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Abstract. One of the most challenging tasks of carto~'aphlc map lettering is the optimal placement of region information on a map. We propose as an approach to this task the elastic labeling problem, in which we are given a set of elastic rectangles as labels, each associated with a point in the plane. An elastic rectangle has a specified area but its width and height may vary. The problem then is to choose the height and width of each label, and the corner of the label to place at the associated point, so that no two labels overlap. This problem is known to be NP-hard even when there is no elasticity (just because of the choice of the corners). We show that the problem remains NP-hard when we have elasticity but no choice about which corner of the label to use--we call this the one-corner elastic labeling problem. We give a polynomial time algorithm for the special case of the one-corner elastic labeling problem in which the points lie on the positive x and y axes and the labels lie in the first quadrant. We call this the two-axis labeling problem.

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