diff --git a/.gitea/workflows/compile.yml b/.gitea/workflows/compile.yml index b17f707..7b0c4fb 100644 --- a/.gitea/workflows/compile.yml +++ b/.gitea/workflows/compile.yml @@ -17,7 +17,6 @@ jobs: - uses: http://localhost:3000/sxlxc/gitea-release-action@v1 with: body: '' - prerelease: true name: PDF token: ${{ secrets.RELEASE_TOKEN }} tag_name: latest diff --git a/.zed/tasks.json b/.zed/tasks.json new file mode 100644 index 0000000..7500c78 --- /dev/null +++ b/.zed/tasks.json @@ -0,0 +1,25 @@ +// Static tasks configuration. +// +[ + { + "label": "forward_search", + "command": "/Applications/Skim.app/Contents/SharedSupport/displayline -r -z -b $ZED_ROW $ZED_DIRNAME/$ZED_STEM.pdf", + "allow_concurrent_runs": false, + "reveal": "never", + "hide": "always" + }, + { + "label": "pdflatex_view", + "command": "cd \"$ZED_DIRNAME\" && pdflatex -shell-escape -synctex=-1 \"$ZED_STEM\" && /Applications/Skim.app/Contents/SharedSupport/displayline -r -z -b $ZED_ROW \"$ZED_STEM\".pdf", + "allow_concurrent_runs": false, + "reveal": "no_focus", + "hide": "on_success" + }, + { + "label": "latexmk_view", + "command": "cd \"$ZED_DIRNAME\" && latexmk -pdf \"$ZED_STEM\" && /Applications/Skim.app/Contents/SharedSupport/displayline -r -z -b $ZED_ROW \"$ZED_STEM\".pdf", + "allow_concurrent_runs": false, + "reveal": "no_focus", + "hide": "on_success" + } +] \ No newline at end of file diff --git a/main.tex b/main.tex index a0133df..8b21833 100644 --- a/main.tex +++ b/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, -% bottom=0pt, -% 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}