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The Traveling Salesman Problem

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The Mission: A Tour Around the World

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The Problem: Traveling Costs Money

1795$

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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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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

432

5

1 10

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Polynomial Algorithm for TSP?

What about the greedy strategy:

At any point, choose the closest vertex not explored

yet?

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The Greedy $trategy Fails

5

0

3

1

12

10

2

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The Greedy $trategy Fails

5

0

3

1

12

10

2

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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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TSP is NP-hard

Theorem: HAM-CYCLE p TSP.

Proof: By the straightforward efficient reduction illustrated below:

HAM-CYCLE TSP

1 21

1

1

2 k=|V|

verify!

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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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The Triangle Inequality

Definition: We’ll say the cost function c satisfies the triangle inequality, ifu,v,wV : c(u,v)+c(v,w)c(u,w)

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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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Demonstration and Analysis

The cost of a minimal

Hamiltonian cycle the cost of a

MST

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Demonstration and Analysis

The cost of a preorder walk is twice the cost of

the tree

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Demonstration and Analysis

Due to the triangle inequality, the

Hamiltonian cycle is not worse.

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The Bottom Line

optimal HAM cycle

MSTpreorder

walk

our HAM cycle

= ½· ½·

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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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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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Instances

min cost

|V| |V|

1

1

1

0+1

0

0

1

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What Should an Algorithm for gap-TSP Return?

|V| |V|

YES! NO!

min cost

gap

DON’T-CARE...

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gap-TSP & Approximation

Observation: Efficient approximation of factor for TSP implies an efficient algorithm for gap-TSP[].

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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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The Reduction Illustrated

HAM-CYCLE gap-TSP

1 |V|+11

1

1

|V|+1

Verify (a) correctness (b)

efficiency

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Approximating TSP is NP-hard

gap-TSP[] is NP-hard

Approximating TSP within factor is NP-hard

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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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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.