Gimps offer 4 algorithms for resizing the images: none, linear, cubic, และ Sinc (Lanczos3)
Let's see how well they do on this image (click to see full-resolution image)
Here's the result:
Again, click on the image to see full-resolution image.
For this particular picture, I think sinc (Lanczo3) is the best. It managed to preserve lots of details but minimize, almost as much as 'none', but also kept those stair-step side-effect down.
Cubi-resized result also has lots of details but looks soft compare to 'sinc'.
Anyway, this is a computer-generated image. It would be interesting to see how these algorithm do with the real-world pictures.