On the complexity of finding circumscribed rectangles and squares for a two-dimensional domain

Authors:
Fuxiang Yu;Arthur Chou;Ker-I Ko
Affiliations:
Department of Computer Science, State University of New York at Stony Brook, Stony Brook, NY 11794, USA;Department of Mathematics and Computer Science, Clark University, Worcester, MA 01610, USA;Department of Computer Science, State University of New York at Stony Brook, Stony Brook, NY 11794, USA
Venue:
Journal of Complexity
Year:
2006

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Abstract

We investigate the computational complexity of finding the minimum-area circumscribed rectangle of a given two-dimensional domain. We study this problem in the polynomial-time complexity theory of real functions based on the oracle Turing machine model. We show that a bounded domain S with a polynomial-time computable Jordan curve @C as the boundary may not have a computable minimum-area circumscribed rectangle. We also show that the problem of finding the minimum area of a circumscribed rectangle of a polynomial-time computable Jordan curve @C is equivalent to a discrete @S"2^P-complete problem. The related problem of finding the circumscribed squares of a Jordan curve @C is also studied. We show that for any polynomial-time computable Jordan curve @C, there must exist at least one computable circumscribed square (not necessarily of the minimum area), but this square may have arbitrarily high complexity.