Argthtjtr

How difficult would it be? Unfortunately, your timeline is too tight for any of the easy options to work.

The mass in the asteroid belt is highly concentrated. Between them, Ceres (~30%), Vesta (~10%), Pallas (~8%), and Hygiea (~5%) make up about half the mass of the asteroid belt. Combining these into a single body is (relatively) easy, but slow and energy-intensive: unless you're at least a type II civilization on the Kardashev scale, you're looking at a timeframe of decades or centuries just to get them into the same place through creative use of solar sails. Simply bringing Vesta adjacent to Ceres in under a decade will take on the order of $10^{28}$ joules (equivalent to diverting the total energy output of the Sun for one minute).

Once you've got them together, the only practical way to combine them without scattering debris all over the place is to melt them first. Melting Ceres will require on the order of $2*10^{25}$ joules to heat, and then another $4*10^{24}$ joules for the actual melting. This is on the order of the total solar heating of the Earth in five years. The full asteroid belt will take about three times that (a Kardashev type I civilization couldn't quite do it in the allotted decade). Cooling is out of the question: unless you want to risk thermal shock shattering your new dwarf planet back into an asteroid belt, you're looking at cooling times of millenia.

A domino effect of starting small and letting things work their way up isn't an option. That already happened, and the result was the asteroid belt as you see it today. The clear areas around the orbits of the planets aren't the result of those planets sweeping up the available mass, but the result of those planets ejecting that mass from the solar system. It also takes far too long: current models indicate that the clearing process took around a half-billion years. The asteroid belt exists because Ceres isn't large enough to clear its neighborhood without help.

If you had more time to work with, you could attach an outrageously large solar sail to Ceres and fly it around, using its gravity to throw the rest of the asteroids into unstable resonances with Jupiter, leaving it the only body in the asteroid belt, but this is (a) slow and (b) cheating.

Short answer, no.

The asteroid belt is incredibly sparse, where your chance of seeing another asteroid from the surface of one with the naked eye is pretty low.

There's no simple way to overcome the momentum of every asteroid. They would all require manual adjustments to their orbits, many of them requiring multiple burns to reposition them and collide them with Ceres (the largest asteroid).

Probably not all in one go, but along the way why not have some fun, save your job and give the Emperor something to smile about.

Say you have a sixteen space tugs to work for you - first thing first, order 12 of them to pop out to the Kuiper belt and drag in a dozen matching 100 Km diameter KBO's (essentially comets), whilst this is happening, get the other 4 to grab Ceres, Vesta, Pallas, and Hygiea from the belt , and in the most efficient way place them equidistant around the inner edge of the belt.

Meanwhile the other tugs are draging their loads into equidistant positions between the aforementioned roids.

The tugs can then proceed to push all these objects at optimal speed around the belt to gather mass as they go, gradually increasing the diameter of orbit as they clear debris.

"But, what if the Emperor comes before the job's finished?" I hear you cry in alarm.
Worry not, approaching from the Pole of the solar system (on the Sun's axis of rotation) his magnifience will see: - centre of his view, dark save for the sun shining like the twinkel of humour in his eye, then the inner planets, the sharp inner edge of the belt carved and delineated by the 4 planetisimals and the comets then like a great iris with their tales radiating outward.

The wise and benevolent one will recognise (with judicious prompting) the magnificent sight of a likeness of his own eye gazing benignley upon the universe, and revel in his subjects rejoicing and adulation.

Assumption: There's enough mass in the target asteroid belt to make a planet the desired size.

As amazingly powerful as gravity is, it's quite a bit weaker than kinetic energy unless there's a whomping lot of mass nearby. Consider, for example, the impact of a baseball on a bat. Eventually it falls back to Earth — but for a moment, it's free.

Now let's go to space where we're trying to bring asteroids together and, especially at the beginning, you don't have enough mass to guarantee that the chunks of asteroid that fly off when two come together return to the central mass. What are your options?

You don't have any. This process would be slow.

• You can use an net the size of your final planet to sweep around the orbit and collect stuff. The amount of energy needed to do this is so great that it begs the question "why are we doing this, again?"




