While \Cref{thm:mult-p-hard-result} shows that computing $\rpoly(\prob,\dots,\prob)$ in general is hard it does not rule out the possibility that one can compute this value exactly for a {\em fixed} value of $\prob$. Indeed, it is easy to check that one can compute $\rpoly(\prob,\dots,\prob)$ exactly in linear time for $\prob\in\inset{0,1}$. In this section, we show that these two are the only possibilities:
Fix $\prob\in(0,1)$. Then assuming \Cref{conj:graph} is true, any algorithm that computes $\rpoly_{G}^3(\prob,\dots,\prob)$ from $G$ exactly has to run in time $\Omega\inparen{m^{1+\eps_0}}$, where $\eps_0$ is as defined in \Cref{conj:graph}.
%\begin{proof}[Proof of Corollary ~\ref{th:single-p-gen-k}]
%Consider $\poly^3_{G}$ and $\poly' = 1$ such that $\poly'' = \poly^3_{G} \cdot \poly'$. By \Cref{th:single-p}, query $\poly''$ with $\kElem = 4$ has $\Omega(\numvar^{\frac{4}{3}})$ complexity.
The above shows the hardness for a very specific query polynomial but it is easy to come up with an infinite family of hard query polynomials by `embedding' $\rpoly_{G}^3$ into an infinite family of trivial query polynomials.
Unlike \Cref{thm:mult-p-hard-result} the above result does not show that computing $\rpoly_{G}^3(\prob,\dots,\prob)$ for a fixed $\prob\in(0,1)$ is \sharpwonehard.
However, in \Cref{sec:algo} we show that if we are willing to compute an approximation that this problem (and indeed solving our problem for a much more general setting) is in linear time.
If we can compute $\rpoly_{G}^3(\prob,\dots,\prob)$ exactly in $T(\numedge)$ time, then we can exactly compute $\numocc{G}{\tri}$%count the number of triangles, 3-paths, and 3-matchings in $G$
Fix $\prob\in(0,1)$. Given $\rpoly_{\graph{\ell}}^3(\prob,\dots,\prob)$ for $\ell\in[2]$, we can compute in $O(m)$ time a vector $\vct{b}\in\mathbb{R}^3$ such that
The lower bounds of \cref{thm:mult-p-hard-result} and \cref{th:single-p-hard} hold with respect to $\timeOf{\abbrStepOne}$.
\end{Corollary}
\begin{proof}
We start by showing that there exists an algorithm that computes $\poly$ in $\bigO{\numedge}$ time for our hard query in \cref{def:qk}. Assume that the edges $E$ of graph $G$ are encoded in a relation $R$. Then a simple table scan of $\rel$ will iterate through the entire set of $\numedge$ edges computing summation of $\poly$ in $\numedge$ steps, and we can replicate this sum $k$ times to output the final $\poly$, in at most $\numedge+ k =\bigO{\numedge}$ steps.
This implies that $\numedge\geq\Omega\inparen{\timeOf{\abbrStepOne}}$, and since the results of \cref{thm:mult-p-hard-result} and \cref{th:single-p-hard} are in the number of edges $\numedge$, then it follows that our lower bounds hold with respect to $\timeOf{\abbrStepOne}$.