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\sum_{e\in p} x_e&\geq y_i & &\forall p\in \mathcal{P}_{s_i,t_i}, \forall i\\ & & x_e,y_i&\in \{0,1\} \end{aligned} \end{equation} @@ -86,7 +86,7 @@ s.t.& & \sum_{e\in p} x_e&\geq y_i & &\forall \mathcal{P} \begin{aligned} \min& & \sum_e c_e x_e& & &\\ s.t.& & \sum_i D_iy_i&=1 & &\\ - & & \sum_{e\in p} x_e&\geq y_i & &\forall \mathcal{P}_{s_i,t_i}, \forall i\\ + & & \sum_{e\in p} x_e&\geq y_i & &\forall p\in \mathcal{P}_{s_i,t_i}, \forall i\\ & & x_e,y_i&>0 \end{aligned} \end{equation} @@ -120,7 +120,7 @@ s.t.& & \sum_i D_i d(s_i,t_i)&=1 & &\\ \begin{enumerate} \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, the optimal solution to \lp{} also induces a feasible metric. Consider a solution $x_e,y_i$ to \lp{}. Let $d$ be the shortest path metric on $V$ using edge length $x_e$. It suffices to show that $y_i$ is the shortest path distance fron $s_i$ to $t_i$. Suppose $y_i\leq d(s_i,t_i)$... +\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, the optimal solution to \lp{} also induces a feasible metric. Consider a solution $x_e,y_i$ to \lp{}. Let $d_x$ be the shortest path metric on $V$ using edge length $x_e$. It suffices to show that $y_i=d_x(s_i,t_i)$. This can be seen from a reformulation of \lp{}. The constraint $\sum_i D_i y_i=1$ can be removed and the objective becomes $\sum_e c_e x_e / \sum_i D_i y_i$. This reformulation does not change the optimal solution. Now suppose in the optimal solution to \lp{} there is some $y_i$ which is strictly smaller than $d_x(s_i,t_i)$. Then the denominator $\sum_i D_i y_i$ in the objective of our reformulation can be larger, contradicting to the optimality of solution $x_e,y_i$. \end{enumerate} \bibliographystyle{plainnat}