Design an algorithm to solve the traveling salesman problem, which involves finding the shortest possible route through a given set of cities, visiting each city exactly once, and returning to the starting city. Explain your approach and any assumptions made.

The Traveling Salesman Problem (TSP) is an NP-hard problem, which means that there is no known algorithm to solve it optimally in polynomial time. However, we can design approximation algorithms or heuristics to find near-optimal solutions for the TSP.

One such heuristic is the Nearest Neighbor Algorithm. This algorithm doesn't guarantee the optimal solution but usually provides a good approximation. Here's how the Nearest Neighbor Algorithm works:

1. Start at any city as the current city.
2. Find the nearest unvisited city and move to it.
3. Mark the new city as visited.
4. Repeat steps 2-3 until all cities are visited.
5. Return to the starting city.

This algorithm has a time complexity of O(n^2), where n is the number of cities.

Another approach is using the Minimum Spanning Tree (MST) based heuristic. The idea is to first construct an MST and then traverse it using Depth First Search (DFS). This algorithm also doesn't guarantee the optimal solution but generally performs better than the Nearest Neighbor Algorithm. Here's how the MST-based heuristic works:

1. Choose any city as the root and create a minimum spanning tree (MST) using Prim's or Kruskal's algorithm.
2. Perform a depth-first search (DFS) traversal on the MST.
3. As you visit each city during DFS, add it to the tour.
4. After visiting all cities, return to the starting city.

The time complexity of this algorithm is O(n^2) due to the construction of the MST.

Assumptions made:
1. The distance between cities satisfies the triangle inequality, i.e., the direct path between two cities is never longer than the sum of the paths through other cities.
2. The distances between cities are symmetric, i.e., the distance from city A to city B is the same as the distance from city B to city A.

These algorithms do not guarantee an optimal solution, but they are relatively fast and provide a good approximation for the TSP. In practice, more advanced techniques like branch-and-bound or genetic algorithms can be used to find even better approximations or sometimes even optimal solutions depending on the problem size and constraints.