What is the travelling salesman problem in graph theory?

The travelling salesman problem is a problem in graph theory that seeks to find the shortest possible route that visits every vertex exactly once.

The travelling salesman problem (TSP) is a classic problem in graph theory that seeks to find the shortest possible route that visits every vertex exactly once. It is a well-known NP-hard problem, meaning that it is computationally difficult to solve for large instances. The problem is often stated as follows: given a complete weighted graph G = (V, E), where V is the set of vertices and E is the set of edges, find a Hamiltonian cycle of minimum weight. A Hamiltonian cycle is a cycle that visits every vertex exactly once.

There are several algorithms that can be used to solve the TSP, including brute force, dynamic programming, and heuristics. Brute force involves checking every possible permutation of vertices, which is computationally infeasible for large graphs. Dynamic programming involves breaking the problem down into smaller subproblems and solving them recursively. Heuristics are approximate algorithms that provide a good solution but do not guarantee optimality.

One popular heuristic algorithm for the TSP is the nearest neighbour algorithm. This algorithm starts at a random vertex and repeatedly visits the nearest unvisited vertex until all vertices have been visited. Another heuristic algorithm is the 2-opt algorithm, which involves iteratively removing two edges from the cycle and reconnecting them in a different way if it results in a shorter cycle.

In conclusion, the travelling salesman problem is a challenging problem in graph theory that seeks to find the shortest possible route that visits every vertex exactly once. While there are several algorithms that can be used to solve the problem, it remains a difficult problem for large instances.