10,273 research outputs found
The Satisfiability Threshold of Random 3-SAT Is at Least 3.52
We prove that a random 3-SAT instance with clause-to-variable density less
than 3.52 is satisfiable with high probability. The proof comes through an
algorithm which selects (and sets) a variable depending on its degree and that
of its complement
Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems
Semidefinite programming is a powerful tool in the design and analysis of
approximation algorithms for combinatorial optimization problems. In
particular, the random hyperplane rounding method of Goemans and Williamson has
been extensively studied for more than two decades, resulting in various
extensions to the original technique and beautiful algorithms for a wide range
of applications. Despite the fact that this approach yields tight approximation
guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and
Max-DiCut, the tight approximation ratio is still unknown. One of the main
reasons for this is the fact that very few techniques for rounding semidefinite
relaxations are known.
In this work, we present a new general and simple method for rounding
semi-definite programs, based on Brownian motion. Our approach is inspired by
recent results in algorithmic discrepancy theory. We develop and present tools
for analyzing our new rounding algorithms, utilizing mathematical machinery
from the theory of Brownian motion, complex analysis, and partial differential
equations. Focusing on constraint satisfaction problems, we apply our method to
several classical problems, including Max-Cut, Max-2SAT, and MaxDiCut, and
derive new algorithms that are competitive with the best known results. To
illustrate the versatility and general applicability of our approach, we give
new approximation algorithms for the Max-Cut problem with side constraints that
crucially utilizes measure concentration results for the Sticky Brownian
Motion, a feature missing from hyperplane rounding and its generalization
A Continuous-Discontinuous Second-Order Transition in the Satisfiability of Random Horn-SAT Formulas
We compute the probability of satisfiability of a class of random Horn-SAT
formulae, motivated by a connection with the nonemptiness problem of finite
tree automata. In particular, when the maximum clause length is 3, this model
displays a curve in its parameter space along which the probability of
satisfiability is discontinuous, ending in a second-order phase transition
where it becomes continuous. This is the first case in which a phase transition
of this type has been rigorously established for a random constraint
satisfaction problem
Heuristic average-case analysis of the backtrack resolution of random 3-Satisfiability instances
An analysis of the average-case complexity of solving random 3-Satisfiability
(SAT) instances with backtrack algorithms is presented. We first interpret
previous rigorous works in a unifying framework based on the statistical
physics notions of dynamical trajectories, phase diagram and growth process. It
is argued that, under the action of the Davis--Putnam--Loveland--Logemann
(DPLL) algorithm, 3-SAT instances are turned into 2+p-SAT instances whose
characteristic parameters (ratio alpha of clauses per variable, fraction p of
3-clauses) can be followed during the operation, and define resolution
trajectories. Depending on the location of trajectories in the phase diagram of
the 2+p-SAT model, easy (polynomial) or hard (exponential) resolutions are
generated. Three regimes are identified, depending on the ratio alpha of the
3-SAT instance to be solved. Lower sat phase: for small ratios, DPLL almost
surely finds a solution in a time growing linearly with the number N of
variables. Upper sat phase: for intermediate ratios, instances are almost
surely satisfiable but finding a solution requires exponential time (2 ^ (N
omega) with omega>0) with high probability. Unsat phase: for large ratios,
there is almost always no solution and proofs of refutation are exponential. An
analysis of the growth of the search tree in both upper sat and unsat regimes
is presented, and allows us to estimate omega as a function of alpha. This
analysis is based on an exact relationship between the average size of the
search tree and the powers of the evolution operator encoding the elementary
steps of the search heuristic.Comment: to appear in Theoretical Computer Scienc
Threshold Saturation in Spatially Coupled Constraint Satisfaction Problems
We consider chains of random constraint satisfaction models that are
spatially coupled across a finite window along the chain direction. We
investigate their phase diagram at zero temperature using the survey
propagation formalism and the interpolation method. We prove that the SAT-UNSAT
phase transition threshold of an infinite chain is identical to the one of the
individual standard model, and is therefore not affected by spatial coupling.
We compute the survey propagation complexity using population dynamics as well
as large degree approximations, and determine the survey propagation threshold.
We find that a clustering phase survives coupling. However, as one increases
the range of the coupling window, the survey propagation threshold increases
and saturates towards the phase transition threshold. We also briefly discuss
other aspects of the problem. Namely, the condensation threshold is not
affected by coupling, but the dynamic threshold displays saturation towards the
condensation one. All these features may provide a new avenue for obtaining
better provable algorithmic lower bounds on phase transition thresholds of the
individual standard model
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