On the limitations of locally robust positive reductions

Abstract

Polynomial-time positive reductions, as introduced by Selman, are by definition globally robust — they are positive with respect to all oracles. This paper studies the extent to which the theory of positive reductions remains intact when their global robustness assumption is removed. We note that two-sided locally robust positive reductions — reductions that are positive with respect to the oracle to which the reduction is made — are sufficient to retain all crucial properties of globally robust positive reductions. In contrast, we prove absolute and relativized results showing that one-sided local robustness fails to preserve fundamental properties of positive reductions, such as the downward closure of NP.