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Yu Cong
2025-09-29 10:41:48 +08:00
parent 763e86f307
commit a276cd658f
2 changed files with 4 additions and 3 deletions
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chao.sty
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\RequirePackage{microtype} \RequirePackage{microtype}
\RequirePackage[mathletters]{ucs} % allow Unicode in .tex file \RequirePackage[mathletters]{ucs} % allow Unicode in .tex file
\RequirePackage[utf8]{inputenc} \RequirePackage[utf8]{inputenc}
\RequirePackage[charter]{mathdesign} % change fonts \RequirePackage[charter]{mathdesign} % change fonts
\RequirePackage{XCharter} % change fonts
\RequirePackage{berasans, beramono} \RequirePackage{berasans, beramono}
\RequirePackage{eucal} \RequirePackage{eucal}
\RequirePackage[nocompress]{cite} % other convenient stuff \RequirePackage[nocompress]{cite} % other convenient stuff

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main.tex
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@@ -196,7 +196,7 @@ We have shown that the budget $B$ in normalized min-cut does not really matter a
% Assume that the graph $G$ is connected. % Assume that the graph $G$ is connected.
For a graph $G=(V,E)$ with edge capacity $c:V\to \Z_+$, the strength $\sigma(G)$ is defined as $\sigma(G)=\min_{\Pi}\frac{c(\delta(\Pi))}{|\Pi|-1}$, where $\Pi$ is any partition of $V$, $|\Pi|$ is the number of parts in the partition and $\delta(\Pi)$ is the set of edges between parts. Note that an alternative formulation of strength (using graphic matroid rank function) is $\sigma(G)=\min_{F\subset E} \frac{c(E-F)}{r(E)-r(F)}$, which in general is the fractional optimum of matroid base packing. For a graph $G=(V,E)$ with edge capacity $c:V\to \Z_+$, the strength $\sigma(G)$ is defined as $\sigma(G)=\min_{\Pi}\frac{c(\delta(\Pi))}{|\Pi|-1}$, where $\Pi$ is any partition of $V$, $|\Pi|$ is the number of parts in the partition and $\delta(\Pi)$ is the set of edges between parts. Note that an alternative formulation of strength (using graphic matroid rank function) is $\sigma(G)=\min_{F\subset E} \frac{c(E-F)}{r(E)-r(F)}$, which in general is the fractional optimum of matroid base packing.
The principal sequence of partitions of $G$ is a piecewise linear concave curve $L(\lambda)=\min_\Pi c(\delta(\Pi))-\lambda |\Pi|$. (alternatively, $L(\lambda)=\min_{F\in E}c(E\setminus F)-\lambda(r(E)-r(F)+1)$) Cunningham used principal partition to computed graph strength\cite{cunningham_optimal_1985}. There is a list of good properties mentioned in \cite[Section 6]{chekuri_lp_2020}(implicated stated in \cite{cunningham_optimal_1985}). The principal sequence of partitions of $G$ is a pwl concave curve $L(\lambda)= \min_\Pi c(\delta(\Pi))-\lambda |\Pi|$. (alternatively, $L(\lambda)=\min_{F\in E}c(E\setminus F)-\lambda(r(E)-r(F)+1)$) Cunningham used principal partition to computed graph strength\cite{cunningham_optimal_1985}. There is a list of good properties mentioned in \cite[Section 6]{chekuri_lp_2020}(implicated stated in \cite{cunningham_optimal_1985}).
\begin{enumerate} \begin{enumerate}
\item We can assume $G$ is connected and deal with the smallest strength component. One can see this by fractional base packing on the direct sum of matroids. Note that on disconnected graphs we should use the edge set definition instead of partitions. \item We can assume $G$ is connected and deal with the smallest strength component. One can see this by fractional base packing on the direct sum of matroids. Note that on disconnected graphs we should use the edge set definition instead of partitions.
\item $L(\lambda)$ is piecewise linear concave since it is the lower envelope of some line arrangement. \item $L(\lambda)$ is piecewise linear concave since it is the lower envelope of some line arrangement.
@@ -356,7 +356,8 @@ for each 2-approx mincut $C$ in $(G,w_\lambda)$:\\
return the optimal $(C,F)$ return the optimal $(C,F)$
\end{algo} \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 $\lambda^*$} $L(\lambda)-b\lambda$ is pwl concave. The number of segments is at most $3^m$. We need almost linear time to find the solution to a fixed $\lambda$. So parametric seach gives complexity $m^{1+o(1)} O(\log 3^m)$.
\note{need to check this}
\paragraph{time for the rest parts} Reweighting takes linear time. \paragraph{time for the rest parts} Reweighting takes linear time.
Finding $<2$-approx mincut takes $\tilde O(n^3)$. FPTAS for knapsack takes $O(\frac{1}{\e}m^2)$. The total complexity is $O(\frac{1}{\e}m^2n^3)$. Finding $<2$-approx mincut takes $\tilde O(n^3)$. FPTAS for knapsack takes $O(\frac{1}{\e}m^2)$. The total complexity is $O(\frac{1}{\e}m^2n^3)$.