328 research outputs found
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
What makes a phase transition? Analysis of the random satisfiability problem
In the last 30 years it was found that many combinatorial systems undergo
phase transitions. One of the most important examples of these can be found
among the random k-satisfiability problems (often referred to as k-SAT), asking
whether there exists an assignment of Boolean values satisfying a Boolean
formula composed of clauses with k random variables each. The random 3-SAT
problem is reported to show various phase transitions at different critical
values of the ratio of the number of clauses to the number of variables. The
most famous of these occurs when the probability of finding a satisfiable
instance suddenly drops from 1 to 0. This transition is associated with a rise
in the hardness of the problem, but until now the correlation between any of
the proposed phase transitions and the hardness is not totally clear. In this
paper we will first show numerically that the number of solutions universally
follows a lognormal distribution, thereby explaining the puzzling question of
why the number of solutions is still exponential at the critical point.
Moreover we provide evidence that the hardness of the closely related problem
of counting the total number of solutions does not show any phase
transition-like behavior. This raises the question of whether the probability
of finding a satisfiable instance is really an order parameter of a phase
transition or whether it is more likely to just show a simple sharp threshold
phenomenon. More generally, this paper aims at starting a discussion where a
simple sharp threshold phenomenon turns into a genuine phase transition
Entropy landscape and non-Gibbs solutions in constraint satisfaction problems
We study the entropy landscape of solutions for the bicoloring problem in
random graphs, a representative difficult constraint satisfaction problem. Our
goal is to classify which type of clusters of solutions are addressed by
different algorithms. In the first part of the study we use the cavity method
to obtain the number of clusters with a given internal entropy and determine
the phase diagram of the problem, e.g. dynamical, rigidity and SAT-UNSAT
transitions. In the second part of the paper we analyze different algorithms
and locate their behavior in the entropy landscape of the problem. For instance
we show that a smoothed version of a decimation strategy based on Belief
Propagation is able to find solutions belonging to sub-dominant clusters even
beyond the so called rigidity transition where the thermodynamically relevant
clusters become frozen. These non-equilibrium solutions belong to the most
probable unfrozen clusters.Comment: 38 pages, 10 figure
Universal Spectrum of Normal Modes in Low-Temperature Glasses: an Exact Solution
We report an analytical study of the vibrational spectrum of the simplest
model of jamming, the soft perceptron. We identify two distinct classes of soft
modes. The first kind of modes are related to isostaticity and appear only in
the close vicinity of the jamming transition. The second kind of modes instead
are present everywhere in the glass phase and are related to the hierarchical
structure of the potential energy landscape. Our results highlight the
universality of the spectrum of normal modes in disordered systems, and open
the way towards a detailed analytical understanding of the vibrational spectrum
of low-temperature glasses.Comment: 6 pages, 3 figures, submitted to PNA
Satisfiability, sequence niches, and molecular codes in cellular signaling
Biological information processing as implemented by regulatory and signaling
networks in living cells requires sufficient specificity of molecular
interaction to distinguish signals from one another, but much of regulation and
signaling involves somewhat fuzzy and promiscuous recognition of molecular
sequences and structures, which can leave systems vulnerable to crosstalk. This
paper examines a simple computational model of protein-protein interactions
which reveals both a sharp onset of crosstalk and a fragmentation of the
neutral network of viable solutions as more proteins compete for regions of
sequence space, revealing intrinsic limits to reliable signaling in the face of
promiscuity. These results suggest connections to both phase transitions in
constraint satisfaction problems and coding theory bounds on the size of
communication codes
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
Source coding by efficient selection of ground states clusters
In this letter, we show how the Survey Propagation algorithm can be
generalized to include external forcing messages, and used to address
selectively an exponential number of glassy ground states. These capabilities
can be used to explore efficiently the space of solutions of random NP-complete
constraint satisfaction problems, providing a direct experimental evidence of
replica symmetry breaking in large-size instances. Finally, a new lossy data
compression protocol is introduced, exploiting as a computational resource the
clustered nature of the space of addressable states.Comment: 4 pages, 4 figure
- …