Minimum Spanning Tree Solving TSP for Metric Graphs

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

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

Special case: Metric TSP 3

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

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. 5

Bridge Finding in Reduced graph 6

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. 7

Implementation 8

Results : MST Running Time 9

Results : TSP Running Time 10

Results : Heuristic vs. Optimum Tour Cost 11