The structure of optimal partitions of orthogonal polygons into fat rectangles

Motivated by a VLSI masking problem, we explore partitions of an orthogonal polygon of n vertices into isothetic rectangles that maximize the shortest rectangle side over all rectangles. Thus no rectangle is “thin”; all rectangles are “fat.” We show that such partitions have a rich structure, more complex than what one might at first expect. For example, for partitions all “cuts” of which are anchored on the boundary, sometimes cuts are needed 1 2 or 1 3 of the distance between two polygon edges, but they are never needed at fractions with a larger denominator. Partitions using cuts without any restrictions seem especially complicated, but we establish a limit on the “depth” of cuts (roughly, how distant from the boundary they “float” in the interior) and other structural constraints that lead to both an O(n) bound on the number of rectangles in an optimal partition, as well as a restriction of the cuts to a polynomial-sized grid. These constraints may be used to develop polynomial-time dynamic programming algorithms for finding optimal partitions under a variety of restrictions.