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A Note on the Multiple-Unicast Network Coding Conjecture

机译：关于多单播网络编码猜想的一个注记

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摘要

Network coding encourages in-network information mixing for enhanced network capacity. An important open problem in network coding, the multiple-unicast network coding (MUNC) conjecture, claims that network coding is equivalent to routing for multiple unicast sessions in an undirected network. Simple and intuitive as it appears, the MUNC conjecture has remained open for a decade. This work studies a special case of MUNC, where intra-session connections are "better" than inter-session connections. In the cost domain, we prove that if each source is closer to its own receiver than to other receivers, then network coding is equivalent to routing. In the throughput domain, an intuitive dual statement is that if each source has a larger max-flow to its own receiver than to other receivers, then network coding is equivalent to routing. We prove that this statement is equivalent to the original MUNC conjecture. An interesting consequence is that an equivalence between the two seemingly dual statements studied in this work, if true, would imply the original MUNC conjecture.
机译：网络编码鼓励网络内信息混合以增强网络容量。网络编码中的一个重要的开放问题是多单播网络编码（MUNC）猜想，它声称网络编码等同于无向网络中多个单播会话的路由。 MUNC猜想看似简单直观，但开放至今已有十年之久。这项工作研究了MUNC的一种特殊情况，其中会话间连接比会话间连接“更好”。在成本方面，我们证明如果每个源比其他接收器更靠近其自己的接收器，则网络编码等效于路由。在吞吐量域中，一个直观的对偶声明是，如果每个源到其自己的接收器的最大流都比其他接收器的流大，那么网络编码等效于路由。我们证明该陈述等同于原始的MUNC猜想。一个有趣的结果是，在这项工作中研究的两个看似对偶的陈述之间的对等关系（如果为真）将暗示原始的MUNC猜想。

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