Seinosuke Toda

PP is as hard as the polynomial-time hierarchy

In this paper, two interesting complexity classes, PP and

Polynomial-time 1-Turing reductions from # PH to # P

\oplus {\text{P}}

Turing machines with few accepting computations and low sets for PP

, are compared with PH, the polynomial-time hierarchy. It is shown that every set in PH is polynomial-time Turing reducible to a set in PP, and PH is included in

On polynomial-time truth-table reducibility of intractable sets to P-selective sets

{\text{BP}} \cdot \oplus {\text{P}}

Graph isomorphism completeness for chordal bipartite graphs and strongly chordal graphs

. As a consequence of the results, it follows that

Turning machines with few accepting computations and low sets for PP

{\text{PP}} \subseteq {\text{PH}}

The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes

(or

The complexity of selecting maximal solutions

\oplus {\text{P}} \subseteq {\text{PH}}

Counting classes are at least as hard as the polynomial-time hierarchy

) implies a collapse of PH. A stronger result is also shown: every set in PP(PH) is polynomial-time Turing reducible to a set in PP.

ON THE COMPLEXITY OF COMPUTING OPTIMAL SOLUTIONS

In this paper, we investigate relative complexity between #P and other classes of functions. Our particular interest is to compare #P with #PH and with PFH by using polynomial-time reducibility and to demonstrate that a weaker notion of polynomial-time reducibility is sufficiently powerful for reducing #PH functions to #P functions. Our main result is stated as follows: Every function in #PH is polynomial-time 1-Turing reducible to some function in #P. That is, #PH⊆PF#P[1]. Some consequences of this result are as follows: Every function in PFH is polynomial-time 1-Turing reducible to some function in #P. If PF#P[1]⊆#PH, then PH collapses to a finite level; furthermore, if either #P⊆PFH or PFH⊆#P, then PH collapses to a finite level. We also give an affirmative answer to an open question posed by Valiant (1979), and we show a generalized result about p-rankability by Hemachandra (1987).