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main.tex

@@ -1,29 +1,10 @@
-\documentclass[12pt]{article}
+\documentclass[a4paper,11pt]{article}
 \usepackage{chao}
 \usepackage{algo}
 
-\geometry{a4paper,margin=2cm}
-% \usepackage[breakable, theorems, skins]{tcolorbox}
-% \tcbset{enhanced}
-% \DeclareRobustCommand{\note}[2][blue]{%
-% \begin{tcolorbox}[
-%         breakable,
-%         left=0pt,
-%         right=0pt,
-%         top=0pt,
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-%         colback=white,
-%         colframe=#1,
-%         width=\dimexpr\textwidth\relax, 
-%         enlarge left by=0mm,
-%         boxsep=5pt,
-%         arc=0pt,outer arc=0pt,
-%         ]
-%         #2
-% \end{tcolorbox}
-% }
+\geometry{margin=2cm}
 
-\title{Connectivity Interdiction Notes}
+\title{Faster FPTAS for Connectivity Interdiction}
 \author{}
 \date{}
 
@@ -40,7 +21,7 @@
 Note that $\mathcal F$ is usually not explicitly given.
 
 \begin{problem}[Normalized knapsack]\label{nknap}
-    Given the same input as \autoref{bfreeknap}, find $\min \limits_{X\in \mathcal F, F\subset E} \frac{w(X\setminus F)}{B-c(F)}$ such that $c(F)\leq b$.
+    Given the same input as \autoref{bfreeknap}, find $\min \limits_{X\in \mathcal F, F\subset E} \frac{w(X\setminus F)}{B-c(F)}\; s.t.\; c(F)\leq b$.
 \end{problem}
 In \cite{vygen_fptas_2024} the normalized min-cut problem use $B=b+1$. Here we use any integer $B>b$ and see how their method works.
 
@@ -364,8 +345,11 @@ It follows directly from the preceding lemma that $\lambda^*$ can be computed in
 
 Reweighting the graph takes linear time.
 Finding $<2$-approx mincut takes $\tilde O(n^3)$. 
-An $1+\e$ approximate solution to knapsack can be found in time $\tilde O(m+\frac{1}{\e^2})$ \cite{10.1145/3618260.3649730}.
-The total complexity is $\tilde O(mn^3+\frac{n^3}{\e^2})$.
+The total complexity is $\tilde O(m^2+n^3 T)$, where $T$ is the running time of FPTAS for 0-1 knapsack.\footnote{An $1+\e$ approximate solution to knapsack can be found in time $\tilde O(m+\frac{1}{\e^2})$ \cite{10.1145/3618260.3649730}.}
+
+\subsection{Avoid the parametric search}
+The $m^2$ term in the complexity is hard to improve.
+So we consider using binary search to find an $(1+\e)$-approximate $\lambda^*$.
 
 \bibliographystyle{plain}
 \bibliography{ref}