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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{\ip}{IP\ref{IP}}
+\newcommand{\dual}{LP\ref{dual}}
+\newcommand{\metric}{LP\ref{metric}}
 \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}
 
 \bibliographystyle{plainnat}