发表机构
Carnegie Mellon University; University of Michigan（卡内基梅隆大学; 密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出一种随机归约方法，通过新颖的迭代树形图采样过程，得到首个近线性时间的有向全局最小割算法，改进了此前的时间复杂度界。
AI 中文摘要
我们给出了一种随机归约方法，将含n个顶点、m条带权边的有向图的全局最小割问题，归约为总规模为Õ(m)的图上的最大流计算。对于多项式有界的整数权重，这得到了首个近线性时间的算法，时间复杂度为m^(1+o(1))[vdBCK+23]，改进了之前的m^(1+o(1))min{√n, n/m^(1/3)}的复杂度界[CLN+22]。此前的研究将最小割的寻找归约为构造1-尊重树形图（即恰好有一条边穿过最小割的有向生成树）[CLN+22]，他们通过从(1+ε)-近似树形图集合中采样来构造此类树，而该过程当前需要超线性时间。我们的算法完全避开了构造该集合的步骤，依赖于一种新颖的迭代树形图采样过程。我们证明，经过O(log n)轮采样后，最终得到的树形图以常数概率1-尊重最小割。
英文摘要
We give a randomized reduction from global minimum cut in a directed graph with $n$ vertices and $m$ weighted edges to maximum-flow computations on graphs of total size $\widetilde{O}(m)$. For polynomially bounded integral weights, this yields the first almost-linear $m^{1+o(1)}$-time algorithm [vdBCK+23], improving the previous $m^{1+o(1)}\min\{\sqrt{n},n/m^{1/3}\}$ bound [CLN+22]. Previous work reduced finding the minimum cut to constructing a 1-respecting arborescence, a directed spanning tree with exactly one edge crossing a minimum cut [CLN+22]. They then construct such a tree by sampling from a $(1+ε)$-approximate arborescence packing, which currently requires superlinear time. Our algorithm sidesteps constructing the packing entirely and relies on a novel iterative arborescence sampling procedure. We show that, after $O(\log n)$ rounds of sampling, our final arborescence 1-respects the minimum cut with constant probability.
Comments11 pages, 4 figures
