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1 ©D.Moshkovitz Complexity The Traveling Salesman Problem

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

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

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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

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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. 3 43 2 5 1 10

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

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7 ©D.Moshkovitz Complexity The Greedy $trategy Fails 5 0 3 1 12 10 2

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8 ©D.Moshkovitz Complexity The Greedy $trategy Fails 5 0 3 1 12 10 2

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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.

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10 ©D.Moshkovitz Complexity TSP is NP-hard Theorem: HAM-CYCLE p TSP. Proof: By the straightforward efficient reduction illustrated below: HAM-CYCLETSP 1 2 1 1 1 2k=|V| verify!

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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.

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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)

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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) 525-527

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

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

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

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

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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

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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|.

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

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

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

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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.

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

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

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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.

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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.

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