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notes.tex

@@ -188,12 +188,12 @@ $L(\lambda^*)+\lambda^*b<2L(\lambda^*)$.
 \begin{figure}  
 \begin{algo}
 \underbar{\textsc{Linear-Minimization-Interdiction}}$(E,\mathcal F,w,c,b)$:\\
-compute a maximizer $\lambda^*$ of $\Phi(\lambda)=L(\lambda)-\lambda b$\\
-compute the truncated weight $w_{\lambda^*}$\\
-enumerate every $S\in\mathcal F$ with $w_{\lambda^*}(S)<2L(\lambda^*)$\\
-for each enumerated $S$:\\
+\;compute a maximizer $\lambda^*$ of $\Phi(\lambda)=L(\lambda)-\lambda b$\\
+\;compute the truncated weight $w_{\lambda^*}$\\
+\;enumerate every $S\in\mathcal F$ with $w_{\lambda^*}(S)<2L(\lambda^*)$\\
+\;for each enumerated $S$:\\
 \;\; compute $g_b(S)=\min\{w(S\setminus R):R\subseteq S,\ c(R)\leq b\}$\\
-return the pair $(S,R)$ with minimum value
+\;return the pair $(S,R)$ with minimum value
 \end{algo}
 \caption{Template for solving interdiction version of linear minimization problem.}
 \label{fig:alg}
@@ -236,7 +236,7 @@ $(G,w_\lambda)$.  Using the deterministic almost-linear time min-cut algorithm
 of Li \cite{Li_2021}, this takes $\tilde O(m)$ time.  Therefore,
 \autoref{lem:para} gives a $\tilde O(m^2)$-time algorithm for computing $\lambda^*$.
 If randomization is allowed, we can find $\lambda^*$ in near linear time.
-Min-cut can be computed with high probability in $O(m\log^2 n)$ work and $O(\log^3 n)$ depth \cite{Anderson_Blelloch_2021}. Brent's law \cite{Gustafson_2011} shows that the running time is $T(n,P)=O(\frac{m\log^2 n}{P}+\log^3 n)$ with $P$ processors. Setting $P=O(m)$ gives a randomized $\tilde O(m)$-time algorithm for $\lambda^*$.
+Min-cut can be computed with high probability in $O(m\log^2 n)$ work and $O(\log^3 n)$ depth \cite{Anderson_Blelloch_2021}. Brent's law \cite{Gustafson_2011} shows that the running time is $T(n,P)=O(\frac{m\log^2 n}{P}+\log^3 n)$ with $P$ processors. Setting $P=O(m)$ and plugging the parallel algorithm into \autoref{lem:para} give a randomized $\tilde O(m)$-time algorithm for $\lambda^*$.
 
 Let $L^*=L(\lambda^*)$.  By \autoref{thm:two-approx-witness}, the optimal
 interdiction cut $C^*$ satisfies
@@ -247,8 +247,8 @@ Thus $C^*$ is one of the strict $2$-approximate min-cuts in
 $(G,w_{\lambda^*})$.  Karger \cite{Karger2000} showed that the number of
 $\alpha$-approximate min-cuts is $O(n^{\floor{2\alpha}})$.  Hence the number of
 cuts with value strictly smaller than $2L^*$ is $O(n^3)$, and they can be
-enumerated in $O(n^3)$ time\footnote{We note that only succinct representation of cuts (with respect to Karger's tree packing) are needed and the cut edges can be recovered while solving the knapsack. However, it is not known if one can deterministically enumerate all $\alpha$-approximate min-cuts in time $O(n^{\floor{2\alpha}})$.}\cite{Karger2000}.
-To sum up, our framework gives a FPRAS for connectivity interdiction with running time $\tilde O(m+n^3(m+\frac{1}{\e^2}))$.
+enumerated in randomized $O(n^3)$ time\footnote{We note that only succinct representation of cuts (with respect to Karger's tree packing) are needed and the cut edges can be recovered while solving the knapsack. However, it is not known if one can deterministically enumerate all $\alpha$-approximate min-cuts in time $O(n^{\floor{2\alpha}})$.}\cite{Karger2000}.
+To sum up, our framework gives an FPRAS for connectivity interdiction with running time $\tilde O(m+n^3(m+\frac{1}{\e^2}))$.
 
 Compared with \cite{huang_fptas_2024}, our analyais is based on LP methods and is more intuitive. An algorithmic improvement is that we use parametric search thus avoiding the enumeration of $\lambda^*$ (the optimum of a normalized min-cut problem in \cite{huang_fptas_2024}).