1 ©D.Moshkovitz Complexity The Traveling Salesman Problem

2 ©D.Moshkovitz Complexity The Mission: A Tour Around the World

3 ©D.Moshkovitz Complexity The Problem: Traveling Costs Money 1795$

4 ©D.Moshkovitz Complexity Introduction Objectives: –To explore the Traveling Salesman Problem. Overview: –TSP: Formal definition & Examples –TSP is NP-hard –Approximation algorithm for special cases –Inapproximability result

5 ©D.Moshkovitz Complexity TSP Instance: a complete weighted undirected graph G=(V,E) (all weights are non-negative). Problem: to find a Hamiltonian cycle of minimal cost

6 ©D.Moshkovitz Complexity Polynomial Algorithm for TSP? What about the greedy strategy: At any point, choose the closest vertex not explored yet?

7 ©D.Moshkovitz Complexity The Greedy $trategy Fails   

8 ©D.Moshkovitz Complexity The Greedy $trategy Fails   

9 ©D.Moshkovitz Complexity TSP is NP-hard The corresponding decision problem: Instance: a complete weighted undirected graph G=(V,E) and a number k. Problem: to find a Hamiltonian path whose cost is at most k.

10 ©D.Moshkovitz Complexity TSP is NP-hard Theorem: HAM-CYCLE  p TSP. Proof: By the straightforward efficient reduction illustrated below: HAM-CYCLETSP k=|V| verify!

11 ©D.Moshkovitz Complexity What Next? We’ll show an approximation algorithm for TSP, with approximation factor 2 for cost functions that satisfy a certain property.

12 ©D.Moshkovitz Complexity The Triangle Inequality Definition: We’ll say the cost function c satisfies the triangle inequality, if  u,v,w  V : c(u,v)+c(v,w)  c(u,w)

13 ©D.Moshkovitz Complexity Approximation Algorithm 1. Grow a Minimum Spanning Tree (MST) for G. 2. Return the cycle resulting from a preorder walk on that tree. COR(B)

14 ©D.Moshkovitz Complexity Demonstration and Analysis The cost of a minimal Hamiltonian cycle  the cost of a MST 

15 ©D.Moshkovitz Complexity Demonstration and Analysis The cost of a preorder walk is twice the cost of the tree

16 ©D.Moshkovitz Complexity Demonstration and Analysis Due to the triangle inequality, the Hamiltonian cycle is not worse.

17 ©D.Moshkovitz Complexity The Bottom Line optimal HAM cycle MST preorder walk our HAM cycle  = ½·  ½·

18 ©D.Moshkovitz Complexity What About the General Case? We’ll show TSP cannot be approximated within any constant factor  1 By showing the corresponding gap version is NP-hard. COR(B) 528

19 ©D.Moshkovitz Complexity gap-TSP[  ] Instance: a complete weighted undirected graph G=(V,E). Problem: to distinguish between the following two cases: There exists a Hamiltonian cycle, whose cost is at most |V|. The cost of every Hamiltonian cycle is more than  |V|.

20 ©D.Moshkovitz Complexity Instances min cost |V|  |V|    0  +1  0 0 1

21 ©D.Moshkovitz Complexity What Should an Algorithm for gap-TSP Return? |V|  |V| YES!NO! min cost gap DON’T-CARE...

22 ©D.Moshkovitz Complexity gap-TSP & Approximation Observation: Efficient approximation of factor  for TSP implies an efficient algorithm for gap-TSP[  ].

23 ©D.Moshkovitz Complexity gap-TSP is NP-hard Theorem: For any constant  1, HAM-CYCLE  p gap-TSP[  ]. Proof Idea: Edges from G cost 1. Other edges cost much more.

24 ©D.Moshkovitz Complexity The Reduction Illustrated HAM-CYCLEgap-TSP 1  |V| Verify (a) correctness (b) efficiency

25 ©D.Moshkovitz Complexity Approximating TSP is NP- hard gap-TSP[  ] is NP-hard Approximating TSP within factor  is NP-hard

26 ©D.Moshkovitz Complexity Summary We’ve studied the Traveling Salesman Problem (TSP). We’ve seen it is NP-hard. Nevertheless, when the cost function satisfies the triangle inequality, there exists an approximation algorithm with ratio-bound 2. 

27 ©D.Moshkovitz Complexity Summary For the general case we’ve proven there is probably no efficient approximation algorithm for TSP. Moreover, we’ve demonstrated a generic method for showing approximation problems are NP-hard. 