Approximation Algorithm of Traveling Salesman Problem

1. Approximation Algorithm of Traveling Salesman Problem By Lin, Jr-Shiun&Chio, Jae Sung Speaker : Lin, Jr-Shiun

2. What is TSP? • Design the shortest, or minimal cost, route for a salesman who wants to travel EVERY cities ONLY ONCE and ,lastly, backs to home city. • In graph, we need to find a tour that starts at a node, visits every other node exactly once, and returns to the starting node.

3. What is TSP • Definition: Find a path through a weighted graph which starts and ends at the same vertex, includes every other vertex exactly once, and minimizes the total cost of edges. (NIST) • TSP is classify as NP-complete problem, that means no polynomial algorithm can guarantee to come within countable(?) times of the shortest tour.

4. Christofides Algorithm • Nicos Christofides find a way to slove TSP problem (1976): • 1: find a minimum spanning tree T. • 2: find a perfect matching M from nodes with odd degree. • 3: combine the edges of M and T to graph G • 4: find an Euler cycle in G by skipping vertices already seen

5. Christofides Algorithm

6. Even nodes with odd degree?? • How can I know that I always have even number of nodes, which have odd degree, for me to do the MATCHING? • From handshaking lemma: in any graph, the sum of all the vertex-degrees is equal to twice the number of edges.

7. Even nodes with odd degree?? • let S (d) = 2m, where m= number of edges. Therefore S(d) is even. • Let Se(d) to be the sum of degrees of the vertices which have even degree, Se(d) is also even. • Therefore S(d)-Se(d) = 2k, k=1,2,…, which means that the sum of degrees of the vertices which have odd degree each is also an even number. Thus there are even numbers of vertices which have odd number of degree. (Dr. Giri Narasimhan)

8. Metric TSP • To convert general TSP to Metric TSP, we need to add weight to each edges. • Why Metric? • in metric TSP, it suffices to find a cyclic tour that visits each vertex at least once, rather than exactly once. Using the triangle inequality, we can always shortcut a vertex that is visited more than once without increasing the cost of the tour. • Metric TSP can be approximated within a factor of 3/2.

9. 3/2-approximation • Let c(TSP) be the cost of the minimum TSP tour, c(MST) be the cost of the MST, and c(M) be the cost of the matching M. • Clearly, c(MST) <=(1 – 1/n)c(TSP ), because n - 1 edges of a TSP are a spanning tree of the graph. Moreover, c(M) <=(TSP)/2 • Because of the triangle inequality, the minimum tour on a elements subset of vertices has cost at most c(TSP).

10. 3/2-approximation (cont.) • If this subset is even cardinality, then the positive minimum cost matching on it has cost at most c(TSP)/2, because the minimum tour is then a disjoint union of two matchings. The set of vertices of odd degree in a spanning tree is of even cardinality.

11. 3/2-approximation (cont.) • An Euler tour exists on the multiset of edges of MST and M because each vertex in the edge induced subgraph has even degree. The cost of the Euler tour is at most [c(MST ) + c(M)] <=(3/2 – 1/n)c(TSP ). (Uriel Feige) • As n goes to infinity, it becomes 3/2 apporximation.