1 ©d.moshkovitz complexity the traveling salesman problem
TRANSCRIPT
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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.