192 research outputs found
The Satisfiability Threshold for k-XORSAT
We consider "unconstrained" random -XORSAT, which is a uniformly random
system of linear non-homogeneous equations in over
variables, each equation containing variables, and also consider a
"constrained" model where every variable appears in at least two equations.
Dubois and Mandler proved that is a sharp threshold for satisfiability
of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform
hypergraph to extend this result to find the threshold for unconstrained
3-XORSAT.
We show that remains a sharp threshold for satisfiability of
constrained -XORSAT for every , and we use standard results on the
2-core of a random -uniform hypergraph to extend this result to find the
threshold for unconstrained -XORSAT. For constrained -XORSAT we narrow
the phase transition window, showing that implies almost-sure
satisfiability, while implies almost-sure unsatisfiability.Comment: Version 2 adds sharper phase transition result, new citation in
literature survey, and improvements in presentation; removes Appendix
treating k=
The Space of Solutions of Coupled XORSAT Formulae
The XOR-satisfiability (XORSAT) problem deals with a system of Boolean
variables and clauses. Each clause is a linear Boolean equation (XOR) of a
subset of the variables. A -clause is a clause involving distinct
variables. In the random -XORSAT problem a formula is created by choosing
-clauses uniformly at random from the set of all possible clauses on
variables. The set of solutions of a random formula exhibits various
geometrical transitions as the ratio varies.
We consider a {\em coupled} -XORSAT ensemble, consisting of a chain of
random XORSAT models that are spatially coupled across a finite window along
the chain direction. We observe that the threshold saturation phenomenon takes
place for this ensemble and we characterize various properties of the space of
solutions of such coupled formulae.Comment: Submitted to ISIT 201
The set of solutions of random XORSAT formulae
The XOR-satisfiability (XORSAT) problem requires finding an assignment of
Boolean variables that satisfy exclusive OR (XOR) clauses, whereby each
clause constrains a subset of the variables. We consider random XORSAT
instances, drawn uniformly at random from the ensemble of formulae containing
variables and clauses of size . This model presents several
structural similarities to other ensembles of constraint satisfaction problems,
such as -satisfiability (-SAT), hypergraph bicoloring and graph coloring.
For many of these ensembles, as the number of constraints per variable grows,
the set of solutions shatters into an exponential number of well-separated
components. This phenomenon appears to be related to the difficulty of solving
random instances of such problems. We prove a complete characterization of this
clustering phase transition for random -XORSAT. In particular, we prove that
the clustering threshold is sharp and determine its exact location. We prove
that the set of solutions has large conductance below this threshold and that
each of the clusters has large conductance above the same threshold. Our proof
constructs a very sparse basis for the set of solutions (or the subset within a
cluster). This construction is intimately tied to the construction of specific
subgraphs of the hypergraph associated with an instance of -XORSAT. In order
to study such subgraphs, we establish novel local weak convergence results for
them.Comment: Published at http://dx.doi.org/10.1214/14-AAP1060 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Ground-state configuration space heterogeneity of random finite-connectivity spin glasses and random constraint satisfaction problems
We demonstrate through two case studies, one on the p-spin interaction model
and the other on the random K-satisfiability problem, that a heterogeneity
transition occurs to the ground-state configuration space of a random
finite-connectivity spin glass system at certain critical value of the
constraint density. At the transition point, exponentially many configuration
communities emerge from the ground-state configuration space, making the
entropy density s(q) of configuration-pairs a non-concave function of
configuration-pair overlap q. Each configuration community is a collection of
relatively similar configurations and it forms a stable thermodynamic phase in
the presence of a suitable external field. We calculate s(q) by the
replica-symmetric and the first-step replica-symmetry-broken cavity methods,
and show by simulations that the configuration space heterogeneity leads to
dynamical heterogeneity of particle diffusion processes because of the entropic
trapping effect of configuration communities. This work clarifies the fine
structure of the ground-state configuration space of random spin glass models,
it also sheds light on the glassy behavior of hard-sphere colloidal systems at
relatively high particle volume fraction.Comment: 26 pages, 9 figures, submitted to Journal of Statistical Mechanic
On the freezing of variables in random constraint satisfaction problems
The set of solutions of random constraint satisfaction problems (zero energy
groundstates of mean-field diluted spin glasses) undergoes several structural
phase transitions as the amount of constraints is increased. This set first
breaks down into a large number of well separated clusters. At the freezing
transition, which is in general distinct from the clustering one, some
variables (spins) take the same value in all solutions of a given cluster. In
this paper we study the critical behavior around the freezing transition, which
appears in the unfrozen phase as the divergence of the sizes of the
rearrangements induced in response to the modification of a variable. The
formalism is developed on generic constraint satisfaction problems and applied
in particular to the random satisfiability of boolean formulas and to the
coloring of random graphs. The computation is first performed in random tree
ensembles, for which we underline a connection with percolation models and with
the reconstruction problem of information theory. The validity of these results
for the original random ensembles is then discussed in the framework of the
cavity method.Comment: 32 pages, 7 figure
The satisfiability threshold for random linear equations
Let be a random matrix over the finite field with
precisely non-zero entries per row and let be a random vector
chosen independently of . We identify the threshold up to which the
linear system has a solution with high probability and analyse the
geometry of the set of solutions. In the special case , known as the
random -XORSAT problem, the threshold was determined by [Dubois and Mandler
2002, Dietzfelbinger et al. 2010, Pittel and Sorkin 2016], and the proof
technique was subsequently extended to the cases [Falke and Goerdt
2012]. But the argument depends on technically demanding second moment
calculations that do not generalise to . Here we approach the problem from
the viewpoint of a decoding task, which leads to a transparent combinatorial
proof
Geometrical organization of solutions to random linear Boolean equations
The random XORSAT problem deals with large random linear systems of Boolean
variables. The difficulty of such problems is controlled by the ratio of number
of equations to number of variables. It is known that in some range of values
of this parameter, the space of solutions breaks into many disconnected
clusters. Here we study precisely the corresponding geometrical organization.
In particular, the distribution of distances between these clusters is computed
by the cavity method. This allows to study the `x-satisfiability' threshold,
the critical density of equations where there exist two solutions at a given
distance.Comment: 20 page
On the cavity method for decimated random constraint satisfaction problems and the analysis of belief propagation guided decimation algorithms
We introduce a version of the cavity method for diluted mean-field spin
models that allows the computation of thermodynamic quantities similar to the
Franz-Parisi quenched potential in sparse random graph models. This method is
developed in the particular case of partially decimated random constraint
satisfaction problems. This allows to develop a theoretical understanding of a
class of algorithms for solving constraint satisfaction problems, in which
elementary degrees of freedom are sequentially assigned according to the
results of a message passing procedure (belief-propagation). We confront this
theoretical analysis to the results of extensive numerical simulations.Comment: 32 pages, 24 figure
The -XORSAT threshold revisited
We provide a simplified proof of the random -XORSAT satisfiability
threshold theorem. As an extension we also determine the full rank threshold
for sparse random matrices over finite fields with precisely non-zero
entries per row. This complements a result from [Ayre, Coja-Oghlan, Gao,
M\"uller: Combinatorica 2020]. The proof combines physics-inspired message
passing arguments with a surgical moment computation
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