8,579 research outputs found
On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint
In the problem of minimum connected dominating set with routing cost
constraint, we are given a graph , and the goal is to find the
smallest connected dominating set of such that, for any two
non-adjacent vertices and in , the number of internal nodes on the
shortest path between and in the subgraph of induced by is at most times that in . For general graphs, the only
known previous approximability result is an -approximation algorithm
() for by Ding et al. For any constant , we
give an -approximation
algorithm. When , we give an -approximation
algorithm. Finally, we prove that, when , unless , for any constant , the problem admits no
polynomial-time -approximation algorithm, improving
upon the bound by Du et al. (albeit under a stronger hardness
assumption)
Bayesian Parameter Estimation for Latent Markov Random Fields and Social Networks
Undirected graphical models are widely used in statistics, physics and
machine vision. However Bayesian parameter estimation for undirected models is
extremely challenging, since evaluation of the posterior typically involves the
calculation of an intractable normalising constant. This problem has received
much attention, but very little of this has focussed on the important practical
case where the data consists of noisy or incomplete observations of the
underlying hidden structure. This paper specifically addresses this problem,
comparing two alternative methodologies. In the first of these approaches
particle Markov chain Monte Carlo (Andrieu et al., 2010) is used to efficiently
explore the parameter space, combined with the exchange algorithm (Murray et
al., 2006) for avoiding the calculation of the intractable normalising constant
(a proof showing that this combination targets the correct distribution in
found in a supplementary appendix online). This approach is compared with
approximate Bayesian computation (Pritchard et al., 1999). Applications to
estimating the parameters of Ising models and exponential random graphs from
noisy data are presented. Each algorithm used in the paper targets an
approximation to the true posterior due to the use of MCMC to simulate from the
latent graphical model, in lieu of being able to do this exactly in general.
The supplementary appendix also describes the nature of the resulting
approximation.Comment: 26 pages, 2 figures, accepted in Journal of Computational and
Graphical Statistics (http://www.amstat.org/publications/jcgs.cfm
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