Geometric Optimization and the Polynomial Hierarchy

We illustrate two different techniques of accurately classifying geometric optimization problems in the polynomial hierarchy. We show that if NP≠Co-NP then there are interesting natural geometric optimization problems (location-allocation problems under minsum) in △ 2 P that are in neither NP nor Co-NP. Hence, all these problems are shown to belong properly to △ 2 P , the second level of the polynomial hierarchy. We also show that if NP≠Co-NP then there are again some interesting geometric optimization problems (location-allocation problems under minmax), properly in △ 2 P and furthermore they are complete for a class DP (which is contained in △ 2 P and contains NP Co-NP).

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