Proof: Given a graph G=(V,E), define a distance table on V as follows: Theorem
Contradiction Argument Suppose r-approximation exists. Then we have a polynomial-time algorithm to solve Hamiltonian Cycle as follow: r-approximation solution < r |V| if and only if G has a Hamiltonian cycle
Special Case Traveling around a minimum spanning tree is a 2-approximation. Theorem
Minimum spanning tree + minimum-length perfect matching on odd vertices is 1.5- approximation Theorem
Minimum perfect matching on odd vertices has weight at most 0.5 opt.