Although Software-Defined Networking (SDN) enables flexible network resource allocations for traffic engineering, current literature mostly focuses on unicast communications. Compared to traffic engineering for multiple unicast flows, multicast traffic engineering for multiple trees is very challenging not only because minimizing the bandwidth consumption of a single multicast tree by solving the Steiner tree problem is already NP-Hard, but the Steiner tree problem does not consider the link capacity constraint for multicast flows and node capacity constraint to store the forwarding entries in Group Table of OpenFlow. In this paper, therefore, we first study the hardness results of scalable multicast traffic engineering in SDN. We prove that scalable multicast traffic engineering with only the node capacity constraint is NP-Hard and not approximable within δ, which is the number of destinations in the largest multicast group. We then prove that scalable multicast traffic engineering with both the node and link capacity constraints is NP-Hard and not approximable within any ratio. To solve the problem, we design a δ-approximation algorithm, named Multi-Tree Routing and State Assignment Algorithm (MTRSA), for the first case and extend it to the general multicast traffic engineering problem. The simulation and implementation results demonstrate that the solutions obtained by the proposed algorithm outperform the shortest-path trees and Steiner trees. Most importantly, MTRSA is computation-efficient and can be deployed in SDN since it can generate the solution with numerous trees in a short time.

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