See: https://en.m.wikipedia.org/wiki/Asteroid_impact_avoidance
See: https://en.m.wikipedia.org/wiki/Asteroid_impact_avoidance
Even nukes have a surprisingly small effect, unless they are buried. We have landed on on an asteroid, but have yet to embed a nuke in one.
The trick about nukes is you have the risk of just making an harder to stop by making a cloud, like say comet levy which turned into 21 pieces, but delivered the same total energy when hitting the atmosphere.
The nice thing about the paint is that it's not that hard, even asteroids have gravity, and we have orbited (very slowly) asteroids before. It doesn't cause the astroid (which might just be a loose collection of rocks) to break up, and you get a huge win per pound of delivered mass, granted it takes months or years to accumulate into a large orbital change.
However there would be more surface area to burn up in the atmosphere, and if you do it far enough out and I'd think it would require very little nudging to move pieces out of the way
Generally if it's a planet killer you want to take no chances. The earlier the detection the better, and changing the albedo will work if it's a sand up to a solid rock.
I've heard similar proposals with a "gravity tug", but it's not clear to me that it would work with a significantly larger change in velocity than just using the fuel directly. Keep in mind that the asteroid that killed the dinosaurs was approximately 10km (some sources) to 15km (other sources) across, and we have no reason to think that future asteroids might not be bigger.
The largest atomic crater I could find was from a 104 kiloton bomb that created a 400m wide and 100M deep. Both the USA and Russia has similar tests, and that would be just a small dent in a 15km across asteroid. Both were from 100-200m deep explosion, and it's not clear if deeper would necessarily be better. Sure you might crack a 15km long asteroid into two pieces, but it wouldn't help much. Vaporizing a 15km asteroid is going to take WAY more than 100 kiloton.
What might work is some kind of nuclear engine that melts stone/sand/ice and ejects it at high velocity, working it's way into the center of the asteroid. It would need a throttle to only fire when it helps and idles when it doesn't.
Being an interstellar object though, it moves faster than the escape velocity to the sun, so it would pack a higher energy per unit of mass than a regular comet or asteroid. Fortunately such objects are also somewhat rare.
Couldn't we detect such things with radar? Given the right frequencies, rocks and dirt will reflect - that's how terrain-following radar works.
So yes, I think we could, but there are some significant problems.
1. 1/r^2 power loss, in both directions. We would need a massive radiator, and very sensitive detectors.
2. The detectors are in the wrong place. You send out your signal, and several hours later the response comes back, by which time the earth has rotated. So where your detector has to be depends on the distance to the object, but you don't know the distance to the object!
So we'd need a bunch of detectors at various points around the globe. And we'd need a huge power source.
Which brings me to solar flares. They emit radio waves. Do they emit enough power at the right frequencies to be useful for this?