 
Summary: On MinimumArea Hulls \Lambda
Esther M. Arkin y YiJen Chiang z Martin Held x Joseph S. B. Mitchell 
Vera Sacristan k Steven S. Skiena \Lambda\Lambda TaeCheon Yang yy
Abstract
We study some minimumarea hull problems that generalize the notion of convex hull to
starshaped and monotone hulls. Specifically, we consider the minimumarea starshaped hull
problem: Given an nvertex simple polygon P , find a minimumarea, starshaped polygon P \Lambda
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 \Lambda are constrained to be vertices of P , and the unrestricted version, in
which the vertices of P \Lambda can be anywhere in the plane. We prove that the restricted problem
falls in the class of ``3sumhard'' (sometimes called ``n 2 hard'') problems, which are suspected
to admit no solutions in o(n 2 ) time. Further, we give an O(n 2 ) time algorithm, improving the
previous bound of O(n 5 ). We also show that the unrestricted problem can be solved in O(n 2 p(n))
time, where p(n) is the time needed to find the roots of two equations in two unknowns, each a
polynomial of degree O(n).
We also consider the case in which P \Lambda is required to be monotone, with respect to an
unspecified direction; we refer to this as the minimumarea monotone hull problem. We give a
matching lower and upper bound of \Theta(n log n) time for computing P \Lambda in the restricted version,
