Traveling Salesman Problem

1. Traveling Salesman Problem DongChul Kim HwangRyol Ryu

2. Introduction • Research Goal • What you will learn …

3. What Is TSP? • Shortest Hamiltonian cycle (i.e. tour). • Grow exponentially • Current Definition of TSP: Given a number of cities and the costs of traveling from one to the other, what is the cheapest round trip route that visits each city and then returns to the starting city?

4. History of TSP • Irish mathematician Sir William Rowan Hamilton and the British mathematician Thomas Penyngton Kirkman Hamilton’s Iconsian game

5. History of TSP (1) • The general form of TSP appeared in 1930s by Karl Menger in Vienna and Havard. • A breakthrough by George Dantzig, Ray Fulkerson, and Selmer Johnson in1954. • 49 - 120 – 550 - 2,392 - 7,397 – 19,509 cities • From year 1954 to year 2001. • 24,098 cities by David Applegate, Robert Bixby, Vasek Chvatal, William Cook, and Keld Helsgaun in May 2004.

6. Branch & Lower Bound • An algorithmic technique to find the optimal solution by keeping the best solution found so far. • Standard to measure performance of TSP heuristics.

7. 2.0 TSP Approximation Algorithm • Double Minimum Spanning Tree • Return a tour of length at most twice the shortest tour. • Algorithm: 1. Construct the minimal spanning tree 2. Duplicate all its edges. This gives us an Euler cycle. 3. Traverse the cycle, but do not visit any node more than once, taking shortcuts when it passes a visited node.

8. 2.0 TSP Approximation Algorithm (2) • 2.0? TSP • 2.0 is TSP version number? • Tour of length is at most twice the length of MST. MST < Euler Cycle = 2 * MST <= 2.0 TSP

9. 1.5 TSP Approximation Algorithm(Known as Christofides Heuristics) • Professor Nicos Christofides extended the 2.0 TSP and published that the worst-case ratio of the extended algorithm was 3/2. • Algorithm: 1. Compute MST graph T. 2. Compute a minimum-weighted matching graph M. 3. Combine T and M as edge set and Compute an Euler Cycle. 4. Traverse each vertex taking shortcuts to avoid visited nodes.

10. 1.5 TSP Approximation Algorithm (2)(Known as Christofides Heuristics) • What is a Minimum-weighted Matching? It creates a MWM on a set of the nodes having an odd degree. • Why odd degree? Property of Euler Cycle • Why 1.5 TSP? MST < Euler Cycle = MWM+MST <= 1.5 TSP (MWM = ½ MST)

11. 1.5 TSP Approximation Algorithm (3)(Known as Christofides Heuristics) • Minimum-weighted Matching example MWM = ½ MST

12. Matching Algorithm • Smile Matching Algorithm • Bad matching • Better matching • Fixed Bad matching problem.

13. Matching Algorithm (2) • Improved Smile Matching Algorithm • Choose the two nodes in the farthest • distance

14. Matching Algorithm (3) • Improved Smile Matching Algorithm 2. Each end node is connected to the node in the closest distance.

15. PTAS Algorithm(Polynomial Time Approximation Scheme) • The status of Euclidean TSP remained open. • PTAS = Polynomial time algorithm, for each c > 1, can approximate the problem within a factor 1 + 1/c.

16. PTAS Algorithm (2) • The central idea of the PTAS is that the plane can be recursively partitioned and by using a dynamic programming on Quadtree, it finds an optimal tour.

17. Other approximation schemes • Minimum Steiner Tree • K-TSP and K-MST • Min Cost Perfect Matching

18. Demonstration