On the Langberg-Médard multiple unicast conjecture

A version of the multiple unicast conjecture, proposed by M. Langberg and M. Médard [4], says that, there exists an undirected fractional multicommodity flow, or simply, multi-flow, with rate (1, 1, · · · , 1) for strongly reachable networks. In this paper, we propose a new type of matrix optimization problem to attack this conjecture: By giving upper and lower bounds on its solution, we prove that there exists a multi-flow with rate ( 8 9 , 8 9 , · · · , 8 9 ) for such networks; on the other hand though, we show that the rate of any multi-flow constructed using this framework cannot exceed (1, 1, · · · , 1).