小札 Maximum Weight Closure of a Graph

1. Introduction

   Define a closure of a directed graph \(G=(V,E)\) as an induced set of vertexes of \(G\), where the out-tended edges in \(E\) start at some vertex in the closure of \(G\) and end at some vertex in the closure, too. More formally, closure is defined as a set of vertexes \(V'\subseteq V\), such that \(\forall u\in V',s.t.\forall\left<u,v\right>\in E,v\in V'\).

   For the Maximum Weight Closure of a Graph, we define it as, with starting out endowing vertexes in \(V'\) costs called \(w_u\), the maximum value of \(\sum\limits_{u\in V'}w_u\).

2. Construction of Algorithm

   Add a source \(s\) and a sink \(t\) to the original graph, endow the origin edges in \(E\) with capacities that \(+\infty\), and connect \(s\) with points with positive costs with the points' costs as capacities on them while \(t\) connects to those with negative costs with the minus points' costs as capacities on them, where \(+\infty\) is defined as any number greater or equal to \(\sum |w_i|\).

   More formally, as explained & illustrated:

\[V_N=V\cup\{s,t\} \\ E_N=E\cup\left\{\left<s,v\right>\mid v\in V,w_v\geqslant0\right\}\cup\left\{\left<v,t\right>\mid v\in V,w_v<0\right\} \\ \begin{cases} c(u,v)=+\infty,\left<u,v\right>\in E \\ \displaystyle c(s,v)=w_v,w_v\geqslant0 \\ \displaystyle c(v,t)=-w_v,w_v<0 \end{cases} \]

   Which make it easy to understand the answer would be \(\text{min-cut}(G_N=\{V_N,E_N\})\).

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[1] Referenced to 最小割模型在信息学竞赛中的应用 by 胡伯涛 Amber.