## 4 Problem Modeling and Solution
### A Ranking Method Based on Path Contribution  
This section introduces a novel path contribution ranking approach to identify and select paths based on their contribution to the overall transmission capacity, rather than relying solely on simple metrics like hop count or bottleneck link bandwidth. The method works by selecting paths that maximize overall transmission capacity and ranking them according to the order in which they are selected. This process eliminates redundant paths and ensures that high-contribution paths are prioritized. The earlier a path is selected, the greater its contribution and the higher its rank. These path rankings then serve as a reference for the subsequent multi-topology construction. A mathematical model for this problem is established as follows:  
Objective Function: Maximize the flow capacity \(F_{t}\) that can be transmitted through multiple paths from node \(s\) to node \(t\) in a unit of time:  
\[Maximize\ F_{t}=\sum_{p\in P_{t}}f_{p} \tag{5}\]  
Constraints:  
Link Bandwidth Constraint: For each link \(l\), the amount of traffic that can be transmitted in a unit of time must not exceed its available bandwidth. \[\sum_{p\in P_{t}l\in L_{p}}f_{p}\leq b_{l},\forall l\in L \tag{6}\]  
Non-Negativity Constraint: Ensure that the amount of traffic is non-negative. \[f_{p}\geq 0,\forall p\in P_{t} \tag{7}\]  
where \(f_{p}\) may equal 0, indicating that path \(p\) is not in use. From this mathematical model, we can see that, compared to the overall problem model, we have not considered the flow entry constraints of the switches as they are difficult to define using the variable \(f_{p}\). We will address this issue later. Additionally, in the overall problem model, the objective is to maximize the average transmission capacity from the source node to all destination nodes. In contrast, this problem aims to maximize the transmission capacity from the source node to a single destination node. Therefore, the number of variables in this problem is the number of paths between one node pair, while the number of variables in the overall problem is the sum of the number of paths of all node pairs. This problem is a subproblem of the original problem. In most multipath selection schemes, such as those in [22; 23], if two paths share an overlapping node or link, one of them is typically removed. This aims to avoid the situation where both paths become unavailable due to the failure of the shared node or link. However, this requirement is too strict for maximizing network transmission capacity in this algorithm. The bottleneck link \(l_{bn}\) of path \(p\) is defined as follows:  
\[l_{bn}=argmin_{l}(b_{l}),\hskip 28.452756 ptl\in L_{p} \tag{8}\]If the shared link of two paths is not the bottleneck link of them, including the other path can still increase network transmission capacity. Therefore, the focus should be on ensuring that the selected paths do not overlap on bottleneck links; that is, if multiple paths have the same bottleneck link, only one of them can be selected. In this case, the shorter path is retained, as shorter paths generally result in lower data transmission delays. The details of the ranking method are given below Algorithm 1.