In all of the movies involving asteroids you see a group of astronauts blowing the hell of the big rock and saving the day. You never see them shooting it a zillion paintballs. Here's the idea.
"That’s the idea of the winning entry in this year’s Move an Asteroid Technical Paper Competition, sponsored by the United Nations’ Space Generation Advisory Council. Sung Wook Paek, a graduate student in MIT’s Department of Aeronautics and Astronautics, says if timed just right, pellets full of paint powder, launched in two rounds from a spacecraft at relatively close distance, would cover the front and back of an asteroid, more than doubling its reflectivity, or albedo. The initial force from the pellets would bump an asteroid off course; over time, the sun’s photons would deflect the asteroid even more."