Lecture 21 Approximation Algorithms

Introduction

Earlier Results on Approximations

• Vertex-Cover

• Traveling Salesman Problem

• Knapsack Problem

Performance Ratio

MAX.for )(

)(sup

MINfor )(

)(sup

inputApprox

inpuOptr

inputOpt

inputApproxr

input

input

Constant-Approximation

• c-approximation is a polynomial-time approximation satisfying:

1 < approx(input)/opt(input) < c for MIN

or

1 < opt(input)/approx(input) < c for MAX

Vertex Cover

• Given a graph G=(V,E), find a minimum subset C of vertices such that every edge is incident to a vertex in C.

Vertex-Cover

• The vertex set of a maximal matching gives 2-approximation, i.e.,

approx / opt < 2

Traveling Salesman

• Given n cities with a distance table, find a minimum total-distance tour to visit each city exactly once.

Traveling Salesman with triangular inequality

• Traveling around a minimum spanning tree is a 2-approximation.

Traveling Salesman with Triangular Inequality

• Minimum spanning tree + minimum-length perfect matching on odd vertices is 1.5-approximation

Minimum perfect matching on odd verticeshas weight at most 0.5 opt.

Lower Bound

1+ε 1+ε

1+ε

1 1

1+ε 1+ε

n

n

nn as

2

3

)1)(12(2

)1(2

Traveling Salesman without Triangular Inequality

Theorem For any constant c> 0, TSP has no c-approximation unless NP=P.

Given a graph G=(V,E), define a distance table on V as follows:

EvucV

Evuvud

),( if ,||

),( if ,1),(

Contradition Argument

• Suppose c-approximation exists. Then we have a polynomial-time algorithm to solve Hamiltonian Cycle as follow:

C-approximation solution < cn

if and only if

G has a Hamiltonian cycle

Knapsack

}. ,max{Output

. such that Choose

.Sort

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2-approximation

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PTAS

• A problem has a PTAS (polynomial-time approximation scheme) if for any ε > 0, it has a (1+ε)-approximation.

Knapsack has PTAS

• Classify: for i < m, ci < a= cG,

for i > m+1, ci > a.• Sort

• For

.2

2

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1

m

m

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c

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Fully PTAS

• A problem has a fully PTAS if for any ε>0, it has (1+ε)-approximation running in time poly(n,1/ε).

Fully FTAS for Knapsack

exists.subset such no if ),(

}}.,...,1{

, | {min

and ,

such that i} ..., {1,in indeces ofsubset a is ),(

),(

),(),(

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Pseudo Polynomial-time Algorithm for Knapsak

• Initially,

otherwise

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1

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cj

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Time

• outside loop: O(n)• Inside loop: O(nM) where M=max ci

• Core: O(n log (MS))

• Total O(n M log (MS))

• Since input size is O(n log (MS)), this is a pseudo-polynomial-time due to M=2

3

log M

h

nh

nhhh

h

i

nn

nn

ii

ch

opt

xcxcxcc

ShnhnO

x

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M

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)1

1(

show We.Let

)).)1(log()1((

in time computed becan solution optimal Its

}.1 ,0{

s.t.

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Consider

)1('Set

2211

4

2211

2211

)1

11(

1

)1()*)1

)1((*)1

)1(((

)1()*'*'(

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*** Suppose

11

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hopt

h

Mopt

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nn

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nh

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Lecture 3 Complexity of Approximation

• L-reduction

• Two subclass of PTAS

• Set-cover ((ln n)-approximation)

• Independent set (n -approximation)c