## 1 Introduction  
With the expansion of network scale and the continuous emergence of high-traffic services, such as data center services and cloud-based services [1; 2; 3], explosive traffic carried by the optical networks has led to an urgent demand for bandwidth resources. Currently, researchers are exploring more advanced optical transmission and networking technologies to improve network capacity [4; 5; 6]. In this case, flexible grid technology has attracted extensive attention due to its fine-grained, high flexibility and efficient network resource-utilization characteristics [7; 8]. Compared with the fixed-grid wavelength division multiplexing (WDM) optical networks, the emerging optical networks architecture, which is based on flexible-grid technology, can overcome the limitation of rigid capacity for optical channel and provide efficient on-demand spectrum allocation with fine granularity spectrum slots, such as 12.5 GHz [9], which can greatly improve spectrum resource utilization [10; 11]. Additionally, new types of optical networks components, such as, flex-grid optical cross-connects built on Liquid Crystal on Silicon-based wavelength selective switches (WSSs) [12] and Sliceable Bandwidth-Variable Transponders [13] have been introduced to optical networks to support flexible grid technology, which greatly foster the development of novel network architecture. The flex-grid optical networks use technologies such as optical cross-connects with variable bandwidth, wavelength selective switches, and sliceable bandwidth transponders to transmit network data. Flex-grid optical networks abandon the wavelength constraints, and provide more flexible and fine-grained services through on-demand spectrum resources allocation and the adaptive selection of modulation formats compared with fixed-grid optical networks. Super-channels [14] can be established between flex-grid nodes to achieve ultra-high-speed optical channel transmission, which improves spectrum efficiency and network transmission capacity. However, due to technical cost and feasibility constraints, it is unrealistic to complete the upgrade deployment of the entire network at one time. The transition technology called fixed/flex-grid optical networks where both fixed- and flex-grid nodes co-exist is more practical before complete migration to flex-grid network. Through the deployment of the flexible network node, the capacity of a network can be increased, while preserving the already made investment. The fixed/flex-grid optical networks presents new challenges and limitations at the network level, since the spectrum is shared as a common resource. Different from the fixed-grid ROADMs, for which the central frequencies and spectrum grids are defined as 50 or 100 GHz, the flex-grid ROADMs have finer granularity, e.g., 6.25 or 12.5 GHz. Figure 1 illustrates four cases of spectrum allocation in fixed/flex-grid optical networks scenario. The slot-width of fixed-grid node and flex-grid node are defined as 50 GHz and the multiple of 12.5 GHz, respectively. In Figure 1a, a light path request of 100 Gb/s between two fixed-gird nodes traverses another fixed-grid node. Due to the characteristics of the fixed-grid nodes, a 50 GHz channel which represents one fixed-grid slot is needed on both of the links to carry the 100 Gb/s request. In Figure 1b, for the request carried by the second link, only three flex-grid slots are sufficient because the established super optical channel between two flex-grid nodes has finer granularity. For the 200 Gb/s request in Figure 1c, only 75 GHz which represents six flex-grid slots is required on both links. In Figure 1d, to carry the 40 Gb/s request, two flex-grid slots are required for the first link and one fixed-grid slot is required for the following fixed-grid link. Figure 1: Different mixed-grid scenarios. (**a**) The 1 st case (**b**) The 2 nd case (**c**) The 3 rd case (**d**) The 4 th case (**e**) 9-node topology.