10,873 research outputs found
Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference
We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF
inference problems. The core of our method is a very efficient bounding
procedure, which combines scalable semidefinite programming (SDP) and a
cutting-plane method for seeking violated constraints. In order to further
speed up the computation, several strategies have been exploited, including
model reduction, warm start and removal of inactive constraints.
We analyze the performance of the proposed method under different settings,
and demonstrate that our method either outperforms or performs on par with
state-of-the-art approaches. Especially when the connectivities are dense or
when the relative magnitudes of the unary costs are low, we achieve the best
reported results. Experiments show that the proposed algorithm achieves better
approximation than the state-of-the-art methods within a variety of time
budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page
Propagating Conjunctions of AllDifferent Constraints
We study propagation algorithms for the conjunction of two AllDifferent
constraints. Solutions of an AllDifferent constraint can be seen as perfect
matchings on the variable/value bipartite graph. Therefore, we investigate the
problem of finding simultaneous bipartite matchings. We present an extension of
the famous Hall theorem which characterizes when simultaneous bipartite
matchings exists. Unfortunately, finding such matchings is NP-hard in general.
However, we prove a surprising result that finding a simultaneous matching on a
convex bipartite graph takes just polynomial time. Based on this theoretical
result, we provide the first polynomial time bound consistency algorithm for
the conjunction of two AllDifferent constraints. We identify a pathological
problem on which this propagator is exponentially faster compared to existing
propagators. Our experiments show that this new propagator can offer
significant benefits over existing methods.Comment: AAAI 2010, Proceedings of the Twenty-Fourth AAAI Conference on
Artificial Intelligenc
Constraint optimization and landscapes
We describe an effective landscape introduced in [1] for the analysis of
Constraint Satisfaction problems, such as Sphere Packing, K-SAT and Graph
Coloring. This geometric construction reexpresses these problems in the more
familiar terms of optimization in rugged energy landscapes. In particular, it
allows one to understand the puzzling fact that unsophisticated programs are
successful well beyond what was considered to be the `hard' transition, and
suggests an algorithm defining a new, higher, easy-hard frontier.Comment: Contribution to STATPHYS2
Lagrangian Relaxation for MAP Estimation in Graphical Models
We develop a general framework for MAP estimation in discrete and Gaussian
graphical models using Lagrangian relaxation techniques. The key idea is to
reformulate an intractable estimation problem as one defined on a more
tractable graph, but subject to additional constraints. Relaxing these
constraints gives a tractable dual problem, one defined by a thin graph, which
is then optimized by an iterative procedure. When this iterative optimization
leads to a consistent estimate, one which also satisfies the constraints, then
it corresponds to an optimal MAP estimate of the original model. Otherwise
there is a ``duality gap'', and we obtain a bound on the optimal solution.
Thus, our approach combines convex optimization with dynamic programming
techniques applicable for thin graphs. The popular tree-reweighted max-product
(TRMP) method may be seen as solving a particular class of such relaxations,
where the intractable graph is relaxed to a set of spanning trees. We also
consider relaxations to a set of small induced subgraphs, thin subgraphs (e.g.
loops), and a connected tree obtained by ``unwinding'' cycles. In addition, we
propose a new class of multiscale relaxations that introduce ``summary''
variables. The potential benefits of such generalizations include: reducing or
eliminating the ``duality gap'' in hard problems, reducing the number or
Lagrange multipliers in the dual problem, and accelerating convergence of the
iterative optimization procedure.Comment: 10 pages, presented at 45th Allerton conference on communication,
control and computing, to appear in proceeding
Statistical mechanics of optimization problems
Here I will present an introduction to the results that have been recently
obtained in constraint optimization of random problems using statistical
mechanics techniques. After presenting the general results, in order to
simplify the presentation I will describe in details the problems related to
the coloring of a random graph.Comment: proceedings of the conference SigmaPhi di Crete 2005, 10 pages, one
figur
Turduckening black holes: an analytical and computational study
We provide a detailed analysis of several aspects of the turduckening
technique for evolving black holes. At the analytical level we study the
constraint propagation for a general family of BSSN-type formulation of
Einstein's field equations and identify under what conditions the turducken
procedure is rigorously justified and under what conditions constraint
violations will propagate to the outside of the black holes. We present
high-resolution spherically symmetric studies which verify our analytical
predictions. Then we present three dimensional simulations of single distorted
black holes using different variations of the turduckening method and also the
puncture method. We study the effect that these different methods have on the
coordinate conditions, constraint violations, and extracted gravitational
waves. We find that the waves agree up to small but non-vanishing differences,
caused by escaping superluminal gauge modes. These differences become smaller
with increasing detector location.Comment: Minor changes to match the final version to appear in PR
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