This isn't so much an "algorithm" as a "trick" but it could be automated in code. It works mostly for stereo tracks with where the vocals are centered. If the vocals are centered, they manifest equally in both tracks. If you invert one of the tracks and then merge them back together, the wave forms of the center vocals cancel out and are virtually removed. You can do this manually with most good audio editors like audacity. It doesn't give you perfect results and the rest of the audio suffers a bit too but it makes for great karaoke tracks :)
