1. 1 Minimum Spanning Tree: Solving TSP for Metric Graphs using MST Heuristic Soheil Shafiee Shabnam Aboughadareh

2. 2 Problem definition: A Salesman wishes to travel around a given set of cities, and return to the beginning, covering the smallest total distance

3. 3 Special case: Metric TSP

4. 4 Approximation Algorithms - Constructing minimum spanning tree. - Duplicating all edges in MST. - Constructing Eulerian tour using Fluery’s algorithm. - Converting Euerian tour to Hamiltonian tour. 4 0 1 2 4 3

5. 5 Fluery’s Algorithm - Pick one vertex as starting vertex. - Pick an edge from the picked vertex. (No bridge in reduced graph!) - Mark the edge as used edge to be a reminder that we can’t traverse it again. - Travel that edge and come into next vertex. - Repeating the procedure above until all edges will be traversed * Reduced graph is the original graph minus used edges. * Bridge is an edge whose deletion will increase the number of connected components in graph.

6. 6 Bridge Finding in Reduced graph

7. 7 Analysis 2 approximation algorithm MST < Eulerian Tour = 2 * MST <= 2.0 TSP Is there any better solution? Yes, Christofide Algorithm. (1.5 approximation Algorithm) Instead of duplicating MST edges combine MST with Minimum Weight Perfect Matching.

8. 8 Implementation

9. 9 Results : MST Running Time

10. 10 Results : TSP Running Time

11. 11 Results : Heuristic vs. Optimum Tour Cost