We study threshold minimum cut problems with a distinguished root vertex, a set of terminals, and a quota. In the threshold minimum edge cut problem (\TMEC), the goal is to find a minimum-cost edge cut that disconnects at least $k$ terminals from the root. In the threshold minimum node cut problem (\TMNC), the goal is to delete a minimum-cost set of nonterminal, nonroot vertices so that at least $k$ terminals become disconnected from the root. We prove three approximation guarantees. First, undirected general-graph \TMEC{} admits a randomized polynomial-time expected $O(\log n)$ approximation via a Räcke-style cut-dominating tree decomposition and an exact dynamic program on trees. A standard repetition argument gives the same asymptotic ratio with high probability. Second, planar \TMEC{} admits a factor-$2$ approximation by reducing the threshold condition to planar weighted balanced cut. Third, bounded-degree planar \TMNC{} admits a $2Δ$-approximation, where $Δ$ is the maximum degree of a deletable vertex, by reducing the node-cost problem to the planar edge-cut problem on the same graph. The results separate exact-quota guarantees from bicriteria small-set-expansion-type guarantees and identify the unbounded-degree planar node-cut case as the main remaining obstacle.

翻译：我们研究带有特定根顶点、终端集和配额的阈值最小割问题。在阈值最小边割问题（\TMEC）中，目标是找到一条最小代价边割，使得至少$k$个终端与根断开连接。在阈值最小点割问题（\TMNC）中，目标是删除一组最小代价的非终端、非根顶点，使得至少$k$个终端与根断开连接。我们证明了三种近似保证。首先，无向一般图上的\TMEC通过Räcke风格的割支配树分解和树上的精确动态规划，在随机多项式时间内具有期望$O(\log n)$近似比，通过标准重复论证可在高概率下获得相同渐近比。其次，平面图上的\TMEC通过将阈值条件归约为平面加权平衡割，具有因子$2$近似。第三，有界度平面图上的\TMNC通过将点代价问题归约为同一图上的平面边割问题，具有$2Δ$近似，其中$Δ$是可删除顶点的最大度。这些结果将精确配额保证与双准则小扩张型保证区分开来，并指出无界度平面点割情形是主要遗留障碍。