Added hardness result for k-matchings

aaron.bib

@@ -168,3 +168,24 @@ series = {PODS '07}
 }
 
 
+@inproceedings{k-match,
+  author    = {Radu Curticapean},
+  editor    = {Fedor V. Fomin and
+               Rusins Freivalds and
+               Marta Z. Kwiatkowska and
+               David Peleg},
+  title     = {Counting Matchings of Size k Is W[1]-Hard},
+  booktitle = {Automata, Languages, and Programming - 40th International Colloquium,
+               {ICALP} 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part {I}},
+  series    = {Lecture Notes in Computer Science},
+  volume    = {7965},
+  pages     = {352--363},
+  publisher = {Springer},
+  year      = {2013},
+  url       = {https://doi.org/10.1007/978-3-642-39206-1\_30},
+  doi       = {10.1007/978-3-642-39206-1\_30},
+  timestamp = {Tue, 14 May 2019 10:00:44 +0200},
+  biburl    = {https://dblp.org/rec/conf/icalp/Curticapean13.bib},
+  bibsource = {dblp computer science bibliography, https://dblp.org}
+}
+

mult_distinct_p.tex

@@ -2,7 +2,14 @@
 
 \subsection{Multiple Distinct $\prob$ Values}
 
-We would like to argue for a compressed version of $\poly(\vct{w})$, in general $\expct_{\vct{w}}\pbox{\poly(\vct{w})}$ cannot be computed in linear time.
+We would like to argue for a compressed version of $\poly(\vct{w})$, in general $\expct_{\vct{w}}\pbox{\poly(\vct{w})}$ cannot be computed in linear time. 
+\AR{Added the hardness result below.}
+The hardness result is based on the following hardness result:
+\begin{theorem}[\cite{k-match}]
+\label{thm:k-match-hard}
+Given a positive integer $k$ and  an undirected graph $G$ with no self-loops of parallel edges, couting the number of $k$-matchings in $G$ is $\#W[1]$-hard.
+\end{theorem}
+The above result means that we cannot hope to count the number of $k$-matchings in $G=(V,E)$ in time $f(k)\cdot |V|^{O(1)}$ for any function $f$. In fact, all known algorithms to solve this problem takes time $|V|^{\Omega(k)}$.
 
 To this end, consider the following graph $G(V, E)$, where $|E| = \numedge$, $|V| = \numvar$, and $i, j \in [\numvar]$.