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ABSTRACT For analysing maps of the cosmic microwave background sky, it is necessary to mask out the region around the Galactic equator where the parasitic foreground emission is strongest as well as the brightest compact sources. Since many of the analyses of the data, particularly those searching for non-Gaussianity of a primordial origin, are most straightforwardly carried out on full-sky maps, it is of great interest to develop efficient algorithms for filling in the missing information in a plausible way. In this paper, we explore algorithms for filling in based on constrained Gaussian realizations. The non-trivial step in such a filling-in procedure is to find the maximum likelihood configuration for the masked pixels given the data for the unmasked pixels. We reformulate the problem in terms of an integral equation involving a short-range kernel and show that in the far interior of a masked region the maximum likelihood configuration very nearly obeys the Laplace equation. Although carrying out constrained Gaussian realizations is in principle straightforward, for finely pixelized maps as will be required for the Planck analysis, a direct brute force method is not numerically tractable. Here we present some concrete solutions to this problem, both on a spatially flat sky with periodic boundary conditions and on the pixelized sphere. One approach is to solve the linear system using an appropriately preconditioned conjugate gradient method. While this approach was successfully implemented on a rectangular domain with periodic boundary conditions and converged rapidly even for very wide masked regions, we found that the method failed on the pixelized sphere for reasons that we explain here. We present an approach that works for full-sky pixelized maps on the sphere involving a kernel-based multiresolution Laplace solver followed by a series of conjugate gradient corrections near the boundary of the mask. [ABSTRACT FROM AUTHOR]