2. There is a Postman who delivers mail to a certain neighborhood of streets. The postman is unwilling to walk far so he wants to find the shortest route possible through the whole neighborhood. He must start and end at the same spot and walk down each street at least once. How can he accomplish this task?

3.  First proposed in 1962 by Mei-Ku Kuan, Chinese mathematician. › Named by Alan Goldman of the U.S. National Bureau of Standards.  Kuan gave an algorithm in 1968: › It was designed to give the shortest route that was required for any network

4.  Graph  Euler Circuit  Degree (of a vertex)  Weighted Graph

5.  Euler's Theorem › A connected graph G possesses an Euler circuit if and only if G contains exactly zero vertices of odd degree.

6.  Proof of Euler's theorem : › Assume that G has zero vertices of odd degree. › It can then be shown that this is a necessary and a sufficient condition for an Euler circuit to exist. › It is necessary because any Euler circuit drawn on the graph must always enter a vertex through some edge and leave through another and all edges on the graph must be used exactly once. › Thus, an even number of incident edges is required for every vertex on the graph.

7.  What if the neighborhood (graph) contains odd degree vertices?  Chinese Postman Problem Algorithm: › Lemma: The number of vertices of odd degree in an undirected graph G is always even.

8. 1) Identify all vertices of odd degree in graph G. Say there are m of them, where m is an even number according to the lemma. 2) Find a minimum-length pair wise matching of the m odd- degree vertices and identify the m/2 shortest paths between the two vertices composing each of the m/2 pairs. 3) For each of the pairs of odd-degree vertices in the minimum-length pair wise matching found in Step 2, add to the graph G the edges of the shortest path between the two vertices in the pair. The graph G 1 thus obtained contains no vertices of odd degree. 4) Find an Euler circuit on G 1. This Euler circuit is an optimal solution to the Chinese Postman Problem on the original graph G. The length of the optimal circuit is equal to the total length of the edges in G plus the total length of the edges in the minimum-length matching.

9.  Using this weighted graph. The graph contains four vertices of odd degree (a, b, d, and e). Thus, there are three possible pair wise matchings of the odd-degree vertices: a-b and d-e; a-d and b-e; and ace and bcd.

10.  The augmented networks that would result from the addition of the artificial edges corresponding to each of these three matchings are shown to the right.  Of the three matchings, the optimal is obviously the one shown in (c). It adds only 12 units of length to the circuit as opposed to 16 for (b)and 20 for(d).  Thus, the graph shown on (c) should be the result of the procedure outlined in Steps 2 and 3 of Algorithm.

11.  a b e d a c e b c d a  8+6+8+6+5+5+6+5+5+6=60

12.  Deliveries  Site Seeing  Collecting Money  Shopping

13.  http://web.mit.edu/urban_or_book/www /book/chapter6/6.4.1.html http://web.mit.edu/urban_or_book/www /book/chapter6/6.4.1.html