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main.tex

@@ -43,7 +43,7 @@
 \maketitle
 \section{Connectivity interdiction}
 \begin{frame}{Connectivity interdiction}
-\begin{problem}[connectivity interdiction {[Zenklusen ORL'14]}]
+\begin{problem}[connectivity interdiction {[Zenklusen, ORL'14]}]
 Let $G=(V,E)$ be a graph with edge weights $w:E\to\Z_+$ and edge removal costs $c:E\to\Z_+$ and let $B\in \mathbb Z_+$ be the budget.
 
 Find $F\subset E$ with $w(F)\leq B$ s.t. the min-cut in $G-F$ is minimized.
@@ -74,10 +74,10 @@ Find the cut with minimum weight $w'$.
 \toprule
 unit cost $w(\cdot)=1$ & general case & random? & ref \\
 \midrule
-$O(m^2n^4\log n)$ & $O(m^{2+1/\e}n^4\log n)$ & $\times$  & [Zenklusen ORL'14] \\
-$O(mn^4\log^2 n)$ & $O(m^{1+1/\e}n^4\log^2 n)$ & $\checkmark$ & [Zenklusen ORL'14] \\
-$\tilde O(m+n^4 B)$ & $O(\frac{m^2n^4}{\e}\log(Bnw_{\max}))$ & $\times$ &   [Huang \etal{} IPCO'24]\\
-$\tilde O(mn^3\log w_{\max})$ & - & $\checkmark$ &   [Drange \etal{} AAAI'26]\\
+$O(m^2n^4\log n)$ & $O(m^{2+1/\e}n^4\log n)$ & $\times$  & [Zenklusen, ORL'14] \\
+$O(mn^4\log^2 n)$ & $O(m^{1+1/\e}n^4\log^2 n)$ & $\checkmark$ & [Zenklusen, ORL'14] \\
+$\tilde O(m+n^4 B)$ & $O(\frac{m^2n^4}{\e}\log(Bnw_{\max}))$ & $\times$ &   [Huang \etal{}, IPCO'24]\\
+$\tilde O(mn^3\log w_{\max})$ & - & $\checkmark$ &   [Drange \etal{}, AAAI'26]\\
 $\tilde O(m^2+n^3B)$ & $\tilde O(m^2+mn^3+n^3/\e)$ & $\times$ &  this work\\
 $\tilde O(m+n^3B)$ & $\tilde O(mn^3+n^3/\e)$ & $\checkmark$ &  this work\\
 \bottomrule
@@ -86,18 +86,18 @@ $\tilde O(m+n^3B)$ & $\tilde O(mn^3+n^3/\e)$ & $\checkmark$ &  this work\\
 \end{table}
 \end{frame}
 
-\begin{frame}{Method in [Huang \etal{} IPCO'24]}
+\begin{frame}{Method in [Huang \etal{}, IPCO'24]}
 
-\begin{problem}[\emph{normalized min-cut} {[Chalermsook \etal{} ICALP'22]}]
+\begin{problem}[\emph{normalized min-cut} {[Chalermsook \etal{}, ICALP'22]}]
 Given the same input as connectivity interdiction, find
 \[\argmin_{\text{cut $C$}, F\subset C} \frac{w(C-F)}{B-c(F)+1} \quad s.t. \quad 0\leq c(F)\leq B.\]
 \end{problem}
 
 \p
-First considered in [Chalermsook \etal{} ICALP'22] as a subproblem in MWU framework when solving minimum $k$-edge connected spanning subgraph (ECSS) problem.
+First considered in [Chalermsook \etal{}, ICALP'22] as a subproblem in MWU framework when solving minimum $k$-edge connected spanning subgraph (ECSS) problem.
 \end{frame}
 
-\begin{frame}{Method in [Huang \etal{} IPCO'24]}
+\begin{frame}{Method in [Huang \etal{}, IPCO'24]}
 Let $\tau$ be a $(1+\e)$-approximation to the normalized min-cut and let $C^*$ be the optimal $B$-free cut.
 \begin{lemma}
 $C^*$ is a $(2+2\e)$-approximate min-cut in $(G,w_\tau)$, 
@@ -171,18 +171,18 @@ P & & P & & P & & \textcolor{BrickRed}{\text{NP-Hard}}
 \]
 
 \p
-\begin{conjecture}[{[Geelen \etal{} Ann. Comb. 2015]}]
+\begin{conjecture}[{[Geelen \etal{}, Ann. Comb. 2015]}]
 For any proper minor-closed class $\mathcal M$ of binary matroids, there is a polynomial time algorithm for computing the girth of matroids in $\mathcal M$.
 \end{conjecture}
 \end{frame}
 
 \begin{frame}{Perturbed graphic matroid}
-\begin{theorem}[{[Geelen \etal{} Ann. Comb. 2015]}]
+\begin{theorem}[{[Geelen \etal{}, Ann. Comb. 2015]}]
 For any proper minor-closed class $\mathcal M$ of binary matroids, there exists two constants $k,t\in \Z_+$ such that, for each vertically $k$-connected matroid $M\in \mathcal M$, there exist matrices $A,P\in \F_2^{r\times n}$ such that $A$ is the incidence matrix of a graph, $\mathrm{rank}(P)\leq t$, and either $M$ or $M^*$ is isomorphic to $M(A+P)$.
 \end{theorem}
 
 \p
-\begin{theorem}[{[Geelen \& Kapadia Combinatorica'17]}]
+\begin{theorem}[{[Geelen \& Kapadia, Combinatorica'17]}]
 There are polynomial-time randomized algorithms for computing the girth and the cogirth of $M(A+P)$.
 \end{theorem}
 
@@ -215,13 +215,13 @@ $M(A)$ is a \emph{$(1,t)$-signed graft}.
 \begin{frame}{Previous works}
 Results on computing the cogirth of $(1,t)$-signed grafts:
 \begin{itemize}
-\item $O(r^5n)$ random algorithm [Geelen \& Kapadia Combinatorica'17]
-\item $n^{O(1)}$ deterministic algorithm when the $\set{s}$-indexed row is all 0 [Nägele \etal{} SODA'18]
+\item $O(r^5n)$ random algorithm [Geelen \& Kapadia, Combinatorica'17]
+\item $n^{O(1)}$ deterministic algorithm when the $\set{s}$-indexed row is all 0 [Nägele \etal{}, SODA'18]
 \end{itemize}
 \end{frame}
 
 \begin{frame}{Method}
-\begin{theorem}[{[Chekuri \etal{} SOSA'19]}]
+\begin{theorem}[{[Chekuri \etal{}, SOSA'19]}]
 Given a matroid $M$, let $\lambda(M)$ be its cogirth and let $\sigma(M)$ be the fractional base packing number.
 If there is some constant $c$ that $\frac{\lambda(M)}{\sigma(M)}<c$, then the cogirth of $M$ can be computed deterministically in polynomial number of calls to the independence oracle.
 \end{theorem}