12,806 research outputs found
Clustering in Hilbert space of a quantum optimization problem
The solution space of many classical optimization problems breaks up into
clusters which are extensively distant from one another in the Hamming metric.
Here, we show that an analogous quantum clustering phenomenon takes place in
the ground state subspace of a certain quantum optimization problem. This
involves extending the notion of clustering to Hilbert space, where the
classical Hamming distance is not immediately useful. Quantum clusters
correspond to macroscopically distinct subspaces of the full quantum ground
state space which grow with the system size. We explicitly demonstrate that
such clusters arise in the solution space of random quantum satisfiability
(3-QSAT) at its satisfiability transition. We estimate both the number of these
clusters and their internal entropy. The former are given by the number of
hardcore dimer coverings of the core of the interaction graph, while the latter
is related to the underconstrained degrees of freedom not touched by the
dimers. We additionally provide new numerical evidence suggesting that the
3-QSAT satisfiability transition may coincide with the product satisfiability
transition, which would imply the absence of an intermediate entangled
satisfiable phase.Comment: 11 pages, 6 figure
Glassy Behavior and Jamming of a Random Walk Process for Sequentially Satisfying a Constraint Satisfaction Formula
Random -satisfiability (-SAT) is a model system for studying
typical-case complexity of combinatorial optimization. Recent theoretical and
simulation work revealed that the solution space of a random -SAT formula
has very rich structures, including the emergence of solution communities
within single solution clusters. In this paper we investigate the influence of
the solution space landscape to a simple stochastic local search process {\tt
SEQSAT}, which satisfies a -SAT formula in a sequential manner. Before
satisfying each newly added clause, {\tt SEQSAT} walk randomly by single-spin
flips in a solution cluster of the old subformula. This search process is
efficient when the constraint density of the satisfied subformula is
less than certain value ; however it slows down considerably as
and finally reaches a jammed state at . The glassy dynamical behavior of {\tt SEQSAT} for probably is due to the entropic trapping of various communities in
the solution cluster of the satisfied subformula. For random 3-SAT, the jamming
transition point is larger than the solution space clustering
transition point , and its value can be predicted by a long-range
frustration mean-field theory. For random -SAT with , however, our
simulation results indicate that . The relevance of this
work for understanding the dynamic properties of glassy systems is also
discussed.Comment: 10 pages, 6 figures, 1 table, a mistake of numerical simulation
corrected, and new results adde
Solution space heterogeneity of the random K-satisfiability problem: Theory and simulations
The random K-satisfiability (K-SAT) problem is an important problem for
studying typical-case complexity of NP-complete combinatorial satisfaction; it
is also a representative model of finite-connectivity spin-glasses. In this
paper we review our recent efforts on the solution space fine structures of the
random K-SAT problem. A heterogeneity transition is predicted to occur in the
solution space as the constraint density alpha reaches a critical value
alpha_cm. This transition marks the emergency of exponentially many solution
communities in the solution space. After the heterogeneity transition the
solution space is still ergodic until alpha reaches a larger threshold value
alpha_d, at which the solution communities disconnect from each other to become
different solution clusters (ergodicity-breaking). The existence of solution
communities in the solution space is confirmed by numerical simulations of
solution space random walking, and the effect of solution space heterogeneity
on a stochastic local search algorithm SEQSAT, which performs a random walk of
single-spin flips, is investigated. The relevance of this work to glassy
dynamics studies is briefly mentioned.Comment: 11 pages, 4 figures. Final version as will appear in Journal of
Physics: Conference Series (Proceedings of the International Workshop on
Statistical-Mechanical Informatics, March 7-10, 2010, Kyoto, Japan
Optimal Gossip with Direct Addressing
Gossip algorithms spread information by having nodes repeatedly forward
information to a few random contacts. By their very nature, gossip algorithms
tend to be distributed and fault tolerant. If done right, they can also be fast
and message-efficient. A common model for gossip communication is the random
phone call model, in which in each synchronous round each node can PUSH or PULL
information to or from a random other node. For example, Karp et al. [FOCS
2000] gave algorithms in this model that spread a message to all nodes in
rounds while sending only messages per node
on average.
Recently, Avin and Els\"asser [DISC 2013], studied the random phone call
model with the natural and commonly used assumption of direct addressing.
Direct addressing allows nodes to directly contact nodes whose ID (e.g., IP
address) was learned before. They show that in this setting, one can "break the
barrier" and achieve a gossip algorithm running in
rounds, albeit while using messages per node.
We study the same model and give a simple gossip algorithm which spreads a
message in only rounds. We also prove a matching lower bound which shows that this running time is best possible. In
particular we show that any gossip algorithm takes with high probability at
least rounds to terminate. Lastly, our algorithm can be
tweaked to send only messages per node on average with only
bits per message. Our algorithm therefore simultaneously achieves the optimal
round-, message-, and bit-complexity for this setting. As all prior gossip
algorithms, our algorithm is also robust against failures. In particular, if in
the beginning an oblivious adversary fails any nodes our algorithm still,
with high probability, informs all but surviving nodes
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
Monte Carlo algorithms are very effective in finding the largest independent set in sparse random graphs
The effectiveness of stochastic algorithms based on Monte Carlo dynamics in
solving hard optimization problems is mostly unknown. Beyond the basic
statement that at a dynamical phase transition the ergodicity breaks and a
Monte Carlo dynamics cannot sample correctly the probability distribution in
times linear in the system size, there are almost no predictions nor intuitions
on the behavior of this class of stochastic dynamics. The situation is
particularly intricate because, when using a Monte Carlo based algorithm as an
optimization algorithm, one is usually interested in the out of equilibrium
behavior which is very hard to analyse. Here we focus on the use of Parallel
Tempering in the search for the largest independent set in a sparse random
graph, showing that it can find solutions well beyond the dynamical threshold.
Comparison with state-of-the-art message passing algorithms reveals that
parallel tempering is definitely the algorithm performing best, although a
theory explaining its behavior is still lacking.Comment: 14 pages, 12 figure
Exact Clustering of Weighted Graphs via Semidefinite Programming
As a model problem for clustering, we consider the densest k-disjoint-clique
problem of partitioning a weighted complete graph into k disjoint subgraphs
such that the sum of the densities of these subgraphs is maximized. We
establish that such subgraphs can be recovered from the solution of a
particular semidefinite relaxation with high probability if the input graph is
sampled from a distribution of clusterable graphs. Specifically, the
semidefinite relaxation is exact if the graph consists of k large disjoint
subgraphs, corresponding to clusters, with weight concentrated within these
subgraphs, plus a moderate number of outliers. Further, we establish that if
noise is weakly obscuring these clusters, i.e, the between-cluster edges are
assigned very small weights, then we can recover significantly smaller
clusters. For example, we show that in approximately sparse graphs, where the
between-cluster weights tend to zero as the size n of the graph tends to
infinity, we can recover clusters of size polylogarithmic in n. Empirical
evidence from numerical simulations is also provided to support these
theoretical phase transitions to perfect recovery of the cluster structure
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