alg and complexity

algo.sty

@@ -0,0 +1,14 @@
+\def\begin@lg{\begin{minipage}{1in}\begin{tabbing}
+        \quad\=\qquad\=\qquad\=\qquad\=\qquad\=\qquad\=\qquad\=\kill}
+\def\end@lg{\end{tabbing}\end{minipage}}
+
+\newenvironment{algorithm}
+{\begin{tabular}{|l|}\hline\begin@lg}
+{\end@lg\\\hline\end{tabular}}
+
+\newenvironment{algo}
+{\begin{center}\begin{algorithm}}
+{\end{algorithm}\end{center}}
+
+\def\argmax{\operatornamewithlimits{arg\,max}}
+\def\argmin{\operatornamewithlimits{arg\,min}}

main.tex

@@ -1,6 +1,6 @@
 \documentclass[12pt]{article}
 \usepackage{chao}
-
+\usepackage{algo}
 
 \geometry{a4paper,margin=2cm}
 % \usepackage[breakable, theorems, skins]{tcolorbox}
@@ -345,6 +345,22 @@ We can also recover IPCO alg's 2-approximation.
 
 Then $\opt(IP)\in [L(\lambda^*),2L(\lambda^*)]$.
 
+\subsection{complexity}
+
+\begin{algo}
+\underbar{\textsc{$b$-free MinCut}}($G,w,c,b$):\\
+compute $\lambda^*$ using parametric search\\
+reweight $G$ with $w_\lambda$\\
+for each 2-approx mincut $C$ in $(G,w_\lambda)$:\\
+\quad run FPTAS for knapsack to compute $\min \set{w(C-F)| F\subset C, c(F)\leq b}$\\
+return the optimal $(C,F)$
+\end{algo}
+
+\paragraph{time for $\lambda^*$} $L(\lambda)-b\lambda$ is pwl concave. The number of segments is at most the number of lines which has a trivial upperbound of $2^m 2^m$. We need almost linear time to find the solution to a fixed $\lambda$. So parametric seach gives complexity $m^{1+o(1)} O(\log 4^m)$.
+
+\paragraph{time for the rest parts} Reweighting takes linear time.
+Finding 2-approx mincut takes $\tilde O(n^4)$. FPTAS for knapsack takes $O(\frac{1}{\e}m^2)$. The total complexity is $O(\frac{1}{\e}m^2n^4)$.
+
 \bibliographystyle{plain}
 \bibliography{ref}