Update main.tex

main.tex

@@ -294,24 +294,12 @@ Given a hypergraph with $<k$ components, we can add at most $O(|V|)$ dummy hyper
 
 \autoref{thm:kcut} reduces minimum $k$-cut problem on hypergraphs to minimum $k$-cocycle on hypergraphic matroids. Now we try to apply the ideal base packing framework on hypergraphic matroids.
 
-\begin{lemma}
+\begin{conjecture}
 	Let $H$ be a parititon-connected hypergraph and let $M$ be the hypergraphic matroid on $H$.
-	There is a distribution $\mathcal D$ on $k$-cocycles such that for any base in the ideal base packing, the expected size of their intersection is at most $2k$.
+	There is a distribution $\mathcal D$ on $k$-cocycles such that for any base in the ideal base packing, the expected size of their intersection is at most $\max\set{\gamma,2k}$.
-\end{lemma}
+\end{conjecture}
-\begin{proof}
+This is not true. Consider a hypergraph with $\gamma\in[h-k-1,h+1]$ and an ideal base that every inter-component hyperedge has size $\gamma$. One can see that $e_B(\mathcal Q)$ has an upperbound $h$.
-	Let flat $F\in \argmin_{F'\subset E}\frac{|E-F'|}{r(E)-r(F')}$ be the optimal set for matroid strength.
+I think in general one cannot do better than $\gamma k$.
-	Each flat in $M$ induces a vertex partition $\mathcal P_F$ by its forest representation.
-	One can see that $|\mathcal P|=r(E)-r(F)+1$. Now we randomly pick $r(E)-r(F)+1-k$ parts and merge them into a new component.
-	Let $\mathcal P'$ be the resulting partition and let $B$ be any ideal base of $M$.
-	We have $e_B(\mathcal P)=r(E)-r(F)$ since $B$ is ideal. Now we count the expected number of $e_B(\mathcal P')$,
-	\[
-    	\begin{aligned}
-    		\E_{\mathcal D}[e_B(\mathcal P')] &\leq e_B(\mathcal P)\Pr[\text{a hyperedge is not contained in the new part}] \\
-					&\leq (r(E)-r(F))\cdot ???
-        \end{aligned}
-	\]
-	Note that the first inequality is because $\mathcal P'$ may induce a $\geq k$-cocycle, the second one ...
-\end{proof}
 
 \section{Greedy base packing}