Geometric optimization and \(D^ P\)-completeness (Q1106665)

The task of classifying the complexity of optimization problems accurately in the polynomial hierarchy is one of continuing interest and importance. We show that a large number of geometric optimization problems, that arise naturally from the optimal placement of simple geometric objects are complete for a class \(D^ P\). The class of \(D^ p\) is defined as follows: L is in \(D^ p\) iff L is an intersection of \(L_ 1\) and \(L_ 2\) such that \(L_ 1\) is in NP and \(L_ 2\) is in Co- NP. The class \(D^ p\) contains both NP and Co-NP and is contained in \(\Delta_ 2^ P=P^{NP}\). Completeness in \(D^ p\) is exhibited under many-one and positive reductions. These results also prove the existence of natural geometric optimization problems that are proper in \(\Delta^ P_ 2=P^{NP}\). Further OptP(O(log n)) results are exhibited for some of these optimization problems.