Content Delivery Networks (CDN) are witnessing the outburst of video streaming (e.g., personal live streaming or Video-on-Demand) where the video content, produced or accessed by mobile phones, must be quickly transferred from a point to another of the network. Whenever a user requests a video not directly available at the edge server, the CDN network must 1) identify the best location in the network where the content is stored, 2) set up a connection and 3) deliver the video as quickly as possible. For this reason, existing CDNs are adopting an overlay structure to reduce latency, leveraging the flexibility introduced by the Software Defined Networking (SDN) paradigm. In order to guarantee a satisfactory Quality of Experience (QoE) to users, the connection must respect several Quality of Service (QoS) constraints. In this paper, we focus on the sub-problem 2), by presenting an approach to efficiently compute and maintain paths in the overlay network. Our approach allows to speed up the transfer of video segments by finding minimum delay overlay paths under constraints on hop count, jitter, packet loss and relay processing capacity. The proposed algorithm provides a near-optimal solution, while drastically reducing the execution time. We show on traces collected in a real CDN that our solution allows to maximize the number of fast video transfers.