All-Pairs Minimum Cut (APMC) is a fundamental graph problem that asks to find a minimum $s,t$-cut for every pair of vertices $s,t$. A recent line of work on fast algorithms for APMC has culminated with a reduction of APMC to $\mathrm{polylog}(n)$-many max-flow computations. But unfortunately, no fast algorithms are currently known for exact max-flow in several standard models of computation, such as the cut-query model and the fully-dynamic model. Our main technical contribution is a sparsifier that preserves all minimum $s,t$-cuts in an unweighted graph, and can be constructed using only approximate max-flow computations. We then use this sparsifier to devise new algorithms for APMC in unweighted graphs in several computational models: (i) a randomized algorithm that makes $\tilde{O}(n^{3/2})$ cut queries to the input graph; (ii) a deterministic fully-dynamic algorithm with $n^{3/2+o(1)}$ worst-case update time; and (iii) a randomized two-pass streaming algorithm with space requirement $\tilde{O}(n^{3/2})$. These results improve over the known bounds, even for (single pair) minimum $s,t$-cut in the respective models.

翻译：全对最小割（APMC）是一个基础图问题，要求为每对顶点$s,t$找到最小$s,t$割。近期关于APMC快速算法的系列研究最终将APMC归约为$\mathrm{polylog}(n)$次最大流计算。但不幸的是，在若干标准计算模型（如割查询模型和全动态模型）中，目前尚无精确最大流的快速算法。我们的主要技术贡献是一种稀疏化器，它能保留无权重图中所有最小$s,t$割，且仅需近似最大流计算即可构建。随后我们利用该稀疏化器在多种计算模型中为无权重图设计了新的APMC算法：（i）一种随机算法，对输入图进行$\tilde{O}(n^{3/2})$次割查询；（ii）一种确定性全动态算法，最坏情况更新时间为$n^{3/2+o(1)}$；（iii）一种随机两遍流算法，空间需求为$\tilde{O}(n^{3/2})$。这些结果改进了已知界限，即便在各自模型中针对（单对）最小$s,t$割也是如此。