AI 中文总结
该研究针对Steiner森林的半整Forest-BCR，优化递归取整框架，将近似比从16/9降至8/5，且证明了该框架的投影密度界渐近紧。
AI 中文摘要
我们研究了Steiner森林的根分配双向割松弛（Forest-BCR）所提供的半整可行解的取整问题。Byrka、Grandoni和Traub[IPCO 2025]证明了一种递归框架的近似比为16/9，该框架会对线性规划（LP）点进行归一化，选择具有最大投影LP密度的顶点集，在该顶点集上购买最小生成树，收缩该树并进行递归。我们证明同一框架的近似比可达8/5。新的分析保留了每个投影半单位LP质量的方向和根标签。在简单投影中，度为2的终端处的割约束决定了与其关联的每个需求的根分配向量，而半整性仅留下两种可能：向一个根分配单位，这会在该根的支撑内部产生过剩出度；或向两个根分配拆分，这会在它们的支撑之间产生重叠。对于拆分根图的每个连通分量C，这得出β_C ≥ L_C/2，其中β_C是C中根支撑并集的回路秩，L_C是全投影中度为2的顶点数。将该不等式得到的密度证书与普通度和进行平衡，得到密度至少为5/8的顶点集，而继承的收缩引理将其转化为8/5的取整结果。对于每个q≥3，我们还构造了一个归一化半整点，其最大投影密度恰好为5q/[2(4q-1)]，因此通用投影密度界是渐近紧的。
英文摘要
We study the rounding of a supplied half-integral feasible solution of the root-assignment bidirected cut relaxation for Steiner Forest (Forest-BCR). Byrka, Grandoni, and Traub [IPCO 2025] proved a $16/9$ guarantee for a recursive framework that normalizes the LP point, selects a vertex set of maximum projected LP density, buys a minimum spanning tree on that set, contracts it, and recurses. We prove that the same framework has guarantee $8/5$. The new analysis keeps the orientation and the root label of each projected half-unit of LP mass. In a simple projection, the cut constraints at a terminal of degree two determine the root-assignment vector of every demand incident with it, and half-integrality leaves only two possibilities: a unit assignment to one root, which forces excess outdegree inside that root's support, or a split assignment to two roots, which forces overlap between their supports. For every connected component $C$ of the split-root graph this yields $ β_C\ \ge\ \frac{L_C}{2}$, where $β_C$ is the circuit rank of the union of the root supports in $C$ and $L_C$ is the number of its vertices of degree two in the full projection. Balancing the density certificate obtained from this inequality against the ordinary degree sum gives a vertex set of density at least $5/8$, and the inherited contraction lemma turns that into the $8/5$ rounding. For every $q\ge3$ we also construct a normalized half-integral point whose maximum projected density is exactly $5q/[2(4q-1)]$, so the universal projected-density bound is asymptotically tight.
