kcut and cocycle equivalence.

main.tex

@@ -184,10 +184,10 @@ For general matroids, we want to show the following.
 	If \autoref{conj:dist} is true for matroid $M$, then one can compute minimum $k$-cocycle in polynomial time.
 \end{theorem}
 \begin{proof}
-    It follows from \autoref{conj:dist} that the minimum $k$-cocycle $C^*_k$ shares at most $h=O(1)$ elements with some base in the ideal base packing. The number of bases we need in the ideal base packing is polynomial (see next subsection).
-    We enumerate all bases in this set and for each base $B$ enumerate all subsets with size in range $[r-h,r-k]$.
-    Such a subset $I$ must be indenpendent and be contained in the flat $\overline{C^*_k}$. Then for such subset we further enumerate another subset $X$ such that $I\cup X$ is a rank-$r-k$ independent set. Thus, $\overline{\cl(I\cup X)}$ is a $k$-cocycle and we take the minimum one among all enumerations.
-    One can check that each part of the enumeration can be done in polynomial time.
+	It follows from \autoref{conj:dist} that the minimum $k$-cocycle $C^*_k$ shares at most $h=O(1)$ elements with some base in the ideal base packing. The number of bases we need in the ideal base packing is polynomial (see next subsection).
+	We enumerate all bases in this set and for each base $B$ enumerate all subsets with size in range $[r-h,r-k]$.
+	Such a subset $I$ must be indenpendent and be contained in the flat $\overline{C^*_k}$. Then for such subset we further enumerate another subset $X$ such that $I\cup X$ is a rank-$r-k$ independent set. Thus, $\overline{\cl(I\cup X)}$ is a $k$-cocycle and we take the minimum one among all enumerations.
+	One can check that each part of the enumeration can be done in polynomial time.
 \end{proof}
 
 Certainly \autoref{conj:dist} does not hold on any matroid.
@@ -261,8 +261,38 @@ Consider an edge $(u,v)$ and one round.
 Then the probability that edge $(u,v)$ survives in the end is at most $\prod_{k=3}^n \frac{(n+1)(n-2)}{n(n-1)}=\frac{n+1}{3(n-1)}$.
 Then the number of remaining edges in any spanning tree is at most $(n+1)/3$.
 
-\subsection{Hypergraphic matroid}
-Hypergraphic matroids are not closed under contraction (cf. Tamás Király's thesis).
+\subsection{Hypergraphic matroid cocycle and hypergraph k-cut}
+
+Let $H=(V,E)$ be a hypergraph and let $M=(E,\mathcal I)$ be a hypergraphic matroid on the hyperedge set $E$. A subset $I$ of hyperedges is independent in $M$ if the union of any subset $I'\subseteq I$ has at least $|I'|+1$ vertices.
+One can see that hypergraphic matroid is a count matroid induced by $|E[V]|\leq |V|-1$.
+The rank of a hypergraphic matroid is given by $\min\set{|V|-|\mathcal P|+e_H(\mathcal P):\text{$\mathcal P$ is a partition of $V$}}$, where $e_H(\mathcal P)$ is the number of inter-component hyperedges.
+Hypergraphic matroids are not closed under contraction.\footnote{see Tamás Király's thesis \url{https://tkiraly.web.elte.hu/pub/tkiraly_thesis.pdf}}
+
+Let $\mathcal P=\set{V_1,\ldots,V_k}$ be a non-empty $k$-partition of $V$.
+Then the $k$-cut of a hypergraph is the set of hyperedges intersecting at least 2 parts of $\mathcal P$.
+We say $H$ is partition connected if $e_H(\mathcal P)\geq |\mathcal P|-1$ for any partition $\mathcal P$.
+It follows from the rank function that $\rank(M(H))=|V|-1$ iff $H$ is partition-connected.
+Given a hypergraph with $<k$ components, we can add at most $O(|V|)$ dummy hyperedges with zero cost to make it partition connected. We always assume the input hypergraph is partition connected since adding zero-cost hyperedges does not affect the $k$-cut cost.
+
+\begin{theorem}\label{thm:kcut}
+	Let $H$ be a partition-connected hypergraph and let $M$ be the hypergraphic matroid on $H$.
+	The minimum $k$-cut of $H$ is the same as the minimum $(k-1)$-cocycle of $M$.
+\end{theorem}
+\begin{proof}
+	First consider any $k$-cut $\delta(\mathcal P)$ induced by partition $\mathcal P=\set{V_1,\ldots,V_k}$. Let $X=E\setminus \delta(\mathcal P)$. The rank of $X$ is
+	\[
+	    \rank(X)= \min_{\mathcal P'} \set{|V|-|\mathcal P'|+e_X(\mathcal P')}\leq |V|-|\mathcal P|+e_X(\mathcal P)\leq |V|-k.
+	\]
+	So $\delta(\mathcal P)$ must contain a $(k-1)$-cocycle.
+	
+	On the other hand, let $C$ be any $(k-1)$-cocycle of $M$. Let $F$ be the complement of $C$.
+	$F$ is a flat of $M$ with rank $r-k+1$.
+	Take any maximal independent set $I_F\subset F$. It is known that one can idenfity two vertices as a graph edge in each hyperedge of an independent set, such that the resulting graph is a forest. Let $T=(V[H],E)$ be such a forest of $I_F$. Note that we include isolated vertices.
+	So the number of components of $I_F$ is exactly $k$. Let $\mathcal Q$ be the partition of $V[H]$ into components of $T$.
+	It follows from the definition that $\delta(\mathcal Q)$ is a $k$-cut.
+\end{proof}
+
+\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.
 
 \section{Greedy base packing}