Exponential-size numbers in NP completeness reduction
Changing the encoding of the inputs changes the problem.
When inputs are given succinctly the problem can become much harder.
On the surface of what you are saying 
they are proving the NP-hardness of a succinct version of the problem,
not the original problem.

You can also think of it as a reduction from Partition 
where numbers are given in unary to the original problem.
The unary version of Partition is in P.
So does not show the problem is NP-hard.
(It might be possible to use 3-Partion in place of Partition 
which remains NP-hard even when numbers in the input are encoded in unary.)

The issue is that it is not clear what are the inputs to the problem.
They seem to be working with a version of the problem where $\gamma$s are not 
part of the input to the problem 
but are  determined from $\tau_e$ which is part of the input.
In the theorem it seems that 
$\gamma_e$ is fixed to be $2^{c\tau_e}$ (where $c$ is a fixed constant)
and are not part of the input.
If $\gamma$ is not part of the input and fixed to be determined from $\tau$ as stated
then it doesn't matter that they are exponentially large.

In summery, the paper is not very clear in its treatment of the issue. 
It seems that in the theorem 
they are considering those numbers as not being part of the input to the problem (determined from $\tau_e$).
Or that they are given in a succinct form of $a2^e$.
The paragraph after the theorem 
where they say that it is open whether the results holds in case the usual binary encoding of numbers is used seems to support that.
So they are not proving that the problem where $\gamma$ is part of the input
and encoded as normal binary numbers is NP-hard.
That is still open according to them.
They are proving that variants of the problem mentioned above are NP-hard.