82 research outputs found
Pairs of SAT Assignment in Random Boolean Formulae
We investigate geometrical properties of the random K-satisfiability problem
using the notion of x-satisfiability: a formula is x-satisfiable if there exist
two SAT assignments differing in Nx variables. We show the existence of a sharp
threshold for this property as a function of the clause density. For large
enough K, we prove that there exists a region of clause density, below the
satisfiability threshold, where the landscape of Hamming distances between SAT
assignments experiences a gap: pairs of SAT-assignments exist at small x, and
around x=1/2, but they donot exist at intermediate values of x. This result is
consistent with the clustering scenario which is at the heart of the recent
heuristic analysis of satisfiability using statistical physics analysis (the
cavity method), and its algorithmic counterpart (the survey propagation
algorithm). The method uses elementary probabilistic arguments (first and
second moment methods), and might be useful in other problems of computational
and physical interest where similar phenomena appear
The Satisfiability Threshold for Non-Uniform Random 2-SAT
Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. Its worst-case hardness lies at the core of computational complexity theory, for example in the form of NP-hardness and the (Strong) Exponential Time Hypothesis. In practice however, SAT instances can often be solved efficiently. This contradicting behavior has spawned interest in the average-case analysis of SAT and has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures.
Despite a long line of research and substantial progress, most theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a non-uniform distribution of the variables, which can result in distributions closer to industrial SAT instances.
We study satisfiability thresholds of non-uniform random 2-SAT with n variables and m clauses and with an arbitrary probability distribution (p_i)_{i in[n]} with p_1 >=slant p_2 >=slant ... >=slant p_n>0 over the n variables. We show for p_{1}^2=Theta (sum_{i=1}^n p_i^2) that the asymptotic satisfiability threshold is at {m=Theta ((1-{sum_{i=1}^n p_i^2})/(p_1 * (sum_{i=2}^n p_i^2)^{1/2}))} and that it is coarse. For p_{1}^2=o (sum_{i=1}^n p_i^2) we show that there is a sharp satisfiability threshold at m=(sum_{i=1}^n p_i^2)^{-1}. This result generalizes the seminal works by Chvatal and Reed [FOCS 1992] and by Goerdt [JCSS 1996]
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
Optimal Testing for Planted Satisfiability Problems
We study the problem of detecting planted solutions in a random
satisfiability formula. Adopting the formalism of hypothesis testing in
statistical analysis, we describe the minimax optimal rates of detection. Our
analysis relies on the study of the number of satisfying assignments, for which
we prove new results. We also address algorithmic issues, and give a
computationally efficient test with optimal statistical performance. This
result is compared to an average-case hypothesis on the hardness of refuting
satisfiability of random formulas
On the critical exponents of random k-SAT
There has been much recent interest in the satisfiability of random Boolean
formulas. A random k-SAT formula is the conjunction of m random clauses, each
of which is the disjunction of k literals (a variable or its negation). It is
known that when the number of variables n is large, there is a sharp transition
from satisfiability to unsatisfiability; in the case of 2-SAT this happens when
m/n --> 1, for 3-SAT the critical ratio is thought to be m/n ~ 4.2. The
sharpness of this transition is characterized by a critical exponent, sometimes
called \nu=\nu_k (the smaller the value of \nu the sharper the transition).
Experiments have suggested that \nu_3 = 1.5+-0.1, \nu_4 = 1.25+-0.05,
\nu_5=1.1+-0.05, \nu_6 = 1.05+-0.05, and heuristics have suggested that \nu_k
--> 1 as k --> infinity. We give here a simple proof that each of these
exponents is at least 2 (provided the exponent is well-defined). This result
holds for each of the three standard ensembles of random k-SAT formulas: m
clauses selected uniformly at random without replacement, m clauses selected
uniformly at random with replacement, and each clause selected with probability
p independent of the other clauses. We also obtain similar results for
q-colorability and the appearance of a q-core in a random graph.Comment: 11 pages. v2 has revised introduction and updated reference
Computational Complexity and Phase Transitions
Phase transitions in combinatorial problems have recently been shown to be
useful in locating "hard" instances of combinatorial problems. The connection
between computational complexity and the existence of phase transitions has
been addressed in Statistical Mechanics and Artificial Intelligence, but not
studied rigorously.
We take a step in this direction by investigating the existence of sharp
thresholds for the class of generalized satisfiability problems defined by
Schaefer. In the case when all constraints are clauses we give a complete
characterization of such problems that have a sharp threshold.
While NP-completeness does not imply (even in this restricted case) the
existence of a sharp threshold, it "almost implies" this, since clausal
generalized satisfiability problems that lack a sharp threshold are either
1. polynomial time solvable, or
2. predicted, with success probability lower bounded by some positive
constant by across all the probability range, by a single, trivial procedure.Comment: A (slightly) revised version of the paper submitted to the 15th IEEE
Conference on Computational Complexit
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