We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism, denoted CHI, which has running time (2 b N) O(1), where the parameter b is the maximum size of the color classes of the given hypergraphs and N is the input size. We also describe an fpt algorithm for a parameterized coset intersection problem that is used as a subroutine in our algorithm for CHI.

For any operatorτon integer-valued functions, we say that #P isclosed under τ in the context ofPF∘#P if, for everyf∈#P,τ[f] belongs to PF∘num;P. For several operatorsτ, it is shown that the closure properties of #P underτin the above sense is closely related to the relationships between P#P[1]and higher classes such as PHPPand PPPP.

In this paper, it is shown that many natural counting classes, such as PP, $C_ = P$, and ${\text{MOD}}_k {\text{P}}$, are at least as computationally hard as PH (the polynomial-time hierarchy) in the following sense: for each ${\bf K}$ of the counting classes above, every set in ${\bf K}$(PH) is polynomial-time randomized many-one reducible to a set in ${\bf K}$ with two-sided exponentially small error probability. As a consequence of the result, it is seen that all the counting classes above are computationally harder than PH unless PH collapses to a finite level. Some other consequences are also shown.

In this paper, two interesting complexity classes, PP and $ \oplus {\text{P}}$, 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 ${\text{BP}} \cdot \oplus {\text{P}}$. As a consequence of the results, it follows that ${\text{PP}} \subseteq {\text{PH}}$ (or $\oplus {\text{P}} \subseteq {\text{PH}}$) 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.

PF( Hash P) is characterized in a manner similar to M.W. Krentel's (1988) characterization of Pf(NP). If MidP is the class of functions that give the medians in the outputs of metric Turing machines, then it is shown that every function in PF( Hash P) is polynomial time 1-Turing reducible to a function in MidP and MidP contained in PF( Hash P); that is, PF( Hash P)=PF(MidP(1)). Intuitively, finding medians is as hard computationally as PF( Hash P); this forms a contrast to an intuitive interpretation of Krentel's result that finding maxima (or minima) is as hard as PF(NP). Several applications of the result are shown.>

Two complexity classes, PP and (+)P, are compared with PH (the polynomial-time hierarchy). The main results are as follows: (1) every set in PH is reducible in a certain sense to a set in PP, an (2) every set in PH is reducible to a set in (+)P under randomized polynomial-time reducibility with two-sided bounded error probability. It follows from these results that neither PP nor (+)P is a subset of or equivalent to PH unless PH collapses to a finite level. This is strong evidence that both classes are strictly harder than PH.>