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Yu Cong 2025-05-13 17:33:37 +08:00
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commit 9ccff75dea
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@ -115,8 +115,12 @@ s.t.& & \sum_i D_i d(s_i,t_i)&=1 & &\\
\newcommand{\lp}{LP\ref{LP}} \newcommand{\lp}{LP\ref{LP}}
\newcommand{\ip}{IP\ref{IP}} \newcommand{\ip}{IP\ref{IP}}
\newcommand{\dual}{LP\ref{dual}}
\newcommand{\metric}{LP\ref{metric}}
\begin{enumerate} \begin{enumerate}
\item \ip $\geq$ \lp. Given any feasible solution to \ip, we can \item \ip{} $\geq$ \lp{}. Given any feasible solution to \ip{}, we can scale all $x_e$ and $y_i$ simultaneously with factor $1/\sum_i D_i y_i$. The scaled solution is feasible for \lp{} and gets the same objective value.
\item \lp{} $=$ \dual{}. by duality.
\item \metric{} $=$ \lp{}. It is easy to see \metric{} $\geq$ \lp{} since any feasible metric to \metric{} induces a feasible solution to \lp{}. In fact, any solution to \lp{} also induces a feasible metric.
\end{enumerate} \end{enumerate}
\bibliographystyle{plainnat} \bibliographystyle{plainnat}