To improve traffic management ability, Internet Service Providers (ISPs) are gradually upgrading legacy network devices to programmable devices that support Software-Defined Networking (SDN). The coexistence of legacy and SDN devices gives rise to a hybrid SDN. Existing hybrid SDNs do not consider the potential performance issues introduced by a centralized SDN controller: flow requests processed by a highly loaded controller may experience long tail processing delay; inappropriate multi-controller deployment could increase the propagation delay of flow requests.In this paper, we propose to jointly consider the deployment of SDN switches and their controllers for hybrid SDNs. We formulate the joint problem as an optimization problem that maximizes the number of flows that can be controlled and managed by the SDN and minimizes the propagation delay of flow requests between SDN controllers and switches under a given upgrade budget constraint. We show this problem is NP-hard. To efficiently solve the problem, we propose some techniques (e.g., strengthening the constraints and adding additional valid inequalities) to accelerate the global optimization solver for solving the problem for small networks and an efficient heuristic algorithm for solving it for large networks. The simulation results from real network topologies illustrate the effectiveness of the proposed techniques and show that our proposed heuristic algorithm uses a small number of controllers to manage a high amount of flows with good performance. Z. Guo and Z.-L. Complexity analysis, controller deployment, heuristic algorithm, hybrid SDN, switch upgrade, upgrade budget I. INTRODUCTION Software-Defined Networking (SDN) has been widely studied and gradually adapted for campus networks [1], data center networks [2], Wide Area Networks (WANs) [3], enterprise networks [4], and Internet exchange points [5]. Due to the cost and operational considerations, SDN technology is usually deployed in an incremental fashion. In particular, at each time of network upgrade, only a set of selected legacy network devices (i.e., layer-3 routers and layer-2 switches) are upgraded to SDN switches. AT&T converted 34% of its network to SDN by the end of 2016 and virtualized 55% of its network to software by the end 2017. Its final goal is to reach 75% softwarization of its network by 2020 [6]. Therefore, the legacy network devices and SDN switches may coexist for a long time. In this paper, we will refer to such a network as a hybrid SDN.A WAN usually consists of many network devices at geo-distributed locations. A straightforward method to upgrade legacy network devices in WANs to SDN switches is based on the locations of network devices, for example, upgrade a part of WAN with network devices in proximity. The partial upgraded network can enjoy the benefit of SDN, but the performance improvement of the entire network is limited. An efficient solution for network providers is to spread the benefit of upgraded SDNs in the entire network. Based on this consideration, existing ...