## 1 Introduction  
Visible light communication (VLC) is an emerging wireless communication technology, which simultaneously achieves communication and illumination [1; 2] and has the advantage of high speeds [3]. Due to the ubiquitous indoor light sources, visible light communication has become an attractive alternative to conventional radio communications for indoor environments [4]. However, the modulation bandwidth of commercial light emitting diodes (LEDs) is only 20 MHz and the bandwidth resource is limited [5; 6]. Therefore, a key issue in this field is the design of an efficient resource allocation algorithm for visible light communication. The visible light communication networks are typical ultra-dense networks (UDNs) that have three characteristics: high terminal density, high traffic density requirement and high access point (AP) density. The reference values for the three characters in VLC-UDNs are provided as follows [7]: the terminal density is higher than 0.25 terminals per m\({}^{2}\); the traffic density is higher than 10 Mbps/m\({}^{2}\), and the distance between the APs is less than 10 m. Being different from the radio signals in the traditional radio frequency ultra-density networks (RF-UDNs), the visible light in VLC-UDNs has a line of sight (LoS) and it is easily blocked, which interrupts communication [8]. The strong directivity of light sources limits the coverage of a single light source and results in the user channel quality being closely related to the terminal's spatial location [9]. Compared with radio frequency networks, VLC-UDNs have a higher AP density, which would cause increase the prominence of interference in overlapping areas. It is crucial to design an efficient resource allocation algorithm to overcome these challenges and satisfy the requirements of the terminal. There are many classical research solutions for the resource allocation problem in multi-cell scenarios for indoor VLC-UDNs. References [10; 11; 12] used the Markov chain model to allocate the visible light multi-color channel resources to terminals. Based on the concept of channel reservation, reference [13] proposed an adaptive bandwidth allocation method based on traffic priority. Although reference [10; 11; 12; 13] improved the system performance, the algorithm design deviates from the actual spatial scene of VLC-UDNs. References [14; 15] allocated resources based on the priority of terminals, which was measured using the fuzzy logic (FL) theory. Although the system throughput and the terminal satisfaction performance were improved, the algorithm complexity increased due to the utilization of centroid method during the defuzzification process. Moreover, the design of fuzzy logic rule table was subjective. References [16; 17; 18] used the required data rate proportion (RDR-PA) method to allocate resources. Although the requirements of terminals were considered, the channel quality was ignored and the resource utilization efficiency needed to be improved. Reference [19] used the satisfaction degree proportion method to allocate resources for terminals, but the method was applied only to time resources. Although the algorithms in reference [16; 17; 18; 19] had low complexity, the optimal throughput performance was not achieved. Reference [20] used the gradient method to resolve the optimal resource allocation problem and the optimal system throughput was realized. However, when the objective function is a complex nonlinear function, the gradient method has a slow convergence rate. In summary, the optimality of the throughput performance and the low complexity of the resource allocation algorithm are a pair of contradictory indicators. There is the need to design an efficient resource allocation algorithm with optimal throughput and low complexity.