this model will allow us to
generate random instances similar to industrial instances, of interest for
testing purposes
Phase Transitions of the Typical Algorithmic Complexity of the Random Satisfiability Problem Studied with Linear Programming
Here we study the NP-complete -SAT problem. Although the worst-case
complexity of NP-complete problems is conjectured to be exponential, there
exist parametrized random ensembles of problems where solutions can typically
be found in polynomial time for suitable ranges of the parameter. In fact,
random -SAT, with as control parameter, can be solved quickly
for small enough values of . It shows a phase transition between a
satisfiable phase and an unsatisfiable phase. For branch and bound algorithms,
which operate in the space of feasible Boolean configurations, the empirically
hardest problems are located only close to this phase transition. Here we study
-SAT () and the related optimization problem MAX-SAT by a linear
programming approach, which is widely used for practical problems and allows
for polynomial run time. In contrast to branch and bound it operates outside
the space of feasible configurations. On the other hand, finding a solution
within polynomial time is not guaranteed. We investigated several variants like
including artificial objective functions, so called cutting-plane approaches,
and a mapping to the NP-complete vertex-cover problem. We observed several
easy-hard transitions, from where the problems are typically solvable (in
polynomial time) using the given algorithms, respectively, to where they are
not solvable in polynomial time. For the related vertex-cover problem on random
graphs these easy-hard transitions can be identified with structural properties
of the graphs, like percolation transitions. For the present random -SAT
problem we have investigated numerous structural properties also exhibiting
clear transitions, but they appear not be correlated to the here observed
easy-hard transitions. This renders the behaviour of random -SAT more
complex than, e.g., the vertex-cover problem.Comment: 11 pages, 5 figure
Community structure in industrial SAT instances
Modern SAT solvers have experienced a remarkable progress on solving industrial instances. It is believed that most of these successful techniques exploit the underlying structure of industrial instances. Recently, there have been some attempts to analyze the structure of industrial SAT instances in terms of complex networks, with the aim of explaining the success of SAT solving techniques, and possibly improving them.
In this paper, we study the community structure, or modularity, of industrial SAT instances. In a graph with clear community structure, or high modularity, we can find a partition of its nodes into communities such that most edges connect variables of the same community. Representing SAT instances as graphs, we show that most application benchmarks are characterized by a high modularity. On the contrary, random SAT instances are closer to the classical Erdös-Rényi random graph model, where no structure can be observed. We also analyze how this structure evolves by the effects of the execution of a CDCL SAT solver, and observe that new clauses learned by the solver during the search contribute to destroy the original structure of the formula. Motivated by this observation, we finally present an application that exploits the community structure to detect relevant learned clauses, and we show that detecting these clauses results in an improvement on the performance of the SAT solver. Empirically, we observe that this improves the performance of several SAT solvers on industrial SAT formulas, especially on satisfiable instances.Peer ReviewedPostprint (published version
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