The Power of Non-Rectilinear Holes

Four multiconnected-polygon partition problems are shown to be NP-hard. bltroduction One of the main topics of computational geometry is the problem of optimally partitioning figures into simpler ones. Pioneers in this field mention at least two reasons for the interest : (1) such a partition may give us an efficient description of the original figure, and (2) many efficient algorithms may be applied only to simpler figures . Besides inherent applications to computational geometry [CI], the partition problems have a variety of applications in such domains as database systems [LLMPL], VLSI and architecture design [LPRS] . Among others, the three following partition problems have been investigated : MNRP ( Minimum Number Rectangular Partition ) . Given a rectilinear polygon with rectilincar polygon holes, partition the figure into a minimum number of rectangles. MNCP1 ( Mininmm Number Convex Partition 1 ) . Given a polygon, partition it into a minimum number of convex parts . MNDT1 ( Minimum Number Diagonal Triangulation 1 ) . Given a polygon, partition it into a minimum number of triangles, by drawing not-intersecting diagonals . In the above definitions, as in the course of the entire paper, we assume the following conventions. A polygon means a simple polygon ( see [ SH ] ), given by a sequence of pairs of integer-coordinate points in the plane, representing its edges. A rectilinear polygon is a polygon, all of whose edges are either horizontal or vertical . A polygon with polygon holes is a figure Consisting of a polygon and a collection of not-overlapping, notdegenerate polygons lying inside i t . The perimeter of the outer polygon and the contours of the inner polygons form boundaries of the figure, enclosing its inside equal to the inside of the outer polygon minus the boundaries and insides of the inner polygons. A diagonal of a planar figure is a line segment lying inside it and joining two of its non-adjacent vertices. At first sight, MNRP and MNCP1 seem to be NP-hard. Smgrisingly, both arc solvable in time O(n3), where n is the number of corners of the input figure ( see [ LLMPL ] and [ C, CD] ). The O(n 3) time algorithm for MNRP uses a matching technique, lhat for MNCP1 is an example of a sophisticated dynamic progrmnming approach. MNDT1 is also solvable in time O(n3), by a straightforward, dynamic progrmnming procedure * *. In contrast to these results, we show the following problems to be NP-hard : ~ r ' T f i ' / ~ was supported by NSF grants MCS-8006938 and MCS-7805849 . 9 * The known triangulation algorithm of time complexity O(nlogn) [GJPrl] divides the input into n-2 triangles which is not always optimal [P].