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That is, G is a complete graph on the set V of vertices, and the function w assigns a nonnegative real weight to every edge of G. Then the algorithm can be described in pseudocode as follows. Let O be the set of vertices with odd degree in T. By the handshaking lemma , O has an even number of vertices. Find a minimum-weight perfect matching M in the induced subgraph given by the vertices from O. Combine the edges of M and T to form a connected multigraph H in which each vertex has even degree.
Form an Eulerian circuit in H. Make the circuit found in previous step into a Hamiltonian circuit by skipping repeated vertices shortcutting.
To prove this, let C be the optimal traveling salesman tour. Next, number the vertices of O in cyclic order around C, and partition C into two sets of paths: the ones in which the first path vertex in cyclic order has an odd number and the ones in which the first path vertex has an even number.
Each set of paths corresponds to a perfect matching of O that matches the two endpoints of each path, and the weight of this matching is at most equal to the weight of the paths. Since these two sets of paths partition the edges of C, one of the two sets has at most half of the weight of C, and thanks to the triangle inequality its corresponding matching has weight that is also at most half the weight of C.
All remaining edges of the complete graph have distances given by the shortest paths in this subgraph.
CHRISTOFIDES TSP PDF
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Traveling Salesman Algorithms