多单播猜想：超越割度量
On the Multiple-Unicast Conjecture: Beyond Cut Metrics
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中文总结 AI 辅助
本文构建统一图度量框架研究多单播猜想，证明三类新无向网络的猜想成立，给出编码节点≤6的网络的猜想成立的新证明，还推导了Γ₃,₃与多会话终端位置的关联结论。
中文摘要 AI 辅助
网络编码允许中间节点在传输前对接收的消息进行编码。多单播猜想断言，在任何无向网络中，针对独立单播会话，编码在吞吐量上并不优于分数路由。尽管经过二十多年的持续研究，这个核心开放问题仍未解决。该猜想与计算复杂性密切相关：若能证明它成立，将为基础问题提供长期寻求的下界。为研究该猜想，本文从编码与路由比较的基本对象不只是割，而是图度量这一视角出发，构建了一个统一的度量框架。利用该框架，我们证明了三类新无向网络的猜想成立：(a) 终端位置最多为5个的网络；(b) 终端位置位于至多3个指定面边界上，且每个会话端点均在其中一个面上的平面图；(c) 节点和终端位置数量任意，但会话端点受结构限制的网络。我们还给出了一个新证明，表明编码节点最多为6个的网络中猜想成立，且无需计算机辅助搜索；并证明若猜想在Γ₃,₃上成立，则当不存在三个会话具有六个不同终端位置时，猜想均成立。
英文摘要
Network coding allows intermediate nodes to encode received messages before transmission. The multiple-unicast conjecture asserts that coding has no throughput advantage over fractional routing for independent unicast sessions in any undirected network. Despite more than two decades of sustained study, this central open problem remains unresolved. The conjecture is deeply connected to computational complexity: a proof would yield long-sought lower bounds for fundamental problems. To study the conjecture, this paper develops a unified metric framework from the perspective that the basic objects behind the comparison between coding and routing are not cuts alone, but graph metrics. Using this framework, we prove the conjecture for three new classes of undirected networks: (a) networks with at most five terminal locations; (b) planar networks whose terminal locations lie on the boundaries of at most three designated faces, with each session's endpoints on one such face; and (c) networks with arbitrarily many nodes and terminal locations under a structural restriction on session endpoints. We give a new proof that the conjecture holds for networks with at most six coding nodes, without computer-aided search, and show that if the conjecture holds on $Γ_{3,3}$, then it holds whenever no three sessions have six distinct terminal locations.
