The Coin Collector's Problem.

In the country of Binatopia, the face value of every coin is a power of two
times the face value of the smallest coin.

As in other countries, a young coin collector must sometimes part with some
cherished coins to buy something.

A Binatopian coin collector has a collection of coins, each of which has
a face value (see above) but also a numismatic value, which can be any
number.  The coin collector wishs to purchase a bicycle, which costs a
certain amount, but the bicycle shop has no appreciation of the numismatic
values of coins, and demands a certain amount of money, measured by the
face value of each coin.  How can the coin collector find that subset of
the collection which has the minimum total numismatic value, given that it
must have a face value equal to the price of the bicycle?