543 research outputs found
A variational description of the ground state structure in random satisfiability problems
A variational approach to finite connectivity spin-glass-like models is
developed and applied to describe the structure of optimal solutions in random
satisfiability problems. Our variational scheme accurately reproduces the known
replica symmetric results and also allows for the inclusion of replica symmetry
breaking effects. For the 3-SAT problem, we find two transitions as the ratio
of logical clauses per Boolean variables increases. At the first one
, a non-trivial organization of the solution space in
geometrically separated clusters emerges. The multiplicity of these clusters as
well as the typical distances between different solutions are calculated. At
the second threshold , satisfying assignments disappear
and a finite fraction of variables are overconstrained and
take the same values in all optimal (though unsatisfying) assignments. These
values have to be compared to obtained
from numerical experiments on small instances. Within the present variational
approach, the SAT-UNSAT transition naturally appears as a mixture of a first
and a second order transition. For the mixed -SAT with , the
behavior is as expected much simpler: a unique smooth transition from SAT to
UNSAT takes place at .Comment: 24 pages, 6 eps figures, to be published in Europ. Phys. J.
The random K-satisfiability problem: from an analytic solution to an efficient algorithm
We study the problem of satisfiability of randomly chosen clauses, each with
K Boolean variables. Using the cavity method at zero temperature, we find the
phase diagram for the K=3 case. We show the existence of an intermediate phase
in the satisfiable region, where the proliferation of metastable states is at
the origin of the slowdown of search algorithms. The fundamental order
parameter introduced in the cavity method, which consists of surveys of local
magnetic fields in the various possible states of the system, can be computed
for one given sample. These surveys can be used to invent new types of
algorithms for solving hard combinatorial optimizations problems. One such
algorithm is shown here for the 3-sat problem, with very good performances.Comment: 38 pages, 13 figures; corrected typo
The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective
Among various algorithms designed to exploit the specific properties of
quantum computers with respect to classical ones, the quantum adiabatic
algorithm is a versatile proposition to find the minimal value of an arbitrary
cost function (ground state energy). Random optimization problems provide a
natural testbed to compare its efficiency with that of classical algorithms.
These problems correspond to mean field spin glasses that have been extensively
studied in the classical case. This paper reviews recent analytical works that
extended these studies to incorporate the effect of quantum fluctuations, and
presents also some original results in this direction.Comment: 151 pages, 21 figure
Unstructured Randomness, Small Gaps and Localization
We study the Hamiltonian associated with the quantum adiabatic algorithm with
a random cost function. Because the cost function lacks structure we can prove
results about the ground state. We find the ground state energy as the number
of bits goes to infinity, show that the minimum gap goes to zero exponentially
quickly, and we see a localization transition. We prove that there are no
levels approaching the ground state near the end of the evolution. We do not
know which features of this model are shared by a quantum adiabatic algorithm
applied to random instances of satisfiability since despite being random they
do have bit structure
Quantum adiabatic optimization and combinatorial landscapes
In this paper we analyze the performance of the Quantum Adiabatic Evolution
algorithm on a variant of Satisfiability problem for an ensemble of random
graphs parametrized by the ratio of clauses to variables, . We
introduce a set of macroscopic parameters (landscapes) and put forward an
ansatz of universality for random bit flips. We then formulate the problem of
finding the smallest eigenvalue and the excitation gap as a statistical
mechanics problem. We use the so-called annealing approximation with a
refinement that a finite set of macroscopic variables (versus only energy) is
used, and are able to show the existence of a dynamic threshold
starting with some value of K -- the number of variables in
each clause. Beyond dynamic threshold, the algorithm should take exponentially
long time to find a solution. We compare the results for extended and
simplified sets of landscapes and provide numerical evidence in support of our
universality ansatz. We have been able to map the ensemble of random graphs
onto another ensemble with fluctuations significantly reduced. This enabled us
to obtain tight upper bounds on satisfiability transition and to recompute the
dynamical transition using the extended set of landscapes.Comment: 41 pages, 10 figures; added a paragraph on paper's organization to
the introduction, fixed reference
Clustering of solutions in hard satisfiability problems
We study the structure of the solution space and behavior of local search
methods on random 3-SAT problems close to the SAT/UNSAT transition. Using the
overlap measure of similarity between different solutions found on the same
problem instance we show that the solution space is shrinking as a function of
alpha. We consider chains of satisfiability problems, where clauses are added
sequentially. In each such chain, the overlap distribution is first smooth, and
then develops a tiered structure, indicating that the solutions are found in
well separated clusters. On chains of not too large instances, all solutions
are eventually observed to be in only one small cluster before vanishing. This
condensation transition point is estimated to be alpha_c = 4.26. The transition
approximately obeys finite-size scaling with an apparent critical exponent of
about 1.7. We compare the solutions found by a local heuristic, ASAT, and the
Survey Propagation algorithm up to alpha_c.Comment: 8 pages, 9 figure
The Phase Diagram of 1-in-3 Satisfiability Problem
We study the typical case properties of the 1-in-3 satisfiability problem,
the boolean satisfaction problem where a clause is satisfied by exactly one
literal, in an enlarged random ensemble parametrized by average connectivity
and probability of negation of a variable in a clause. Random 1-in-3
Satisfiability and Exact 3-Cover are special cases of this ensemble. We
interpolate between these cases from a region where satisfiability can be
typically decided for all connectivities in polynomial time to a region where
deciding satisfiability is hard, in some interval of connectivities. We derive
several rigorous results in the first region, and develop the
one-step--replica-symmetry-breaking cavity analysis in the second one. We
discuss the prediction for the transition between the almost surely satisfiable
and the almost surely unsatisfiable phase, and other structural properties of
the phase diagram, in light of cavity method results.Comment: 30 pages, 12 figure
Random subcubes as a toy model for constraint satisfaction problems
We present an exactly solvable random-subcube model inspired by the structure
of hard constraint satisfaction and optimization problems. Our model reproduces
the structure of the solution space of the random k-satisfiability and
k-coloring problems, and undergoes the same phase transitions as these
problems. The comparison becomes quantitative in the large-k limit. Distance
properties, as well the x-satisfiability threshold, are studied. The model is
also generalized to define a continuous energy landscape useful for studying
several aspects of glassy dynamics.Comment: 21 pages, 4 figure
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