SIAM Journal on Computing, Volume 50, Issue 2, Page 555-601, January 2021.
For an undirected graph or multigraph $G=(V,E)$ and a function $f:V\to \mathbb{Z_+}$, an $f$-factor is a subgraph whose degree function is $f$. For integral edge weights of maximum magnitude $W$ our algorithm finds a maximum weight $f$-factor in time $\tilde{O}(Wf(V)^{\omega})$, where $f(V)=\sum_{v\in V} f(v)$ and $\omega$ is the exponent of matrix multiplication. The algorithm is randomized and has two versions. For worst-case time the algorithm is correct with high probability. For expected time the algorithm is Las Vegas. The algorithm is based on a detailed analysis of the structure of the optimum blossoms. A special case gives a representation for single-source shortest-paths in conservative undirected graphs, generalizing the standard shortest-path tree to a “tree of cycles”. The representation can be constructed by a randomized algorithm with the same time bound as above, or deterministically by an algorithm for maximum weight matching, achieving time $O(n(m + n \log n))$ or $O(\sqrt{n }\ m \log (nW))$.

中文翻译：

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加权匹配泛化算法 II：f 因子和最短路径的特殊情况

SIAM Journal on Computing，第 50 卷，第 2 期，第 555-601 页，2021 年 1 月。
对于无向图或多重图 $G=(V,E)$ 和函数 $f:V\to \mathbb{Z_+}$，$f$-factor 是度函数为 $f$ 的子图。对于最大幅度 $W$ 的积分边权重，我们的算法在时间 $\tilde{O}(Wf(V)^{\omega})$ 中找到最大权重 $f$-factor，其中 $f(V)=\ sum_{v\in V} f(v)$ 和 $\omega$ 是矩阵乘法的指数。该算法是随机的，有两个版本。在最坏情况下，算法正确的概率很高。对于预期时间，算法是拉斯维加斯。该算法基于对最佳花朵结构的详细分析。一个特例给出了保守无向图中单源最短路径的表示，将标准最短路径树推广为“循环树”。