Complexity of Octagonal and Rectangular Cartograms

A cartogram is a type of map used to visualize data. In a map regions are displayed in their true shapes and with their exact relations with the adjacent regions. However, such a map can only be used to demonstrate the actual area values of the regions. Sometimes, we need to display other data on a map, such as population, pollution, electoral votes, production rates, etc. One efficient way to do so is to modify the map such that the area of each shape corresponds to the data to be displayed. A map with given relationships between regions for which each region has pre-specified area is called a cartogram (see [1] for details). There are two major cartogram types: contiguous area cartograms [2, 3, 6, 7, 12], where the regions are deformed but stay connected, and non-contiguous area cartograms [8], where regions preserve their shapes but may lose adjacency relationships. Rectangular cartograms, where every region is a rectangle is a specific type of contigous area cartograms which tries to preserve both the adjacency relations and the shape, but this does not exist for all area values. Kreveld and Speckmann [13] introduced the first automated algorithms for such cartograms. Heilmann et al. proposed RecMap [5] to approximate familiar land covering map region shapes by rectangles. Rahman et al. studied slicing and good slicing graphs and their orthogonal drawings [9], which are similar to orthogonal cartograms. It was left as an open problem whether testing the feasibility of a rectangular cartogram is NP-hard. In this paper, we make significant progress towards answering this question. We first study what we call cartograms of orthogonal octagons where every region is an orthogonal polygon with at most 8 sides. We also assume that the cartogram must be placed within a rectangle of fixed size (a canvas). We show that testing whether a cartogram of orthogonal octagons exists is NP-hard. We then use a very similar reduction to prove NPhardness of a problem where, all faces are rectangles, except for one face corresponding to the “sea” around islands and peninsulas (see the examples in [13]).