## 2 Related Works
[10] addressed the limitations of computational resources in mobile devices and proposed a convex optimization-based approach to solve this problem. [PERSON] et al. [30] proposed a dynamic caching-assisted task offloading scheme (CachOf), which combines collaborative computing of edge servers and dynamic resource allocation strategies. They then applied deep reinforcement learning (DRL) methods to optimize the task offloading strategy. [PERSON] et al. [31] proposed a method based on the multi-objective artificial bee colony algorithm (MOTOCP) to maximize cache hit rate and minimize service delay. The method gradually improves the algorithm's performance by balancing multiple objectives through Pareto optimality. However, the aforementioned methods are either based on deep learning or heuristic algorithms. The former cannot guarantee an offloading solution within polynomial time, while the latter cannot guarantee that each offloading solution obtained will be optimal or near-optimal. In most practical applications, tasks are often dependent on specific services for execution. Additionally, many tasks require low-latency performance. Due to the distance to the server, transmission delay inevitably tends to be relatively high. In the context of approximate algorithms for wireless networks, [PERSON] et al. [34; 35] considered the sum-rate maximization problem by Perron-Frobenius theory. They focused on finding approximately optimal solutions by studying the solutions to two related problems: the signal-to-interference-plus-noise ratio (SINR) approximation power control (SAPC) problem and the max-min weighted SINR problem. For the SAPC problem, the authors proposed an asynchronous iterative algorithm, while for the max-min weighted SINR problem, they developed a fast, two-timescale algorithm and derived a synchronous algorithm. However, the approximation ratio in their work depends on the parameters of the problem instances, and the problem model they consider differs significantly from ours. For the use of approximation algorithms in the task offloading problem, [PERSON] et al. [36] considered the energy minimization problem under latency constraints with linear task dependency, and proposed the Lagrangian Relaxation Based Aggregated Cost algorithm to obtain both optimal and approximate solutions to the constrained optimization problem. However, unlike our approach, they did not provide a clear approximation ratio. [PERSON] et al. [37], aiming to jointly optimize the tradeoff between latency and energy in a multi-server MEC system, proposed an approximation algorithm based on the Markov approximation framework to solve the proposed problem. Additionally, [PERSON] et al. [38] introduced an energy-latency-aware task offloading and approximation computation problem. They decomposed the original problem into three subproblems and used duality techniques and convex optimization to obtain an approximate solution. Although both of these studies proposed approximation algorithms to address the task offloading problem, their approximation ratios are dependent on the objective values of the problem. In contrast, by setting the value of \(\epsilon\), we can guarantee a constant approximation ratio of \((1+\epsilon)\) for each case, which provides a better approximation guarantee in our research. To the best of our knowledge, only [PERSON] [32] has proposed a polynomial-time approximation algorithm to address the delay-minimization problem under a cost constraint. The approximation ratio of this algorithm is also \((1+\epsilon)\), which is the same as ours. However, their approach does not take into account constraints related to service caching, which may lead to situations where offloading solutions are not feasible.