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Transcript of 1 The TSP : Approximation and Hardness of Approximation All exact science is dominated by the idea...
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The TSP : Approximation and Hardness of Approximation
All exact science is dominated by the idea of approximation.-- Bertrand Russell (1872 - 1970)
*
*TSP = Traveling Salesman Problem
Based upon slides of Dana Moshkovitz
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Approximation Algorithms
A “good” algorithm is one whose running time is polynomial in the size of the input.
Any hope of doing something in polynomial time for NP-Complete problems?
Lets look at 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 Hardness of Approximation in general.
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TSP
Given a weighted graph G=(V,E)
V = Vertices = Cities
E = Edges = Distances between cities
Find the shortest tour that visits all cities
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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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Naïve Solution
Try all possible tours and pick the minimum
Dynamic Programming
))!1(( nO
)2( nO
Definitely we need something better
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What can we do?
Give up on polynomial time algorithms?
Try Heuristics by giving up on optimality?
Try approximation algorithms?
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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 Strategy Fails
5
0
3
1
12
10
2
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The Greedy Strategy Fails
5
0
3
1
12
10
2
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Another ExampleGreedy strategy fails
0 1-1 3 7-5-11
Even monkeys can do better than this !!!
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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 cn1
1
1
n = k = |V|
verify!
cn
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What Next?
We will see what are approximation algorithms.
We’ll show an approximation algorithm for TSP,
with approximation factor 2
for cost functions that satisfy a certain property.
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c-approximation algorithm
The algorithm runs in polynomial time The algorithm always produces a
solution which is within a factor of c of the value of the optimal solution
)(
)(
xOPT
xA c
For all inputs x.OPT(x) here denotes the optimal value of the minimization problem
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c-approximation algorithm
The algorithm runs in polynomial time The algorithm always produces a
solution which is within a factor of c of the value of the optimal solution
)(
)(
xOPT
xA c
For all inputs x.OPT(x) here denotes the optimal value of the maximization problem
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So why do we study Approximation Algorithms As algorithms to solve problems which
need a solution As a mathematically rigorous way of
studying heuristics Because they are fun! Because it tells us how hard problems
are
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TSP
Is a minimization problem. We want a 2-approximation algorithm But only for the case when the
cost function
satisfies the triangle inequality.
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The Triangle Inequality
Cost Function: Let c(x,y) be the cost of going from city x to city y.
Triangle Inequality: In most situations, going from x to y directly is no more expensive than going from x to y via an intermediate place z.
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The Triangle Inequality
Definition: We’ll say the cost function c satisfies the triangle inequality, if
x,y,zV : c(x,z)+c(z,y)c(x,y)
x
yz
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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.
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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.
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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.