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

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 Γ 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 Γ is equivalent to a discrete -complete problem. The related problem of finding the circumscribed squares of a Jordan curve Γ is also studied. We show that for any polynomial-time computable Jordan curve Γ, there must exist at least one computable circumscribed square (not necessarily of the minimum area), but this square may have arbitrarily high complexity.