Travelingsalesman ProblemIn the traveling salesman Problem, a salesman must visits n cities. We can say that salesman wishes to make a tour or Hamiltonian cycle, visiting each city exactly once and finishing at the city he starts from. There is a nonnegative cost c (i, j) to travel from the city i to city j. The goal is to find a tour of minimum cost. We assume that every two cities are connected. Such problems are called Travelingsalesman problem (TSP). We can model the cities as a complete graph of n vertices, where each vertex represents a city. It can be shown that TSP is NPC. If we assume the cost function c satisfies the triangle inequality, then we can use the following approximate algorithm. Triangle inequalityLet u, v, w be any three vertices, we have One important observation to develop an approximate solution is if we remove an edge from H*, the tour becomes a spanning tree. Travelingsalesman Problem Intuitively, ApproxTSP first makes a full walk of MST T, which visits each edge exactly two times. To create a Hamiltonian cycle from the full walk, it bypasses some vertices (which corresponds to making a shortcut)
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