127 research outputs found
Simplifying Random Satisfiability Problem by Removing Frustrating Interactions
How can we remove some interactions in a constraint satisfaction problem
(CSP) such that it still remains satisfiable? In this paper we study a modified
survey propagation algorithm that enables us to address this question for a
prototypical CSP, i.e. random K-satisfiability problem. The average number of
removed interactions is controlled by a tuning parameter in the algorithm. If
the original problem is satisfiable then we are able to construct satisfiable
subproblems ranging from the original one to a minimal one with minimum
possible number of interactions. The minimal satisfiable subproblems will
provide directly the solutions of the original problem.Comment: 21 pages, 16 figure
Finite-Connectivity Spin-Glass Phase Diagrams and Low Density Parity Check Codes
We obtain phase diagrams of regular and irregular finite connectivity
spin-glasses. Contact is firstly established between properties of the phase
diagram and the performances of low density parity check codes (LDPC) within
the Replica Symmetric (RS) ansatz. We then study the location of the dynamical
and critical transition of these systems within the one step Replica Symmetry
Breaking theory (RSB), extending similar calculations that have been performed
in the past for the Bethe spin-glass problem. We observe that, away from the
Nishimori line, in the low temperature region, the location of the dynamical
transition line does change within the RSB theory, in comparison with the (RS)
case. For LDPC decoding over the binary erasure channel we find, at zero
temperature and rate R=1/4 an RS critical transition point located at p_c =
0.67 while the critical RSB transition point is located at p_c = 0.7450, to be
compared with the corresponding Shannon bound 1-R. For the binary symmetric
channel (BSC) we show that the low temperature reentrant behavior of the
dynamical transition line, observed within the RS ansatz, changes within the
RSB theory; the location of the dynamical transition point occurring at higher
values of the channel noise. Possible practical implications to improve the
performances of the state-of-the-art error correcting codes are discussed.Comment: 21 pages, 15 figure
The Directed Dominating Set Problem: Generalized Leaf Removal and Belief Propagation
A minimum dominating set for a digraph (directed graph) is a smallest set of
vertices such that each vertex either belongs to this set or has at least one
parent vertex in this set. We solve this hard combinatorial optimization
problem approximately by a local algorithm of generalized leaf removal and by a
message-passing algorithm of belief propagation. These algorithms can construct
near-optimal dominating sets or even exact minimum dominating sets for random
digraphs and also for real-world digraph instances. We further develop a core
percolation theory and a replica-symmetric spin glass theory for this problem.
Our algorithmic and theoretical results may facilitate applications of
dominating sets to various network problems involving directed interactions.Comment: 11 pages, 3 figures in EPS forma
Minimizing energy below the glass thresholds
Focusing on the optimization version of the random K-satisfiability problem,
the MAX-K-SAT problem, we study the performance of the finite energy version of
the Survey Propagation (SP) algorithm. We show that a simple (linear time)
backtrack decimation strategy is sufficient to reach configurations well below
the lower bound for the dynamic threshold energy and very close to the analytic
prediction for the optimal ground states. A comparative numerical study on one
of the most efficient local search procedures is also given.Comment: 12 pages, submitted to Phys. Rev. E, accepted for publicatio
Statistical Physics of Irregular Low-Density Parity-Check Codes
Low-density parity-check codes with irregular constructions have been
recently shown to outperform the most advanced error-correcting codes to date.
In this paper we apply methods of statistical physics to study the typical
properties of simple irregular codes.
We use the replica method to find a phase transition which coincides with
Shannon's coding bound when appropriate parameters are chosen.
The decoding by belief propagation is also studied using statistical physics
arguments; the theoretical solutions obtained are in good agreement with
simulations. We compare the performance of irregular with that of regular codes
and discuss the factors that contribute to the improvement in performance.Comment: 20 pages, 9 figures, revised version submitted to JP
Zero temperature solutions of the Edwards-Anderson model in random Husimi Lattices
We solve the Edwards-Anderson model (EA) in different Husimi lattices. We
show that, at T=0, the structure of the solution space depends on the parity of
the loop sizes. Husimi lattices with odd loop sizes have always a trivial
paramagnetic solution stable under 1RSB perturbations while, in Husimi lattices
with even loop sizes, this solution is absent. The range of stability under
1RSB perturbations of this and other RS solutions is computed analytically
(when possible) or numerically. We compute the free-energy, the complexity and
the ground state energy of different Husimi lattices at the level of the 1RSB
approximation. We also show, when the fraction of ferromagnetic couplings
increases, the existence, first, of a discontinuous transition from a
paramagnetic to a spin glass phase and latter of a continuous transition from a
spin glass to a ferromagnetic phase.Comment: 20 pages, 10 figures (v3: Corrected analysis of transitions. Appendix
proof fixed
Rigorous Inequalities between Length and Time Scales in Glassy Systems
Glassy systems are characterized by an extremely sluggish dynamics without
any simple sign of long range order. It is a debated question whether a correct
description of such phenomenon requires the emergence of a large correlation
length. We prove rigorous bounds between length and time scales implying the
growth of a properly defined length when the relaxation time increases. Our
results are valid in a rather general setting, which covers finite-dimensional
and mean field systems.
As an illustration, we discuss the Glauber (heat bath) dynamics of p-spin
glass models on random regular graphs. We present the first proof that a model
of this type undergoes a purely dynamical phase transition not accompanied by
any thermodynamic singularity.Comment: 24 pages, 3 figures; published versio
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
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
Polynomial iterative algorithms for coloring and analyzing random graphs
We study the graph coloring problem over random graphs of finite average
connectivity . Given a number of available colors, we find that graphs
with low connectivity admit almost always a proper coloring whereas graphs with
high connectivity are uncolorable. Depending on , we find the precise value
of the critical average connectivity . Moreover, we show that below
there exist a clustering phase in which ground states
spontaneously divide into an exponential number of clusters. Furthermore, we
extended our considerations to the case of single instances showing consistent
results. This lead us to propose a new algorithm able to color in polynomial
time random graphs in the hard but colorable region, i.e when .Comment: 23 pages, 10 eps figure
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