To solve the fractional path routing problem efficiently, we can use a randomized algorithm based on the multiplicative weights update method, utilizing the experts framework. Here's how to construct such an algorithm that runs in ( O\left( \frac{k m \log^2 m}{\epsilon^2} \right) ) time and achieves the desired approximation ratio.

Algorithm Overview:

Initialization:
Let ( T = \left\lceil \frac{8k \ln k}{\epsilon^2} \right\rceil ).
Initialize edge lengths ( \ell_e^{(0)} = 1 ) for all edges ( e \in E ).
Initialize edge congestion counts ( c_e^{(0)} = 0 ) for all edges ( e \in E ).
Main Loop (for ( t = 1 ) to ( T )):
Randomly select a terminal pair ( (s_{i_t}, t_{i_t}) ) uniformly from the ( k ) terminal pairs.
Compute edge weights ( w_e^{(t)} = e^{\epsilon c_e^{(t-1)}} ).
Find the shortest path ( P_t ) from ( s_{i_t} ) to ( t_{i_t} ) using weights ( w_e^{(t)} ). This can be done in ( O(m) ) time using Dijkstra's algorithm with non-negative edge weights.
Route one unit of flow along ( P_t ).
Update edge congestion counts:
[
c_e^{(t)} = c_e^{(t-1)} + \begin{cases} 1, & \text{if } e \in P_t \ 0, & \text{otherwise} \end{cases}
]
Edge Length Update (Implicit in Edge Weights):
The edge lengths ( \ell_e^{(t)} ) can be considered updated via ( \ell_e^{(t)} = \ell_e^{(t-1)} e^{\epsilon \delta_{e \in P_t}} ), but we don't need to store ( \ell_e^{(t)} ) explicitly.
Constructing the Routing Solution:
After ( T ) iterations, for each terminal pair ( (s_i, t_i) ), the total number of times it was selected is approximately ( \frac{T}{k}(1 \pm \epsilon) ) with high probability due to the given lemma.
The fractional flow for each terminal pair is given by the frequency of its routing over ( T ) iterations, divided by ( T ).
The flow on each edge ( e ) is ( f_e = \frac{c_e^{(T)}}{T} ).
Analysis:

Runtime:
Each iteration involves selecting a terminal pair (constant time), computing weights (linear in ( m )), and finding a shortest path (also ( O(m) ) with appropriate data structures).
Total runtime is ( O(Tm) = O\left( \frac{k m \ln k}{\epsilon^2} \right) ).
Approximation Guarantee:
Expected Congestion:
The expected congestion on any edge ( e ) is ( E[c_e^{(T)}] = \sum_{t=1}^T P(e \in P_t) ).
Since terminal pairs are selected uniformly at random, and shortest paths are computed with increasing weights on congested edges, the algorithm discourages overuse of any single edge.
Using the multiplicative weights update method's analysis, we can show that:
[
E[c_e^{(T)}] \leq (1 + \epsilon) C^* T + O\left( \frac{\ln m}{\epsilon} \right),
]
where ( C^* ) is the optimal congestion.
Dividing both sides by ( T ) gives:
[
E\left[ \frac{c_e^{(T)}}{T} \right] \leq (1 + \epsilon) C^* + O\left( \frac{\ln m}{\epsilon T} \right).
]
Since ( T ) is large (( T = O\left( \frac{k \ln k}{\epsilon^2} \right) )), the additive term ( O\left( \frac{\ln m}{\epsilon T} \right) ) becomes ( O(\epsilon) ).
High Probability Guarantee:
By the given lemma, with probability at least ( 1 - \frac{1}{k} ), each terminal pair is selected ( \frac{T}{k}(1 \pm \epsilon) ) times.
This ensures that the demands for all terminal pairs are approximately met equally, which is important for balancing the routing and preventing congestion on specific edges.
Conclusion:

The proposed randomized algorithm efficiently approximates the fractional path routing problem. It leverages the multiplicative weights update method to adjust edge weights dynamically based on congestion, routing flow along less congested paths over time. By selecting terminal pairs uniformly at random and routing the same amount of flow for each, the algorithm balances the load across the network.

The algorithm achieves the desired runtime of ( O\left( \frac{k m \ln k}{\epsilon^2} \right) ) and ensures that the expected maximum congestion satisfies ( E[c_{\text{max}}] \leq (1 + \epsilon) C^* + O(\epsilon) ), where ( C^* ) is the optimal congestion.

Note: This algorithm assumes that all edge capacities are normalized appropriately and that ( k \leq n ). The constants in the ( O )-notation may depend on the specific details of the implementation and underlying data structures used for efficiency.