# Spectrum Allocation Using Integer Linear Programming and Kerr Optical Frequency Combs
## 1 Introduction  
The implementation of technologies such as the Internet of Things (IoT), artificial intelligence, and high-performance computing has exponentially increased bandwidth demand, which is critical in optical networks, particularly in passive optical networks (PON). In this context, networks using wavelength division multiplexing (WDM-PON) provide a viable solution to enhance transmission capacity. WDM-PON networks offer significant advantages in meeting quality of service (QoS) standards [1], enabling the simultaneous transmission of multiple optical signals over different wavelengths on a single optical fiber. However, their implementation faces economic challenges due to the requirement of installing a laser for each wavelength and achieving phase synchronization. Optical frequency combs (OFCs) present a promising solution, as they are light sources with equally spaced, phase-coherent frequency lines. In recent years, numerous cutting-edge methods for generating optical frequency combs (OFCs) have made substantial progress, improving their efficiency and expanding their applicability across various fields. One prominent approach is gain modulation in semiconductor lasers, noted for its simplicity and low cost [2]. This technique leverages the dependence of a semiconductor laser's optical gain on the charge carrier density in its active region. By injecting a variable electric current, the carrier density is altered, which in turn modifies the optical gain, resulting in intensity modulation of the emitted light [3]. Another relevant technique is the use of microcavities, which exploit the interactions between light and mechanical vibrations to produce frequency combs with adjustable bandwidths [4; 5; 6]. Electro-optic modulation using Mach-Zehnder devices generates optical frequency combs (OFCs) by inducing periodic phase shifts in a continuous-wave laser. This process leads to the formation of evenly spaced spectral lines resulting from interference effects between two optical paths. However, a key limitation of this approach is its significant sensitivity to temperature variations and electrical instabilities, which can introduce unwanted distortions in the generated comb and consequently limit the system's stability and overall performance [7]. On the other hand, optical microresonators have demonstrated improved power conversion efficiency, making them suitable for large-scale optical systems and supporting transmission speeds of up to 1.84 Pbit/s [5; 6; 7; 8; 9]. Additionally, while some advanced approaches combine both techniques to leverage their respective advantages, these hybrid solutions often come at a higher cost. This approach introduces greater complexity and operational costs due to the need for external signals and additional control variables, which can increase energy consumption and phase noise. In contrast, Kerr-based microresonator OFCs offer distinct advantages, such as lower manufacturing costs and operational simplicity. These combs generate a broad and stable spectrum without external modulating sources, which reduces system complexity and noise levels. This is crucial for telecommunications applications that require precision and stability. Furthermore, Kerr-based OFCs have strong potential to dominate OFC generation methods due to their efficiency, robustness, and scalability. Their compatibility with silicon photonics allows for miniaturization and cost-effective manufacturing, making them an attractive option for applications aiming to maximize integration without significantly altering existing network infrastructures [9; 10]. A critical challenge in spectrum allocation lies in the availability and efficient management of carriers. In this context, optical frequency combs (OFCs) based on Kerr solitons, generated through microring resonators (MRRs) made from nonlinear materials like silicon nitride, have emerged as a promising solution.