44 research outputs found
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
Message passing for vertex covers
Constructing a minimal vertex cover of a graph can be seen as a prototype for
a combinatorial optimization problem under hard constraints. In this paper, we
develop and analyze message passing techniques, namely warning and survey
propagation, which serve as efficient heuristic algorithms for solving these
computational hard problems. We show also, how previously obtained results on
the typical-case behavior of vertex covers of random graphs can be recovered
starting from the message passing equations, and how they can be extended.Comment: 25 pages, 9 figures - version accepted for publication in PR
Belief-Propagation for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions
We consider the general problem of finding the minimum weight \bm-matching
on arbitrary graphs. We prove that, whenever the linear programming (LP)
relaxation of the problem has no fractional solutions, then the belief
propagation (BP) algorithm converges to the correct solution. We also show that
when the LP relaxation has a fractional solution then the BP algorithm can be
used to solve the LP relaxation. Our proof is based on the notion of graph
covers and extends the analysis of (Bayati-Shah-Sharma 2005 and Huang-Jebara
2007}.
These results are notable in the following regards: (1) It is one of a very
small number of proofs showing correctness of BP without any constraint on the
graph structure. (2) Variants of the proof work for both synchronous and
asynchronous BP; it is the first proof of convergence and correctness of an
asynchronous BP algorithm for a combinatorial optimization problem.Comment: 28 pages, 2 figures. Submitted to SIAM journal on Discrete
Mathematics on March 19, 2009; accepted for publication (in revised form)
August 30, 2010; published electronically July 1, 201
Next nearest neighbour Ising models on random graphs
This paper develops results for the next nearest neighbour Ising model on
random graphs. Besides being an essential ingredient in classic models for
frustrated systems, second neighbour interactions interactions arise naturally
in several applications such as the colour diversity problem and graphical
games. We demonstrate ensembles of random graphs, including regular
connectivity graphs, that have a periodic variation of free energy, with either
the ratio of nearest to next nearest couplings, or the mean number of nearest
neighbours. When the coupling ratio is integer paramagnetic phases can be found
at zero temperature. This is shown to be related to the locked or unlocked
nature of the interactions. For anti-ferromagnetic couplings, spin glass phases
are demonstrated at low temperature. The interaction structure is formulated as
a factor graph, the solution on a tree is developed. The replica symmetric and
energetic one-step replica symmetry breaking solution is developed using the
cavity method. We calculate within these frameworks the phase diagram and
demonstrate the existence of dynamical transitions at zero temperature for
cases of anti-ferromagnetic coupling on regular and inhomogeneous random
graphs.Comment: 55 pages, 15 figures, version 2 with minor revisions, to be published
J. Stat. Mec
Region graph partition function expansion and approximate free energy landscapes: Theory and some numerical results
Graphical models for finite-dimensional spin glasses and real-world
combinatorial optimization and satisfaction problems usually have an abundant
number of short loops. The cluster variation method and its extension, the
region graph method, are theoretical approaches for treating the complicated
short-loop-induced local correlations. For graphical models represented by
non-redundant or redundant region graphs, approximate free energy landscapes
are constructed in this paper through the mathematical framework of region
graph partition function expansion. Several free energy functionals are
obtained, each of which use a set of probability distribution functions or
functionals as order parameters. These probability distribution
function/functionals are required to satisfy the region graph
belief-propagation equation or the region graph survey-propagation equation to
ensure vanishing correction contributions of region subgraphs with dangling
edges. As a simple application of the general theory, we perform region graph
belief-propagation simulations on the square-lattice ferromagnetic Ising model
and the Edwards-Anderson model. Considerable improvements over the conventional
Bethe-Peierls approximation are achieved. Collective domains of different sizes
in the disordered and frustrated square lattice are identified by the
message-passing procedure. Such collective domains and the frustrations among
them are responsible for the low-temperature glass-like dynamical behaviors of
the system.Comment: 30 pages, 11 figures. More discussion on redundant region graphs. To
be published by Journal of Statistical Physic
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
Distribution of partition function zeros of the model on the Bethe lattice
The distribution of partition function zeros is studied for the model
of spin glasses on the Bethe lattice. We find a relation between the
distribution of complex cavity fields and the density of zeros, which enables
us to obtain the density of zeros for the infinite system size by using the
cavity method. The phase boundaries thus derived from the location of the zeros
are consistent with the results of direct analytical calculations. This is the
first example in which the spin glass transition is related to the distribution
of zeros directly in the thermodynamical limit. We clarify how the spin glass
transition is characterized by the zeros of the partition function. It is also
shown that in the spin glass phase a continuous distribution of singularities
touches the axes of real field and temperature.Comment: 23 pages, 12 figure
The number of matchings in random graphs
We study matchings on sparse random graphs by means of the cavity method. We
first show how the method reproduces several known results about maximum and
perfect matchings in regular and Erdos-Renyi random graphs. Our main new result
is the computation of the entropy, i.e. the leading order of the logarithm of
the number of solutions, of matchings with a given size. We derive both an
algorithm to compute this entropy for an arbitrary graph with a girth that
diverges in the large size limit, and an analytic result for the entropy in
regular and Erdos-Renyi random graph ensembles.Comment: 17 pages, 6 figures, to be published in Journal of Statistical
Mechanic
Exhaustive enumeration unveils clustering and freezing in random 3-SAT
We study geometrical properties of the complete set of solutions of the
random 3-satisfiability problem. We show that even for moderate system sizes
the number of clusters corresponds surprisingly well with the theoretic
asymptotic prediction. We locate the freezing transition in the space of
solutions which has been conjectured to be relevant in explaining the onset of
computational hardness in random constraint satisfaction problems.Comment: 4 pages, 3 figure
Optimization hardness as transient chaos in an analog approach to constraint satisfaction
Boolean satisfiability [1] (k-SAT) is one of the most studied optimization
problems, as an efficient (that is, polynomial-time) solution to k-SAT (for
) implies efficient solutions to a large number of hard optimization
problems [2,3]. Here we propose a mapping of k-SAT into a deterministic
continuous-time dynamical system with a unique correspondence between its
attractors and the k-SAT solution clusters. We show that beyond a constraint
density threshold, the analog trajectories become transiently chaotic [4-7],
and the boundaries between the basins of attraction [8] of the solution
clusters become fractal [7-9], signaling the appearance of optimization
hardness [10]. Analytical arguments and simulations indicate that the system
always finds solutions for satisfiable formulae even in the frozen regimes of
random 3-SAT [11] and of locked occupation problems [12] (considered among the
hardest algorithmic benchmarks); a property partly due to the system's
hyperbolic [4,13] character. The system finds solutions in polynomial
continuous-time, however, at the expense of exponential fluctuations in its
energy function.Comment: 27 pages, 14 figure