On Minimum-Area Hulls

Abstract. We study some minimum-area hull problems that generalize the notion of convex hull to starshaped and monotone hulls. Specifically, we consider the minimum-area star-shaped hull problem: Given an n-vertex simple polygon P, find a minimum-area, star-shaped polygon P∗ containing P. This problem arises in lattice packings of translates of multiple, nonidentical shapes in material layout problems (e.g., in clothing manufacture), and has been recently posed by Daniels and Milenkovic. We consider two versions of the problem: the restricted version, in which the vertices of P∗ are constrained to be vertices of P, and the unrestricted version, in which the vertices of P∗ can be anywhere in the plane. We prove that the restricted problem falls in the class of “3SUM-hard” (sometimes called “n2-hard”) problems, which are suspected to admit no solutions in o(n2) time. Further, we give an O(n2) time algorithm, improving the previous bound of O(n5). We also show that the unr...

• Type: Journal
• Year: 1998
• Where: ALGORITHMICA
• Authors: Esther M. Arkin, Yi-Jen Chiang, Martin Held, Joseph S. B. Mitchell, Vera Sacristan, Steven Skiena, Tae-Heng Yang
• Keywords: ALGORITHMICA 1998, Hull Problem, Monotone Hull, Polygons