On Different Reducibility Notions for Function Classes

Heribert Vollmer

Abstract: This paper continues research of Toda (The Complexity of Finding Medians, 31st Symposium on Foundations of Computer Science (1990), pp. 778–787) on problems complete for function classes like FP#P and Mid P under Krentel's metric reductions. We first show that metric reductions wipe out the difference between Mid P and other related classes of functions which are probably different from Mid P. In order to obtain a more detailed classification of naturally arising functional problems we then define and examine a stricter notion of reducibility and show that a number of problems, among them those proved by Toda to be hard for Mid P under metric reductions, are complete for different classes of median functions related to Mid P under our stricter reducibility. Finally, we use these results to exhibit new natural complete sets for the well studied classes of sets PP, PPNP, and PPP.