Strong nondeterministic Turing reduction--a Technique for Proving Intractability

Selman and Long introduced the concept of a strong nondeterministic reduction (denoted by ≥
t
ⁱⁿ
) as a generalization of Adleman-Manders' γ-reduction. This reduction has the property that if A ≤
t
ⁱⁿ
B, then A ∈ NP
^B
⋒ co-NP
^B
. Thus if Β is a problem in NP such that A ≤
t
ⁱⁿ
for all A ∈ NP, then Β is not Ρ unless NP = co-NP, providing a strong evidence for the intractability of B. We say in this case that A is ≤
t
ⁱⁿ
-complete for NP. Our main result is the proof of ≤
t
ⁱⁿ
-completeness for NP of a combinatorial problem arising in the analysis of sorting-type networks. This is the first example (to the best of authors' knowledge) of a non number-theoretic problem which is shown to be intractable using a nondeterministic reduction. It is not known if this problem is NP-complete. Earlier, Adleman and Manders used nondeterministic reductions to show the intractability of some number-theoretic problems. Our problem can be stated as follows. ‘Given a network, does it possess a given property?’, e.g., ‘is it a sorting network?’. We show that testing of any property that has an exponentially large test set is ≤
t
ⁱⁿ
-complete for NP. On our way to the main result, we also show the NP-completeness or coNP-completeness of some fundamental network testing problems.