On solving geometric optimization problems using shortest paths

Abstract

We have developed techniques which contribute to efficient algorithms for certain geometric optimization problems involving simple polygons: computing minimum separators, maximum inscribed triangles, a minimum circumscribed concave quadrilateral, or a maximum contained triangle. The structure for our algorithms is as follows: a) decompose the initial problem into a low-degree polynomial number of easy optimization problems; b) solve each individual subproblem in constant time using the methods of cealcuIns, standard methods of numerical analysis, or linear programming. The decomposition step uses shorteat path trees inside simple polygons (Guibas et. al.~ 1987) and, in the case of inscribed triangles, produces a new class of polygons, the fan-shaped polygon. By extending the shortest-path algorithm to splinegons, we also generate splinegon-versions of the algorithms for some of the optimization problems. The problems we discuss fall into four subgroups: Separa tors : If two points z and y lie on the boundary of simple polygon P and define a directed line segment zy C_ P that separates P into two sets PL and Pa, then zy is called a separator.