From fdd6afa5f93c49cebb6e069f13cbbe1212c20c8a Mon Sep 17 00:00:00 2001 From: Tommy Chiang Date: Sat, 7 Oct 2023 21:29:44 +0800 Subject: [PATCH] Fix typo --- codes/Matching_and_Flow/FlowModels.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/codes/Matching_and_Flow/FlowModels.tex b/codes/Matching_and_Flow/FlowModels.tex index aff3c988..f2daa31d 100644 --- a/codes/Matching_and_Flow/FlowModels.tex +++ b/codes/Matching_and_Flow/FlowModels.tex @@ -83,7 +83,7 @@ \item $\forall X \subseteq Y \subseteq V$, $x\notin Y$, $f(X\cup\{x\})-f(X)\geq f(Y\cup\{x\})-f(Y)$. \end{itemize} \item To minimize $\sum_i \theta_i (x_i) + \sum_{i< j} \phi_{ij} (x_i, x_j) + \sum_{i< j< k} \psi_{ijk} (x_i, x_j, x_k)$ - \item If $\theta_i(1)\geq\theta_i(0)$, add edge ($S$, $i$, $\theta_i(1)-\theta_i(0)$) and $\theat_i(0)$ to answer; otherwise, ($i$, $T$, $\theta_i(0)-\theta_i(1)$) and $\theta_i(1)$. + \item If $\theta_i(1)\geq\theta_i(0)$, add edge ($S$, $i$, $\theta_i(1)-\theta_i(0)$) and $\theta_i(0)$ to answer; otherwise, ($i$, $T$, $\theta_i(0)-\theta_i(1)$) and $\theta_i(1)$. \item Add edges ($i$, $j$, $\phi_{ij}(0,1)+\phi_{ij}(1,0)-\phi_{ij}(0,0)-\phi_{ij}(1,1)$). \item Denote $x_{ijk}$ as helper nodes. Let $P=\psi_{ijk}(0,0,0)+\psi_{ijk}(0,1,1)+\psi_{ijk}(1,0,1)+\psi_{ijk}(1,1,0)-\psi_{ijk}(0,0,1)-\psi_{ijk}(0,1,0)-\psi_{ijk}(1,0,0)-\psi_{ijk}(1,1,1)$. Add $-P$ to answer. If $P \geq 0$, add edges ($i$, $x_{ijk}$, $P$), ($j$,$x_{ijk}$,$P$), ($k$,$x_{ijk}$,$P$), ($x_{ijk}$, $T$, $P$); otherwise ($x_{ijk}$,$i$,$-P$), ($x_{ijk}$,$j$,$-P$), ($x_{ijk}$,$k$,$-P$), ($S$, $x_{ijk}$, $-P$). \item The minimum cut of this graph will be the the minimum value of the function above.