A related problem is that of representing integers greater than 1 by unit fractions: it is clear that as the harmonic sum is divergent, such should be always possible. Adding terms till the next one would pass the target, and then proceeding by using the greedy algorithm should give a very inefficient solution, but a better one will have fewer terms.
The best solution could be defined either as the one with the fewest terms, or that with the largest final denominator - I don't know if these are necessarily the same.
Is there any information on how the number of terms is related to the target ?