We consider the problem of finding a minimum cut of a weighted graph presented as a single-pass stream. While graph sparsification in streams has been intensively studied, the specific application of finding minimum cuts in streams is less well-studied. To this end, we show upper and lower bounds on minimum cut problems in insertion-only streams for a variety of settings, including for both randomized and deterministic algorithms, for both arbitrary and random order streams, and for both approximate and exact algorithms. One of our main results is an $\widetilde{O}(n/\varepsilon)$ space algorithm with fast update time for approximating a spectral cut query with high probability on a stream given in an arbitrary order. Our result breaks the $\Omega(n/\varepsilon^2)$ space lower bound required of a sparsifier that approximates all cuts simultaneously. Using this result, we provide streaming algorithms with near optimal space of $\widetilde{O}(n/\varepsilon)$ for minimum cut and approximate all-pairs effective resistances, with matching space lower-bounds. The amortized update time of our algorithms is $\widetilde{O}(1)$, provided that the number of edges in the input graph is at least $(n/\varepsilon^2)^{1+o(1)}$. We also give a generic way of incorporating sketching into a recursive contraction algorithm to improve the post-processing time of our algorithms. In addition to these results, we give a random-order streaming algorithm that computes the {\it exact} minimum cut on a simple, unweighted graph using $\widetilde{O}(n)$ space. Finally, we give an $\Omega(n/\varepsilon^2)$ space lower bound for deterministic minimum cut algorithms which matches the best-known upper bound up to polylogarithmic factors.

翻译：我们研究在单次流中寻找加权图最小割的问题。尽管流中的图稀疏化已得到深入研究，但在流中寻找最小割的具体应用尚未得到充分探讨。为此，我们针对多种设置展示了插入式流中最小割问题的上界与下界，包括随机化与确定性算法、任意顺序流与随机顺序流，以及近似与精确算法。我们的主要成果之一是提出了一种空间复杂度为$\widetilde{O}(n/\varepsilon)$的算法，该算法具有快速更新时间，能够在任意顺序的流上以高概率近似谱割查询。该结果突破了需要同时近似所有割的稀疏化器所要求的$\Omega(n/\varepsilon^2)$空间下界。基于此结果，我们为最小割和近似全对有效电阻问题提供了空间复杂度接近最优（$\widetilde{O}(n/\varepsilon)$）的流算法，并给出了匹配的空间下界。当输入图的边数至少为$(n/\varepsilon^2)^{1+o(1)}$时，我们算法的摊销更新时间为$\widetilde{O}(1)$。我们还提出了一种将草图技术融入递归收缩算法的通用方法，以改进算法的后处理时间。除上述结果外，我们给出了一种随机顺序流算法，该算法使用$\widetilde{O}(n)$空间在简单无权重图上计算{\it 精确}最小割。最后，我们针对确定性最小割算法给出了$\Omega(n/\varepsilon^2)$的空间下界，该下界与已知最佳上界在多项式对数因子范围内相匹配。