An Optimization Approach to the Langberg-Médard Multiple Unicast Conjecture
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The Langberg-Médard multiple unicast conjecture claims that for any strongly reachable k-pair network, there exists a multi-flow with rate (1,1,...,1). In a previous work, through combining and concatenating the so-called elementary flows, we have constructed a multi-flow with rate at least (8/9, 8/9, ..., 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 4 ≠ 2 but also solves the optimization problem for the cases k=3, 4, ..., 10, accordingly yielding multi-flows with the largest rate to date.
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