Done till proof of redux

mult_distinct_p.tex

@@ -41,7 +41,7 @@ Let $G=([n],E)$ be a graph. Then for any $\kElem\ge 1$, define
 \[\poly_{G}^\kElem(X_1,\dots,X_n) = \left(\sum\limits_{(i, j) \in E} X_i \cdot X_j\right)^\kElem.\]
 \end{Definition}
 
-Our hardness results only need TIDB instance and further, we consider the special case when all the tuple probabilities are the same value.
+Our hardness results only need TIDB instance and further, we consider the special case when all the tuple probabilities are the same value. It is not too hard to see that we can encode the above polynomial in an expression tree of size $\Theta(km)$.
 
 Following up on the discussion around Example~\ref{ex:intro}, it is easy to see that $\poly_{G}^\kElem(\vct{X})$ is the query polynomial corresponding to the following query:
 \[\poly:- R(A_1),E(A_1,B_1),R(B_1),\dots,R(A_\kElem),E(A_\kElem,B_\kElem),R(B_\kElem)\]
@@ -54,10 +54,30 @@ Note that this imples that our hard query polynimial can be created from a join-
 \subsection{Multiple Distinct $\prob$ Values}
 \label{sec:multiple-p}
 
+We are now ready to present our main hardness result.
+\begin{Theorem}\label{thm:mult-p-hard-result}
+Computing $\rpoly_G^\kElem(\prob_i,\dots,\prob_i)$ for arbitraryy $G$ and any $(2k+1)$ values $\prob_i$ ($0\le i \le 2k$) is \sharpwonehard.
+\end{Theorem}
+We will prove the above result by reducing the problem of computing the number of $k$-matchings in $G$. Given the current best-known algorithm for this counting problem, our results imply that unless the state of the art $k$-matching algorithms are improved, we cannot hope to have a better runtime to solve our problem in time better than $\Omega_k\inparen{m^{k/2}}$, which is only quadratically faster than expanding $\poly_{G}^\kElem(\vct{X})$ into its SOP form and use~\Cref{cor:expct-sop}. By constrast our approximation algorithm would run in time $O_k\inparen{m}$ on this query (since it runs in linear-time on all query polynomials).
+
+
+As mentioned earlier, we prove our hardness result by presenting a reduction from the problem of couting $\kElem$-matchings in a graph:
 \begin{Lemma}\label{lem:qEk-multi-p}
 Let $\prob_0,\ldots, \prob_{2\kElem}$ be distinct values in $(0, 1]$.  Then given the values $\rpoly_{G}^\kElem(\prob_i,\ldots, \prob_i)$ for $0\leq i\leq 2\kElem$, the number of $\kElem$-matchings in $G$ can be computed in $poly(\kElem)$ time.
 \end{Lemma}
 
+Before we prove the above Lemma, let us use it to prove~\Cref{thm:mult-p-hard-result}:
+\begin{proof}[Proof of Theorem~\ref{thm:mult-p-hard-result}]
+For the sake of contradiction, let us assume we can solve our problem in $f(\kElem)\cdot m^c$ time for some absolute constant $c$. Then given a graph $G$ we can compute the query polynomial $\rpoly_G^\kElem$ (in the obvious way) in $O(km)$ time. Then after we run our algorithm on $\rpoly_G^\kElem$, we get $\rpoly_{G}^\kElem(\prob_i,\ldots, \prob_i)$ for $0\leq i\leq 2\kElem$ in additional $f(\kElem)\cdot m^c$ time. \Cref{lem:qEk-multi-p} then computes the number of $k$-matchings in $G$ in $poly(\kElem)$ time. Thus, overall we have an algorithm for computing the number of $k$-matchings in time
+\begin{align*}
+ O(km) + f(\kElem)\cdot m^c + poly(\kElem)
+&\le \inparen{poly(\kElem) + f(\kElem)}\cdot m^{c+1} \\
+&\le \inparen{poly(\kElem) + f(\kElem)}\cdot n^{2c+2},
+\end{align*}
+which contradicts~\cref{thm:k-match-hard}.
+\end{proof}
+
+Finally, we are rerady to prove~\Cref{lem:qEk-multi-p}:
 \begin{proof}[Proof of ~\cref{lem:qEk-multi-p}]
 %It is trivial to see that one can readily expand the exponential expression by performing the $n^\kElem$ product operations, yielding the polynomial in the sum of products form of the lemma statement.  By definition $\rpoly_{G}^\kElem$ reduces all variable exponents greater than $1$ to $1$.  Thus, a monomial such as $X_i^\kElem X_j^\kElem$ is $X_iX_j$ in $\rpoly_{G}^\kElem$, and the value after substitution is $p_i\cdot p_j = p^2$.  Further, that the number of terms in the sum is no greater than $2\kElem + 1$, can be easily justified by the fact that each edge has two endpoints, and the most endpoints occur when we have $\kElem$ distinct edges (such a subgraph is also known as a $\kElem$-matching), with non-intersecting points, a case equivalent to $p^{2\kElem}$.
 We will show that $\rpoly_{G}^\kElem(\prob,\ldots, \prob) = \sum\limits_{i = 0}^{2\kElem} c_i \cdot \prob^i$.  First, since $\poly_G^\kElem(\vct{X})$ has $\kElem$ products of monomials of degree $2$, it follows that $\poly_G^\kElem(\vct{X})$ has degree $2\kElem$.  We can further write $\poly_{G}^{\kElem}(\vct{X})$ in its expanded SOP form,
@@ -80,14 +100,6 @@ Then, since we have $\kElem!$ duplicates of each distinct $\kElem$-matching, and
 
 \qed
 
-\begin{Corollary}\label{cor:mult-p-hard-result}
-Computing $\rpoly(\vct{X})$ given multiple distinct $\prob$ values is $\#W[1]$-hard.
-\end{Corollary}
-\begin{proof}[Proof of Corollary ~\ref{cor:mult-p-hard-result}]
-The proof follows by ~\cref{thm:k-match-hard} and ~\cref{lem:qEk-multi-p}.
-\end{proof}
-
-\qed