We study the maximum weight perfect $f$-factor problem on any general simple graph $G=(V,E,w)$ with positive integral edge weights $w$, and $n=|V|$, $m=|E|$. When we have a function $f:V\rightarrow \mathbb{N}_+$ on vertices, a perfect $f$-factor is a generalized matching so that every vertex $u$ is matched to $f(u)$ different edges. The previous best algorithms on this problem have running time $O(m f(V))$ [Gabow 2018] or $\tilde{O}(W(f(V))^{2.373}))$ [Gabow and Sankowski 2013], where $W$ is the maximum edge weight, and $f(V)=\sum_{u\in V}f(u)$. In this paper, we present a scaling algorithm for this problem with running time $\tilde{O}(mn^{2/3}\log W)$. Previously this bound is only known for bipartite graphs [Gabow and Tarjan 1989]. The running time of our algorithm is independent of $f(V)$, and consequently it first breaks the $\Omega(mn)$ barrier for large $f(V)$ even for the unweighted $f$-factor problem in general graphs.

中文翻译：

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一般图中加权 $f$-因子的缩放算法

我们在具有正整数边权重 $w$ 和 $n=|V|$, $m=| 的任何一般简单图 $G=(V,E,w)$ 上研究最大权重完美 $f$-因子问题。 E|$。当我们在顶点上有一个函数 $f:V\rightarrow \mathbb{N}_+$ 时，完美的 $f$-factor 是一个广义匹配，因此每个顶点 $u$ 都匹配到不同的 $f(u)$边缘。这个问题以前最好的算法有运行时间 $O(mf(V))$ [Gabow 2018] 或 $\tilde{O}(W(f(V))^{2.373}))$ [Gabow and Sankowski 2013] ]，其中 $W$ 是最大边权重，$f(V)=\sum_{u\in V}f(u)$。在本文中，我们针对这个问题提出了一个运行时间为 $\tilde{O}(mn^{2/3}\log W)$ 的缩放算法。以前，此界限仅适用于二部图 [Gabow 和 Tarjan 1989]。我们算法的运行时间与 $f(V)$ 无关，