14,119 research outputs found
On the Hardness of SAT with Community Structure
Recent attempts to explain the effectiveness of Boolean satisfiability (SAT)
solvers based on conflict-driven clause learning (CDCL) on large industrial
benchmarks have focused on the concept of community structure. Specifically,
industrial benchmarks have been empirically found to have good community
structure, and experiments seem to show a correlation between such structure
and the efficiency of CDCL. However, in this paper we establish hardness
results suggesting that community structure is not sufficient to explain the
success of CDCL in practice. First, we formally characterize a property shared
by a wide class of metrics capturing community structure, including
"modularity". Next, we show that the SAT instances with good community
structure according to any metric with this property are still NP-hard.
Finally, we study a class of random instances generated from the
"pseudo-industrial" community attachment model of Gir\'aldez-Cru and Levy. We
prove that, with high probability, instances from this model that have
relatively few communities but are still highly modular require exponentially
long resolution proofs and so are hard for CDCL. We also present experimental
evidence that our result continues to hold for instances with many more
communities. This indicates that actual industrial instances easily solved by
CDCL may have some other relevant structure not captured by the community
attachment model.Comment: 23 pages. Full version of a SAT 2016 pape
Characterizing the Temperature of SAT Formulas
The remarkable advances in SAT solving achieved in the last years have allowed to use this technology to solve many real-world applications, such as planning, formal verification and cryptography, among others. Interestingly, these industrial SAT problems are commonly believed to be easier than classical random SAT formulas, but estimating their actual hardness is still a very challenging question, which in some cases even requires to solve them. In this context, realistic pseudo-industrial random SAT generators have emerged with the aim of reproducing the main features of these application problems to better understand the success of those SAT solving techniques on them. In this work, we present a model to estimate the temperature of real-world SAT instances. This temperature represents the degree of distortion into the expected structure of the formula, from highly structured benchmarks (more similar to real-world SAT instances) to the complete absence of structure (observed in the classical random SAT model). Our solution is based on the popularity–similarity random model for SAT, which has been recently presented to reproduce two crucial features of application SAT benchmarks: scale-free and community structures. This model is able to control the hardness of the generated formula by introducing some randomizations in the expected structure. Using our regression model, we observe that the estimated temperature of the applications benchmarks used in the last SAT Competitions correlates to their hardness in most of the cases.Juan de la Cierva program, fellowship IJC2019-040489-I, funded by MCIN and AE
What makes a phase transition? Analysis of the random satisfiability problem
In the last 30 years it was found that many combinatorial systems undergo
phase transitions. One of the most important examples of these can be found
among the random k-satisfiability problems (often referred to as k-SAT), asking
whether there exists an assignment of Boolean values satisfying a Boolean
formula composed of clauses with k random variables each. The random 3-SAT
problem is reported to show various phase transitions at different critical
values of the ratio of the number of clauses to the number of variables. The
most famous of these occurs when the probability of finding a satisfiable
instance suddenly drops from 1 to 0. This transition is associated with a rise
in the hardness of the problem, but until now the correlation between any of
the proposed phase transitions and the hardness is not totally clear. In this
paper we will first show numerically that the number of solutions universally
follows a lognormal distribution, thereby explaining the puzzling question of
why the number of solutions is still exponential at the critical point.
Moreover we provide evidence that the hardness of the closely related problem
of counting the total number of solutions does not show any phase
transition-like behavior. This raises the question of whether the probability
of finding a satisfiable instance is really an order parameter of a phase
transition or whether it is more likely to just show a simple sharp threshold
phenomenon. More generally, this paper aims at starting a discussion where a
simple sharp threshold phenomenon turns into a genuine phase transition
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