7,391 research outputs found
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
Biased landscapes for random Constraint Satisfaction Problems
The typical complexity of Constraint Satisfaction Problems (CSPs) can be
investigated by means of random ensembles of instances. The latter exhibit many
threshold phenomena besides their satisfiability phase transition, in
particular a clustering or dynamic phase transition (related to the tree
reconstruction problem) at which their typical solutions shatter into
disconnected components. In this paper we study the evolution of this
phenomenon under a bias that breaks the uniformity among solutions of one CSP
instance, concentrating on the bicoloring of k-uniform random hypergraphs. We
show that for small k the clustering transition can be delayed in this way to
higher density of constraints, and that this strategy has a positive impact on
the performances of Simulated Annealing algorithms. We characterize the modest
gain that can be expected in the large k limit from the simple implementation
of the biasing idea studied here. This paper contains also a contribution of a
more methodological nature, made of a review and extension of the methods to
determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure
The asymptotics of the clustering transition for random constraint satisfaction problems
Random Constraint Satisfaction Problems exhibit several phase transitions
when their density of constraints is varied. One of these threshold phenomena,
known as the clustering or dynamic transition, corresponds to a transition for
an information theoretic problem called tree reconstruction. In this article we
study this threshold for two CSPs, namely the bicoloring of -uniform
hypergraphs with a density of constraints, and the -coloring of
random graphs with average degree . We show that in the large limit
the clustering transition occurs for , , where is the same constant for both models. We
characterize via a functional equation, solve the latter
numerically to estimate , and obtain an analytic
lowerbound . Our
analysis unveils a subtle interplay of the clustering transition with the
rigidity (naive reconstruction) threshold that occurs on the same asymptotic
scale at .Comment: 35 pages, 8 figure
Phylogenetic CSPs are Approximation Resistant
We study the approximability of a broad class of computational problems --
originally motivated in evolutionary biology and phylogenetic reconstruction --
concerning the aggregation of potentially inconsistent (local) information
about items of interest, and we present optimal hardness of approximation
results under the Unique Games Conjecture. The class of problems studied here
can be described as Constraint Satisfaction Problems (CSPs) over infinite
domains, where instead of values or a fixed-size domain, the
variables can be mapped to any of the leaves of a phylogenetic tree. The
topology of the tree then determines whether a given constraint on the
variables is satisfied or not, and the resulting CSPs are called Phylogenetic
CSPs. Prominent examples of Phylogenetic CSPs with a long history and
applications in various disciplines include: Triplet Reconstruction, Quartet
Reconstruction, Subtree Aggregation (Forbidden or Desired). For example, in
Triplet Reconstruction, we are given triplets of the form
(indicating that ``items are more similar to each other than to '')
and we want to construct a hierarchical clustering on the items, that
respects the constraints as much as possible. Despite more than four decades of
research, the basic question of maximizing the number of satisfied constraints
is not well-understood. The current best approximation is achieved by
outputting a random tree (for triplets, this achieves a 1/3 approximation). Our
main result is that every Phylogenetic CSP is approximation resistant, i.e.,
there is no polynomial-time algorithm that does asymptotically better than a
(biased) random assignment. This is a generalization of the results in
Guruswami, Hastad, Manokaran, Raghavendra, and Charikar (2011), who showed that
ordering CSPs are approximation resistant (e.g., Max Acyclic Subgraph,
Betweenness).Comment: 45 pages, 11 figures, Abstract shortened for arxi
Reconstruction of Random Colourings
Reconstruction problems have been studied in a number of contexts including
biology, information theory and and statistical physics. We consider the
reconstruction problem for random -colourings on the -ary tree for
large . Bhatnagar et. al. showed non-reconstruction when and reconstruction when . We tighten this result and show non-reconstruction when and reconstruction when .Comment: Added references, updated notatio
Reconstruction Threshold for the Hardcore Model
In this paper we consider the reconstruction problem on the tree for the
hardcore model. We determine new bounds for the non-reconstruction regime on
the k-regular tree showing non-reconstruction when lambda < (ln
2-o(1))ln^2(k)/(2 lnln(k)) improving the previous best bound of lambda < e-1.
This is almost tight as reconstruction is known to hold when lambda>
(e+o(1))ln^2(k). We discuss the relationship for finding large independent sets
in sparse random graphs and to the mixing time of Markov chains for sampling
independent sets on trees.Comment: 14 pages, 2 figure
Transductive-Inductive Cluster Approximation Via Multivariate Chebyshev Inequality
Approximating adequate number of clusters in multidimensional data is an open
area of research, given a level of compromise made on the quality of acceptable
results. The manuscript addresses the issue by formulating a transductive
inductive learning algorithm which uses multivariate Chebyshev inequality.
Considering clustering problem in imaging, theoretical proofs for a particular
level of compromise are derived to show the convergence of the reconstruction
error to a finite value with increasing (a) number of unseen examples and (b)
the number of clusters, respectively. Upper bounds for these error rates are
also proved. Non-parametric estimates of these error from a random sample of
sequences empirically point to a stable number of clusters. Lastly, the
generalization of algorithm can be applied to multidimensional data sets from
different fields.Comment: 16 pages, 5 figure
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