This is a well known problem with a conjecture due to Karpovsky and Moskalev [1] (I still believe is open) that at most V-1 paths are needed. It is tied to conjector of Erdos, Goodman, and Posa [2].
If you want the state of the art today, I suggest doing a thorough reverse-citation search on [1].
[1] Karpovsky, M. G., and E. A. Moskalev. "Covering of edges of graph by a minimal set of paths." Discr. Math 58.2 (1986): 214. (This is in the "research problems" section)
[2] Erdos, Paul, Adolph W. Goodman, and Louis Pósa. "The representation of a graph by set intersections." Canad. J. Math 18.106-112 (1966): 86.