The Traveling Salesman Problem (TSP) remains an NP-hard problem in computer science, which means that no polynomial-time solution (for all instances) is known, nor is it known that no such solution exists.

**State of the Art (SOTA) for Exact Solutions**:
The state of the art for exact solutions to TSP often involves the use of cutting plane methods, branch and bound, or branch and cut techniques. One of the most popular exact algorithms for TSP is the branch-and-cut algorithm, which starts with a linear programming relaxation and iteratively adds cuts (constraints) to the problem until the solution describes a tour. The Concorde TSP solver, which can solve large instances with tens of thousands of cities, uses this method.

**Asymptotic Complexity**:
The exact complexity for solving TSP with these methods is hard to pin down because it's heavily dependent on the instance specifics, such as the number of cities and their distributions. However, in the worst case, it's still exponential.

**Educational Solutions with Good Asymptotic Complexity**:
If you're looking for solutions that are easier to explain and implement for educational purposes, you might consider approximation algorithms or heuristics. These methods don't guarantee an optimal solution, but they run in polynomial time and often produce solutions that are "close" to optimal. Here are a couple:

1. **Christofides' Algorithm**: This is an approximation algorithm that guarantees a solution within 3/2 times the optimal. It works as follows:
    - Find a minimum spanning tree (MST) of the graph.
    - Find the set of vertices with odd degree in the MST.
    - Compute a minimum-weight perfect matching on this set.
    - Combine the MST and the matching to get a connected multi-graph where every vertex has even degree.
    - Extract an Eulerian circuit from this multi-graph.
    - Convert the Eulerian circuit to a Hamiltonian cycle by skipping repeated vertices.
   
   The complexity of this algorithm is mostly dominated by the steps of finding the MST and the minimum-weight perfect matching. The overall complexity is \(O(n^2 \log n)\) or better, depending on the specific algorithms used for the MST and the matching.

2. **Nearest Neighbor Algorithm**: A heuristic approach where one starts at a city and repeatedly visits the nearest unvisited city until all cities are visited. This does not guarantee any approximation factor but is simple to understand and implement. The complexity is \(O(n^2)\) for a straightforward implementation.

3. **2-Opt Heuristic**: This is a local search method. Start with any tour. Iteratively select two edges and see if swapping them results in a shorter tour. Continue until no such swaps improve the tour. This is also not guaranteed to find an optimal solution but is simple and often used in practice to refine other solutions. The complexity is \(O(n^2)\) for a single pass, but it might take multiple passes to converge.

For educational purposes, introducing the problem with the Nearest Neighbor algorithm can be helpful due to its simplicity. After that, the Christofides' algorithm can give a more structured approach with an approximation guarantee.
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