• You can plop thruster packs on asteroids to speed them up a little, thereby causing them to come together (hopefully gently) with the stuff in front of them. Somewhere around a third the way through^{A complete guess} you could probably dispense with the thruster packs and wait. The final planet would need to swing around the orbit for a very long time (veeeerrrrrryyyyyy looooooonnnngggggg ttiiiiiiimmmmeeeeee) to sweep up the debris.




• You can get them moving faster, but this means debris winging off into space. You'll build the basic planet faster, but it'll take longer to sweep up the debris to finish the job (meteors are not your friend, you must clean up).

But, even if you did these, how long before you have a planet? Pressure would build, heat would form... eventually you'd get the right kind of core, etc. You'd likely need to drop a bunch of methane/water asteroids on the thing to kickstart an atmosphere....

Conclusion

Ten years? The uneducated reader would probably never think twice about it, but it's totally unbelievable. I can't see how you could do it in less than centuries, maybe millennia. You're dealing with so much space, so much mass, so much energy... there's reasons why spacey things happen slowly. When they happen fast, they tend to be catastrophic.

Edit: I didn't think to ask Chasly what's to become of this planet(oid)? Derived from our own asteroid belt, it won't be big enough to develop heat or hold an atmosphere and might not be enough mass to stick together as a cohesive ball. In other words, the dirt clods may simply hang together as dirt clods in space until gravitic distortions from the orbits of other planets start spreading them out again. In astronomical lengths of time, they wouldn't stay together very long.

Nanites

Self replicating nanites/robots eat the asteroids and make more nanites/robots. The nanites have their own propulsion system to track and eat other asteroids and so on.

See Replicators

Unless you have a exponentially growing system system, you will never clear it in time.

With sufficient effort, this could be done. It would require the manual alteration of the orbit of each asteroid.

Bear in mind that the total mass of the asteroid belt is around 4% that of the moon, and that around half of that mass is already contributed by only 4 asteroids. The remainder are tiny and barely significant.

It would be an awful lot of work for basically no gain.

You've specified that you want a pure domino effect rather than a propelled solution for herding / collecting each asteroid, but this isn't possible without flinging the new planetoid cluster(s) into a new orbit and making an even bigger mess long term. If you just need to sweep some debris under the rug for the duration of the royal visit, this may not be a problem.

Why dominoes won't work

As already mentioned, the deadline for the King of the Cosmos Emporer's visit is only 2* orbits around the sun for a given asteroid. Say you stick a giant butterfly net force field on a medium asteroid so it can gently scoop up all the other asteroids Katamari style and eventually stick to Ceres**, and have a giant space tugboat give it a starting push to a new speed.
_{* 2-ish, but let's stick with simple math to illustrate the point.
** Saving the biggest for last saves time and energy.}

Your snow- asteroid-ball either has to move much faster, completing 3 laps instead of 2 just in time to catch the asteroids originally just behind it. Ignoring acceleration times altogether, you have a 50% increase in orbital velocity, and the orbital radius will therefore increase considerably and start interacting more with Jupiter's gravity. Or you could slow down and complete only 1 orbit (50% drop in speed) and give Mars a new moon (or completely re-texture it*). Except that long before you finished either cycle, the initial snowball mass will have already have moved to its new orbit and be far away from the remaining asteroids at the end of the original orbit.
_{* Not necessarily bad - another user already mentioned terraforming. Instant habitable planet, just add water and allow to cool 10,000 years.}

If you are just worried about tidiness and not creating a single mass, you could possibly pull off domino stacking by instead collecting 8 planetoids (or 6 or 12 or His Majesty's favorite number) of equal size and spacing.

Thinking with Portals!

However, you haven't specified your civilization's exact level of technology beyond some very basic clues - obviously they have space flight and, and enough power to get ships around interstellar distances, but probably not FTL or unlimited energy and fabrication time. The minimum power required for any giant problem is a 9V battery and an SEP field, but I'm assuming that's not an option either.

