The literature that I have reviewed shows examples of calculations of known approximation algorithms such as the Christofides' algorithm for the TSP. However, I have not been able to find information regarding the calculation of the approximation factor of an algorithm that has a "complex" loop inside. Is this possible? Does someone know a book or article that does this? Should I try another approach?
For more context, I'm trying to prove an approximation factor for an algorithm for the TSP that creates a k-partition of vertices and then iterates, bringing together 2 disjoint sets of vertices of the partition by iteration; until all disjoint sets of vertices are brought together, resulting in a tour in the original graph. Note that at each iteration the k-partition changes, meaning that the next iteration will have different sets of vertices to join.
If something is not clear, please let me know!