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LP relations

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Yu Cong
2025-05-14 00:18:57 +08:00
parent 8b8dc60d34
commit e68ec80449
2 changed files with 3 additions and 3 deletions
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@@ -76,7 +76,7 @@ For graphs with constant genus, \citep{lee_genus_2010} gives a $O(\sqrt{\log g})
\begin{equation}\label{IP}
\begin{aligned}
\min& & \frac{\sum_e c_e x_e}{\sum_{i} D_i y_i}& & &\\
s.t.& & \sum_{e\in p} x_e&\geq y_i & &\forall \mathcal{P}_{s_i,t_i}, \forall i\\
s.t.& & \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}