单源不可分流中的独立系统实现
Independence-System Realisations in Single-Source Unsplittable Flow
浏览论文内容
中文总结 AI 辅助
本文提出独立系统的路径封闭实现概念，证明每个有限无环独立系统有多项式规模实现，并针对奇环给出精确的加性拥塞阈值，为单源不可分流问题提供统一框架。
中文摘要 AI 辅助
单源不可分流中的加性拥塞约束可以强制稳定集结构。本文隔离并推广了这一机制。我们引入了一种路径封闭的概念，通过有向无环流实例中主终端的零成本选择来实现独立系统。该定义对每条有向源-终端路径进行量化，因此在前缀借用、后缀拼接和混合拼接下仍然有效。主要结果扩展了三角形机制：每个有限无环独立系统都具有多项式规模的实现，以最小禁止集的关联规模衡量。因此，每个有限简单图，更一般地，每个没有单元素禁止超边的超图独立系统，都可以由无环单源小工具表示。然后我们将构造专门化到奇环。对于C_{2k+1}，一个均匀有理族产生一个分数廉价选择向量，该向量违反奇环不等式。势移位将带符号的连接分隔符转换为非负弧成本，并给出精确的保成本加性拥塞阈值τ = 1 - bq。在对称族内，上确界阈值为(k+2)/(2(k+1))，趋向于1/2。对于C5，一个精确证书独立推导出所有源-终端路径，并使用有理算术枚举所有3^10 = 59049种不可分路由。
英文摘要
Additive-congestion constraints in single-source unsplittable flow can enforce stable-set structure. This note isolates and generalises that mechanism. We introduce a path-closed notion of realising an independence system by the zero-cost choices of primary terminals in a directed acyclic flow instance. The definition quantifies over every directed source-terminal path and therefore remains valid under prefix borrowing, suffix splicing, and hybrid routes.Our main result extends the triangle mechanism: every finite loopless independence system has a polynomial-size realisation, measured in the incidence size of its minimal forbidden sets. Hence every finite simple graph, and more generally every hypergraph independence system without singleton forbidden hyperedges, is representable by an acyclic single-source gadget. We then specialise the construction to odd cycles. For C_{2k+1}, a uniform rational family produces a fractional cheap-selection vector that violates the odd-cycle inequality. A potential shift converts a signed connector separator into nonnegative arc costs and gives the exact cost-preserving additive-congestion threshold tau = 1 - bq. Within the symmetric family, the supremum threshold is (k+2)/(2(k+1)), which tends to 1/2. For C5, an exact certificate independently derives all source-terminal paths and enumerates all 3^10 = 59049 unsplittable routings using rational arithmetic.
