"How can I figure out which algorithm to use?"

Form of the input

• Object types: what kind of objects does it handle?

• convex vs. non-convex
• polygons vs. everything else - smooth curves, non-continuous surfaces, volumetric data

• Precision: image space or object space?
• Visible Set Exactness: Does it find exactly the objects that are visible? Or some superset? Or some subset that's "good enough"?
• Clipping (vs. occlusion):
• some algorithms assume a 360-degree viewing angle
• others are tied to a view frustum
• Transparent object support: can it only find the nearest visible object?

• Coherence exploited:

• image-space - objects tend to take up a coherent area of screen real estate
• temporal - two scenes with slightly different viewpoints/view directions will have very similar sets of visible objects ("the incremental oracle")
• object  - objects aren't point clouds - they tend to be continuous
• world - objects tend to be clustered together
• Scene footprint: Do you need to consider every object in the scene?  (the oracle)
• Memory footprint: How much of the scene do you need in memory at each instant?
• Overdraw:  How much work do you have to do for objects you don't end up seeing? Or, how well does it handle clipping and depth complexity?
• Preprocessing effort: How much preprocessing is required?
• Speed/O(time): How does it scale? Tricky because a lot of methods can be combined
 
• Dynamic object support: can it handle objects moving around?