Structural properties of bounded relations with an application to NP optimization problems

We introduce the notion bounded relation which comprises most resource bounded reducibilities which can be found in the literature, including non-uniform bounded reducibilities such as ⩽TP/poly. We state conditions on bounded relations which are again satisfied for most bounded reducibilities and which imply that every countable partial ordering can be embedded into every proper interval of the recursive degrees. As corollaries, we obtain that every countable partial ordering can be embedded into every proper interval of (REC,⩽TP/poly, as well as into every proper interval between either two maximization or two minimization problems in the structures NPO,⩽E and NPO,⩽L. For the two latter structures, we show further that the result about embeddings of partial orderings extends to embeddings of arbitrary countable distributive lattices where in addition the least or the greatest element of the lattice can be preserved. Among other corollaries, we obtain that for both structures every non-trivial NP optimization problem bounds a minimal pair. In connection with embeddings into NPO we introduce a representation of maximization or minimization problems by polynomial time computable functions. We consider decidability issues w.r.t. this representation and show for example that there is no effective procedure which decides for a (subrecursive) index of a polynomial time computable function whether the corresponding maximization problem is approximable within a constant factor.