Cs 598csc: Approximation Algorithms 1 the Traveling Salesperson Problem (tsp) 1.1 Tsp in Undirected Graphs

In the Traveling Salesperson Problem, we are given an undirected graph G = (V,E) and cost c(e) > 0 for each edge e ∈ E. Our goal is to find a Hamiltonian cycle with minimum cost. A cycle is said to be Hamiltonian if it visits every vertex in V exactly once. TSP is known to be NP-complete, and so we cannot expect to exactly solve TSP in polynomial time. What is worse, there is no good approximation algorithm for TSP unless P = NP . This is because if one can give a good approximation solution to TSP in polynomial time, then we can exactly solve the NP-Complete Hamiltonian cycle problem (HAM) in polynomial time, which is impossible unless P = NP . Recall that HAM is the decision problem of deciding whether the given graph G = (V,E) has a Hamiltonian cycle or not.