In 1961, Gomory and Hu showed that the max-flow values of all $n\choose 2$ pairs of vertices in an undirected graph can be computed using only $n-1$ calls to any max-flow algorithm, and succinctly represented them in a tree (called the Gomory-Hu tree later). Even assuming a linear-time max-flow algorithm, this yields a running time of $O(mn)$ for this problem; with current max-flow algorithms, the running time is $\tilde{O}(mn + n^{5/2})$. We break this 60-year old barrier by giving an $\tilde{O}(n^{2})$-time algorithm for the Gomory-Hu tree problem, already with current max-flow algorithms. For unweighted graphs, our techniques show equivalence (up to poly-logarithmic factors in running time) between Gomory-Hu tree (i.e., all-pairs max-flow values) and a single-pair max-flow.
翻译:1961年, Gomory 和 Hu 显示, 一个未定向图表中所有 2 美元双脊椎的最大流量值只能使用 $- 1 来计算, 并且可以在树上( 以后称为 Gomory- Hu 树 ) 简洁地代表它们。 即使假设一个线性时间最大流量算法, 这也会产生一个运行时间为$O( mn) 的问题; 使用当前最大流量算法, 运行时间为$\ tilde{ O}( mn + n ⁇ 5/2} $。 我们通过给 Gomory- Hu 树问题提供 $\ tillde{ O} (n ⁇ 2} ) 和 单面 最大流量算法 来打破这个60 年的屏障 。