Space Information Flow: Multiple Unicast in $l_{p}^{n}$

The multiple-unicast network coding conjecture states that network coding is equivalent to routing for multiple unicast sessions in an undirected network. Despite its seemingly simple and intuitive nature, this conjecture has remained unresolved since its introduction over two decades ago [1], [2], and stands as a well-known open problem in the field of network coding. Recent advances in theoretical computer science further reveal its connection to fundamental open problems in complexity lower bounds. In this work, we address the conjecture from the perspective of space information flow, a paradigm that studies information transmission in geometric spaces [3] [5], aiming to minimize the bandwidth-distance product (network volume) while meeting communication demands between terminals. Our main result demonstrates that network coding is indeed equivalent to routing in $l_{p}^{n}$ spaces for $1 \leq p \leq 2$. Specifically, we provide: (1) a proof of the case $1 \leq p<2$, and (2) a new proof of the case $p=2$. This result partially verifies the original conjecture, while suggesting a possible direction for an eventual proof of the conjecture.