> space based laser
Lasers won't do squat. To move an asteroid you need nukes... either nuclear explosives or nuclear rocket engines. Bombs would be easier... we've got loads of 'em. If'n we could make the General Atomics "Casaba Howitzer" design of nuke, we'd *really* be in business here.
In any event, much as I'm a fan of the Rutans of the space world, this is the sort of thing that would require a Gubmint Mega Project. Build a whole armada of heavy lift launvh vehicles in the Saturn V class, launch a few hundred citybusters and a (probably manned) control system, and you'll be in business. Since I work for the company that makes the Shuttle boosters, I know what I'm going to be arguing for come the new year...
The "good" news is that this puppy passes close to the earth multiple times ... close enough for us to reach it with current tech. If needed we should be able to get a nuke payload to it many years ahead of impact, with plenty of time to position the warheads and blow them at precise the right moment in order to adjust its orbit just enough to miss us. The object is not to try and destroy it ... but, rather, to change its orbit by a fraction of a percent. If done early enough, that should keep it from hitting us.
Anyone ever read Thunderstrike? Good SF book. In that book a huge comet which would have hit the Earth was slowed up by maneuvering smaller rocks (asteroids) into it's path with shaped atomic charges. The impact of these asteroids with the comet caused it to slow by several dozen minutes and, thus, it hit the moon instead of the earth.
Nuclear won't do it either. When Nukes go off on earth, its the shock wave and heat that do the most damage. Out in space even hitting one side of a rather large asteroid may do a little melting but since a nuke in space would be mostly heat and light, there is no kinetic shock energy to break up the asteroid or to make it change course. Even if you could land on the asteroid to bury it or drill deep to place it, there is no air in the rock of said asteroid, just solid matter, so there is no guarantee that a nuke would do more than melt the rock on the inside....though melting the rock to vaporisation from the inside out may cause it to explode from the various gasses and pressures produced. This would create showers of meteors that would be very hard to predict in terms of impact. Smaller asteroids a mile or less may very well be destroyed easily this way but very large ones such as greater than one mile to hundreds of miles would need more more long range work.
Simply throwing missiles at them won't destroy them.