Improving the integrality gap of the LP relaxation for non-linear paging to obtain a randomized polylog(ℓ)-competitive algorithm is a challenging task. One approach to potentially enhance the LP relaxation is to introduce additional constraints or variables that capture the non-linear relationships more accurately. By refining the LP formulation to better represent the underlying structure of the problem, it may be possible to reduce the integrality gap and achieve a tighter bound on the competitive ratio. Furthermore, exploring advanced optimization techniques, such as rounding strategies or dual-fitting algorithms, could help in developing a more efficient and accurate algorithm for non-linear paging with a reduced integrality gap.