## 3 Network Architecture and System Model
### Radroo-Milp Algorithm  
In this paper, the task offloading decision is an integer programming problem, and the additional computational resources requested from CDC follow linear programming, which makes our problem (19) a MILP problem. Hence, we used the branching implicit enumeration method to obtain the optimal offloading decision based on the uncertainty of computational resources and the minimum average cost.  
In the above SDP problem, the X-matrix variables, 0-1 integer variables, the Y-vector variables, and continuous variables, are included. This makes the original problem a mixed-integer linear formulation, which poses a challenge in terms of direct solvability. To simplify the problem, we relax it by treating the discrete variable \(X\) as a continuous variable ranging from 0 to 1, i.e., \(0\leqslant x_{ij}\leqslant 1,\forall x_{ij}\in X\).  
In fact, the above SDP problem can be solved by using the SDPT3 solver through the CVX package in MATLAB. The original MILP problem then needs to be solved; here, we use the branching implicit enumeration method to solve it. This method is a specialized branch and bound technique for 0-1 integer problems, leveraging the fact that the variables can only take values of 0 or 1, allowing for branching for the purpose of hidden enumeration. Each task is divided into m sub-nodes, i.e., each task can only be offloaded to and completely offloaded on a single node.  
Algorithm 1 presents the procedure for calculating the minimum cost of computation offloading. First of all, tasks should be sorted according to the number of CPU cycles they require, which determines the cost spent to a large degree. The solution to the relaxed LP problem is then obtained. Based on this solution, the implicit enumeration method is applied to solve the branching subproblem. During the branch and bound process, the two nodes with the highest values in the \(X_{i}\) vector, representing the most probable nodes for offloading the current task, are selected at each iteration. The process involves constant branching, resulting in a gradually increasing lower bound, and constant delimitation, resulting in a gradually decreasing upper bound.  
```
Input: the convex probability distributional set \(\mathcal{M}^{\sim}\), stochastic sample space \(\mathcal{F}^{\sim}\), trade-off coefficient \(\lambda\), confidence level \(\alpha\), ambiguous set parameters \(\gamma_{1}\) and \(\gamma_{2}\); Output: offload decision \(X\), optimal cost value to satisfy distribution robustness;  Solve the MILP formulation SDP problem with a confidence level of \(\alpha\), to obtain the optimal values of the continuous variables \(X\) and the optimal cost value; for\(i\) from 0 to \(N-1\)do  Sort(\(\mathcal{N}\)) \(\left[opt,\ X\right]=\max(X_{i})\) \(\left[x1_{ij},x2_{ij}\right]=max2(X)\) if\(SDP(X^{\prime},x^{\prime}_{i,j1}=1)\ \leq\ SDP(X^{\prime},x^{\prime}_{i,j2}=1)\)then \(x^{\prime}_{i,j1}=1\) \(min\_cost[i]=SDP(X^{\prime},X^{\prime}_{i,j1}=1)\) else \(x^{\prime}_{i,j2}=1\) \(min\_cost[i]=SDP(X^{\prime},X^{\prime}_{i,j2}=1)\) endif endfor if\(min\_cost[N-1]\!=\ NaN\)then  return\(X^{\prime}\), \(min\_cost[N-1]\) endif
```  
**Algorithm 1** Cost-based MILP algorithms for the DRO task offloading problem.