Edge-cuts Optimized for Average Weight: a new alternative to Ford and Fulkerson

Let $G$ be a directed graph associated with a weight $w: E(G) \rightarrow R^+$\mathrm{.}Foranedge-cut$Q$of$G$,theaverageweightof$Q$ is denoted and defined as $w_{ave}(Q)=\frac{{\sum }_{e\in Q}w(e)}{\mathrm{\mid }Q\mathrm{\mid }}$. An edge-cut of optimal average weight is an edge-cut $Q$ such that $w_{ave}(Q)$ is maximum among all edge-cuts (or minimum, symmetrically). In this paper, a polynomial algorithm for this problem is proved for finding such an optimal edge-cut in a rooted tree, separating the root and the set of all leafs. This algorithm enables us to develop an automatic clustering method with more accurate detection of communities embedded in a hierarchy tree structure.