In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is to approximate cuts in balanced directed graphs. In this problem, the goal is to build a data structure that $(1 \pm \epsilon)$-approximates cut values in graphs with $n$ vertices. For arbitrary directed graphs, such a data structure requires $\Omega(n^2)$ bits even for constant $\epsilon$. To circumvent this, recent works study $\beta$-balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most $\beta$ times that in the other direction. We consider two models: the {\em for-each} model, where the goal is to approximate each cut with constant probability, and the {\em for-all} model, where all cuts must be preserved simultaneously. We improve the previous $\Omega(n \sqrt{\beta/\epsilon})$ lower bound to $\tilde{\Omega}(n \sqrt{\beta}/\epsilon)$ in the for-each model, and we improve the previous $\Omega(n \beta/\epsilon)$ lower bound to $\Omega(n \beta/\epsilon^2)$ in the for-all model. This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is to approximate the global minimum cut in a local query model, where we can only access the graph via degree, edge, and adjacency queries. We improve the previous $\Omega\bigl(\frac{m}{k}\bigr)$ query complexity lower bound to $\Omega\bigl(\min\{m, \frac{m}{\epsilon^2 k}\}\bigr)$ for this problem, where $m$ is the number of edges, $k$ is the size of the minimum cut, and we seek a $(1+\epsilon)$-approximation. In addition, we show that existing upper bounds with slight modifications match our lower bound up to logarithmic factors.

翻译：本文研究大规模图上的两个基本割近似问题。我们证明了这两个问题的新下界，这些下界在对数因子范围内达到最优。第一个问题是在平衡有向图中近似割。该问题的目标是构建一个数据结构，能在具有$n$个顶点的图中$(1 \pm \epsilon)$-近似割值。对于任意有向图，即使$\epsilon$为常数，此类数据结构也需要$\Omega(n^2)$比特。为规避此限制，近期研究关注$\beta$-平衡图，即对于每个有向割，一个方向的边总权重至多为另一方向的$\beta$倍。我们考虑两种模型：{\em 逐例}模型（目标是以恒定概率近似每个割）和{\em 全局}模型（要求同时保留所有割）。在逐例模型中，我们将先前$\Omega(n \sqrt{\beta/\epsilon})$下界改进为$\tilde{\Omega}(n \sqrt{\beta}/\epsilon)$；在全局模型中，将先前$\Omega(n \beta/\epsilon)$下界改进为$\Omega(n \beta/\epsilon^2)$。这解决了(Cen等人，ICALP，2021)的主要开放问题。第二个问题是在局部查询模型中近似全局最小割，其中仅能通过度查询、边查询和邻接查询访问图。我们将该问题先前的$\Omega\bigl(\frac{m}{k}\bigr)$查询复杂度下界改进为$\Omega\bigl(\min\{m, \frac{m}{\epsilon^2 k}\}\bigr)$，其中$m$为边数，$k$为最小割规模，且我们寻求$(1+\epsilon)$-近似解。此外，我们证明经过微调的现有上界与我们的下界在对数因子内匹配。