How did you decide on what resizing algorithm to use?
...and what did you decide on? 
-ewemoa
That was very simple.
There are 2 different resizes happening:
1) Live preview
2) Saved image
The live preview uses low quality, fast resizing for a smoother UI. It reduces the CPU required. The live preview resizes as you do the resizing, so it's important for it to be fast. (I forget if I did another high quality update after the resize...)
For saving, high quality bicubic is used. Basically, maximum quality.
I use it all the time whenever I post a pic to Facebook or need to resize a picture because it's just faster & easier than fiddling with menus and buttons and whatever. I just wanted to get things done fast. I also like high quality resizing.
But, the decisions there were purely about getting maximum quality output and minimum CPU usage when previewing.