The Drawing Challenge - a problem I made up and can't solve!
A closely related problem is strongly NP-hard.

Leung et al. [1] prove that the following decision version of your problem is strongly NP-complete, by reduction from 3-Partition.


  Is it possible to pack $n$ squares of given widths $w_1, w_2, \dots, w_n$ into a larger square of width $W$?


I misread the original question as a generalization of the following decision problem:


  Is it possible to pack $n$ rectangles with given areas $a_1, a_2, \dots, a_n$ into a $H\times W$ rectangle?


The first problem can be reduced to the second as follows: Squares with widths $w_1,\dots,w_n$ can be packed into an outer square of width $W$ if and only if the minimum-perimeter packing of rectangles with areas $w_1^2, \dots, w_n^2$ into a $W\times W$ rectangle uses precisely $4\sum_i w_i$ units of ink.  (Proof: The minimum-perimeter rectangle with any given area is a square.)

However, this reduction does not imply that the actual posted question is NP-hard.


Joseph Y-T. Leung, Tommy W. Tam, C. S. Wong, Gilbert H. Young, and Francis Y. L. Chin.  Packing squares into a square.  J. Parallel and Distributed Computing 10(3):271–275, 1990.