202,826 research outputs found
Exactly solvable models of adaptive networks
A satisfiability (SAT-UNSAT) transition takes place for many optimization
problems when the number of constraints, graphically represented by links
between variables nodes, is brought above some threshold. If the network of
constraints is allowed to adapt by redistributing its links, the SAT-UNSAT
transition may be delayed and preceded by an intermediate phase where the
structure self-organizes to satisfy the constraints. We present an analytic
approach, based on the recently introduced cavity method for large deviations,
which exactly describes the two phase transitions delimiting this adaptive
intermediate phase. We give explicit results for random bond models subject to
the connectivity or rigidity percolation transitions, and compare them with
numerical simulations.Comment: 4 pages, 4 figure
The condensation phase transition in the regular -SAT model
Much of the recent work on random constraint satisfaction problems has been
inspired by ingenious but non-rigorous approaches from physics. The physics
predictions typically come in the form of distributional fixed point problems
that are intended to mimic Belief Propagation, a message passing algorithm,
applied to the random CSP. In this paper we propose a novel method for
harnessing Belief Propagation directly to obtain a rigorous proof of such a
prediction, namely the existence and location of a condensation phase
transition in the random regular -SAT model.Comment: Revised version based on arXiv:1504.03975, version
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
Improving WalkSAT for Random 3-SAT Problems
Stochastic local search (SLS) algorithms are well known for their ability to efficiently find models of random instances of the Boolean satisfiability (SAT) problems. One of the most famous SLS algorithms for SAT is called WalkSAT, which has wide influence and performs well on most of random 3-SAT instances. However, the performance of WalkSAT lags far behind on random 3-SAT instances equal to or greater than the phase transition ratio. Motivated by this limitation, in the present work, firstly an allocation strategy is introduced and utilized in WalkSAT to determine the initial assignment, leading to a new algorithm called WalkSATvav. The experimental results show that WalkSATvav significantly outperforms the state-of-the-art SLS solvers on random 3-SAT instances at the phase transition for SAT Competition 2017. However, WalkSATvav cannot rival its competitors on random 3-SAT instances greater than the phase transition ratio. Accordingly, WalkSATvav is further improved for such instances by utilizing a combination of an improved genetic algorithm and an improved ant colony algorithm, which complement each other in guiding the search direction. The resulting algorithm, called WalkSATga, is far better than WalkSAT and significantly outperforms some previous known SLS solvers on random 3-SAT instances greater than the phase transition ratio from SAT Competition 2017. Finally, a new SAT solver called WalkSATlg, which combines WalkSATvav and WalkSATga, is proposed, which is competitive with the winner of random satisfiable category of SAT competition 2017 on random 3-SAT problem
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