Complexity of octagonal and rectangular cartograms

In this paper, we study the complexity of rectangular cartograms, i.e., maps where every region is a rectangle, and which should be deformed such that given area requirements are satisfied. We study the closely related problem of cartograms with orthogonal octagons, and show that this problem is NP-hard. From our proof, it also follows that rectangular cartograms are NP-hard if we allow for the existence of a “sea”, i.e., a region of arbitrarily high complexity on the outside of the drawing.