## Key Ideas

• The problem of finding this minimum-cost Hamiltonian circuit is called the traveling salesman problem (TSP). It is a common goal in the practice of operations research.
• We use a variety of heuristic (or “fast”) algorithms to find solutions to the TSP. Some are very good, even though they may not be optimal. Heuristic algorithms come close enough to giving optimal solutions to be important for practical use.
• The nearest-neighbor algorithm repeatedly selects the closest neighboring vertex not yet visited in the circuit (with a choice of edges, choose the one with the smallest weight), and returns to the initial vertex at the end of the tour.
• The sorted-edges algorithm is another heuristic algorithm that can lead to a solution that is close to optimal. This algorithm attempts to solve the TSP where the edges added to the circuit being built up are selected in order of increasing cost, but no edge is chosen that would prevent a Hamiltonian circuit from forming. These edges must all be connected at the end, but not necessarily at earlier stages.