Therefore, I propose a middle ground solution (which may or may not fit your particular universe) involving a probably reasonable amount of power, and a fairly large orange and blue portal gun pair of wormgates.

A single ship travels to Ceres and leaves one of the gates floating nearby, then slows down a bit with the other one and take the full 10 years zig-zagging backwards around the belt. Each time you reach an approaching asteroid, power up both gates for a few seconds/minutes and lasso it, leaving it parked in formation right next to Ceres***. Power back down, shift the receiver slightly, and move to the next one while the gates recharge. If you need to move faster than 1 crew can handle, send a handful of collector ships with their own gates (either a separate pair or all tuned to the same destination).

_{*** Crashing everything together would make a debris field that was unsafe to operate your gate in, if not sending pieces back into separate orbits.}

Other posters have focused on two large asteroid belt clearing problems: excessive time and excessive energy are needed to suspend disbelief for the genre of hard science fiction.

Excessive need for space or time seems the lesser issue in a Einsteinian universe.

To suspend excessive space-time, I agree with brichins' idea of using wormgates, a SF technology that harnesses macro wormholes (as opposed to quantum wormholes that seem closer to present physics theory).

I agree, in validly suspended SF, that each end of a sizable wormhole can be placed where needed to move even large planetoids like Ceres. I think all asteroids could be teleported adjacent to each other, with matching momentum for bolting (stone) and welding (iron) into something(s).

Something(s) 4% the size of the moon per Arkenstein XII.

I see no obvious SF reason why wormgates can't teleport other wormgates in a preprogramed pattern around the asteroid belt. What I might call leveraged teleporting, starting with a fleet of cargo starcraft powered by (consensus SF) hyper/warp drive.

Teleport clearing the whole asteroid belt certainly wouldn't be instant, and could take years, because electrons and photons - even in the peripherals of quantum computer circuits needed to solve such an enormous set of calculations - travel at about 95% of light speed (in copper).

One will need a great many individual computers/processors/hardware, and quite at lot of (consensus SF) subspace signaling to project managers in, say, a triangle of bases orbiting the solar system. Debugging committees, even with the help of AI, will have to solve many initial project design issues. That's not going to change in the future.

Mark's delightful analysis of the excessive energetics seems more troublesome.

Einstein seems to allow the cheating of our ordinary calculations of space and time, but not the cheating of mass-energy conservation. Mark probably can calculate how much energy it takes to move objects into higher orbit, if it's 4% of the moon mass, raised from the asteroid belt, to spiral orbits into Jupiter for slow disposal. Jupiter disposal has its own set of problems, but I've read that Jupiter is too small to be set on fusion fire.

I don't know how much recoverable, reusable energy, and lost, heat-dissipated energy it takes to operate a wormgate.

Even if you recover most of the reusable energy after the worm field collapses, to inflate the field, one still has to store a great deal of energy joules in pre-charged capacitors (sized 10^xx farads?).

For story purposes, I think the average reader better comprehends kilowatt-hours (at 20c each) to the equivalent physics joules. Will the Emperor pay for the heat-dissipated energy?

The good news is that your story is premised as tongue-in-cheek SF like Hitchhiker's Guide to the Galaxy. That genre depends more on clever writing than a serious SF suspension.

You've mentioned a bureaucracy, so your story could be one of massively bungling the project, yet in Homer Simpson fashion, all the bungling cancels itself out to result in an unexpectedly successful conclusion.

One way to compress 10 years into a short story is to tell it as a series of dated historic documents, like royal proclamations, emails, radiophone transcripts, corporate invoices, lawsuits, arrest warrants, and news accounts.

You could cover over story holes with puzzled notes from a future historian, writing that he can't find documents to explain how event X occurred improbably - which works with tongue-in-cheek SF.

Impossible

Since the astreroid belt is exactly that due to the gravity of several planets making sure that any "body" would be pulled apart as it forms, or after it is formed. The whole asteroid "belt" is proof of this effect.

So it is impossible without changing the whole solar system this is in. At which point you can no longer speak of an asteroid belt anyways.