An 8/5 Rounding for Half-Integral Forest-BCR via Root Supports and Circuit Rank
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 $ \beta_C\ \ge\ \frac{L_C}{2}$, where $\beta_C$ 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.