41,469 research outputs found
Landscape of solutions in constraint satisfaction problems
We present a theoretical framework for characterizing the geometrical
properties of the space of solutions in constraint satisfaction problems,
together with practical algorithms for studying this structure on particular
instances. We apply our method to the coloring problem, for which we obtain the
total number of solutions and analyze in detail the distribution of distances
between solutions.Comment: 4 pages, 4 figures. Replaced with published versio
A Landscape Analysis of Constraint Satisfaction Problems
We discuss an analysis of Constraint Satisfaction problems, such as Sphere
Packing, K-SAT and Graph Coloring, in terms of an effective energy landscape.
Several intriguing geometrical properties of the solution space become in this
light familiar in terms of the well-studied ones of rugged (glassy) energy
landscapes. A `benchmark' algorithm naturally suggested by this construction
finds solutions in polynomial time up to a point beyond the `clustering' and in
some cases even the `thermodynamic' transitions. This point has a simple
geometric meaning and can be in principle determined with standard Statistical
Mechanical methods, thus pushing the analytic bound up to which problems are
guaranteed to be easy. We illustrate this for the graph three and four-coloring
problem. For Packing problems the present discussion allows to better
characterize the `J-point', proposed as a systematic definition of Random Close
Packing, and to place it in the context of other theories of glasses.Comment: 17 pages, 69 citations, 12 figure
Statistical Mechanics of maximal independent sets
The graph theoretic concept of maximal independent set arises in several
practical problems in computer science as well as in game theory. A maximal
independent set is defined by the set of occupied nodes that satisfy some
packing and covering constraints. It is known that finding minimum and
maximum-density maximal independent sets are hard optimization problems. In
this paper, we use cavity method of statistical physics and Monte Carlo
simulations to study the corresponding constraint satisfaction problem on
random graphs. We obtain the entropy of maximal independent sets within the
replica symmetric and one-step replica symmetry breaking frameworks, shedding
light on the metric structure of the landscape of solutions and suggesting a
class of possible algorithms. This is of particular relevance for the
application to the study of strategic interactions in social and economic
networks, where maximal independent sets correspond to pure Nash equilibria of
a graphical game of public goods allocation
Certifying solution geometry in random CSPs: counts, clusters and balance
An active topic in the study of random constraint satisfaction problems
(CSPs) is the geometry of the space of satisfying or almost satisfying
assignments as the function of the density, for which a precise landscape of
predictions has been made via statistical physics-based heuristics. In
parallel, there has been a recent flurry of work on refuting random constraint
satisfaction problems, via nailing refutation thresholds for spectral and
semidefinite programming-based algorithms, and also on counting solutions to
CSPs. Inspired by this, the starting point for our work is the following
question: what does the solution space for a random CSP look like to an
efficient algorithm?
In pursuit of this inquiry, we focus on the following problems about random
Boolean CSPs at the densities where they are unsatisfiable but no refutation
algorithm is known.
1. Counts. For every Boolean CSP we give algorithms that with high
probability certify a subexponential upper bound on the number of solutions. We
also give algorithms to certify a bound on the number of large cuts in a
Gaussian-weighted graph, and the number of large independent sets in a random
-regular graph.
2. Clusters. For Boolean CSPs we give algorithms that with high
probability certify an upper bound on the number of clusters of solutions.
3. Balance. We also give algorithms that with high probability certify that
there are no "unbalanced" solutions, i.e., solutions where the fraction of
s deviates significantly from .
Finally, we also provide hardness evidence suggesting that our algorithms for
counting are optimal
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
Criticality and Heterogeneity in the Solution Space of Random Constraint Satisfaction Problems
Random constraint satisfaction problems are interesting model systems for
spin-glasses and glassy dynamics studies. As the constraint density of such a
system reaches certain threshold value, its solution space may split into
extremely many clusters. In this paper we argue that this ergodicity-breaking
transition is preceded by a homogeneity-breaking transition. For random K-SAT
and K-XORSAT, we show that many solution communities start to form in the
solution space as the constraint density reaches a critical value alpha_cm,
with each community containing a set of solutions that are more similar with
each other than with the outsider solutions. At alpha_cm the solution space is
in a critical state. The connection of these results to the onset of dynamical
heterogeneity in lattice glass models is discussed.Comment: 6 pages, 4 figures, final version as accepted by International
Journal of Modern Physics
Reweighted belief propagation and quiet planting for random K-SAT
We study the random K-satisfiability problem using a partition function where
each solution is reweighted according to the number of variables that satisfy
every clause. We apply belief propagation and the related cavity method to the
reweighted partition function. This allows us to obtain several new results on
the properties of random K-satisfiability problem. In particular the
reweighting allows to introduce a planted ensemble that generates instances
that are, in some region of parameters, equivalent to random instances. We are
hence able to generate at the same time a typical random SAT instance and one
of its solutions. We study the relation between clustering and belief
propagation fixed points and we give a direct evidence for the existence of
purely entropic (rather than energetic) barriers between clusters in some
region of parameters in the random K-satisfiability problem. We exhibit, in
some large planted instances, solutions with a non-trivial whitening core; such
solutions were known to exist but were so far never found on very large
instances. Finally, we discuss algorithmic hardness of such planted instances
and we determine a region of parameters in which planting leads to satisfiable
benchmarks that, up to our knowledge, are the hardest known.Comment: 23 pages, 4 figures, revised for readability, stability expression
correcte
Taming a non-convex landscape with dynamical long-range order: memcomputing Ising benchmarks
Recent work on quantum annealing has emphasized the role of collective
behavior in solving optimization problems. By enabling transitions of clusters
of variables, such solvers are able to navigate their state space and locate
solutions more efficiently despite having only local connections between
elements. However, collective behavior is not exclusive to quantum annealers,
and classical solvers that display collective dynamics should also possess an
advantage in navigating a non-convex landscape. Here, we give evidence that a
benchmark derived from quantum annealing studies is solvable in polynomial time
using digital memcomputing machines, which utilize a collection of dynamical
components with memory to represent the structure of the underlying
optimization problem. To illustrate the role of memory and clarify the
structure of these solvers we propose a simple model of these machines that
demonstrates the emergence of long-range order. This model, when applied to
finding the ground state of the Ising frustrated-loop benchmarks, undergoes a
transient phase of avalanches which can span the entire lattice and
demonstrates a connection between long-range behavior and their probability of
success. These results establish the advantages of computational approaches
based on collective dynamics of continuous dynamical systems
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