The Langberg-Medard multiple unicast conjecture claims that for any strongly reachable $k$-pair network, there exists a multi-flow with rate $(1,1,dots,1)$. In a previous work, through combining and concatenating the so-called elementary flows, we have constructed a multi-flow with rate at least $(frac{8}{9}, frac{8}{9}, dots, frac{8}{9})$ for any $k$. In this paper, we examine an optimization problem arising from this construction framework. We first show that our previous construction yields a sequence of asymptotically optimal solutions to the aforementioned optimization problem. And furthermore, based on this solution sequence, we propose a perturbation framework, which not only promises a better solution for any $k mod 4 eq 2$ but also solves the optimization problem for the cases $k=3, 4, dots, 10$, accordingly yielding multi-flows with the largest rate to date.
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