Of course given that the HTC is a fictional device you can always imagine arbitrary issues that make it uneconomical. I was considering the HTC just as a computer that had 365x the serial speed of present day computers, and considering whether there would be economically interesting batch (~1 day long) computations to run on it.

Hm, I wasn't, except in the shift to the upload scenario where the speedup is not from executing regular algorithms (presumably anything capable of executing emulated brains at 365x realtime will have much better serial performance than current CPUs). As an ordinary computer there's still heat considerations - how is it taking care of putting out 365x a regular computer's heat even if it's doing 365x the work? And as a pocket universe as specified, heat is an issue - in fact, now that I think about it Stephen Baxter invented an space-faring alien race in his Ring hard sf universe which lives inside tiny pocket universes as the ultimate in heat insulation.

These problems have polynomial time complexity, they don't asymptote. Linear programming, for instance has quadratic worst-case time complexity in the size of problem instance (and O(n^3.5) time complexity in the number of variables).

I was referring to the quality of the solution produced by the approximating algorithms.

For problems related to model checking (circuit value problem, Horn-satisfiability, type inference) approximate solutions don't seem particularly useful.

Quickly googling, there seems to be plenty of work on approximate solutions and approaches in model checking; for example http://cs5824.userapi.com/u11728334/docs/77a8b8880f48/Bernhard_Steffen_Verification_Model_Checking_an.pdf includes a paper:

In this paper, we propose an approximation method to verify quantitative properties on discrete Markov chains. We give a randomized algorithm to approximate the probability that a property expressed by some positive LTL formula is satisfied with high confidence by a probabilistic system. Our randomized algorithm requires only a succinct representation of the system and is based on an execution sampling method. We also present an implementation and a few classical examples to demonstrate the effectiveness of our approach.

I'll admit I don't know much about the model checking field, though.