an introduction to approximation algorithms presented by iman sadeghi

An introduction to An introduction to Approximation AlgorithmsApproximation Algorithms

Presented By Presented By Iman SadeghiIman Sadeghi

Spring 2003Spring 2003 Approximation AlgorithmesApproximation Algorithmes 22

OverviewOverview IntroductionIntroduction Performance ratiosPerformance ratios The vertex-cover problemThe vertex-cover problem Traveling salesman problemTraveling salesman problem Set cover problemSet cover problem


IntroductionIntroduction There are many important NP-There are many important NP-

Complete problemsComplete problems There is no fast solution !There is no fast solution !

But we want the answer …But we want the answer … If the input is small use backtrack.If the input is small use backtrack. Isolate the problem into P-problems !Isolate the problem into P-problems ! Find the Find the Near-OptimalNear-Optimal solution in solution in

polynomial time.polynomial time.


Performance ratiosPerformance ratios We are going to find a Near-We are going to find a Near-

Optimal solution for a given Optimal solution for a given problem.problem.

We assume two hypothesis :We assume two hypothesis : Each potential solution has a positive Each potential solution has a positive

cost.cost. The problem may be either a The problem may be either a

maximization or a minimization maximization or a minimization problem on the cost.problem on the cost.


Performance ratios …Performance ratios … If for any input of size n, the cost C If for any input of size n, the cost C

of the solution produced by the of the solution produced by the algorithm is within a factor of algorithm is within a factor of ρ(n)ρ(n) of the cost C* of an optimal of the cost C* of an optimal solution:solution:

Max ( C/C* , C*/C ) ≤ ρ(n)Max ( C/C* , C*/C ) ≤ ρ(n) We call this algorithm as an ρ(n)-We call this algorithm as an ρ(n)-

approximation algorithm.approximation algorithm.


Performance ratios …Performance ratios … In Maximization problems:In Maximization problems:

0<C≤C* , ρ(n) = C*/C0<C≤C* , ρ(n) = C*/C In Minimization Problems:In Minimization Problems:

0<C*≤C , ρ(n) = C/C*0<C*≤C , ρ(n) = C/C* ρ(n) is never less than 1. ρ(n) is never less than 1. A 1-approximation algorithm is the optimal A 1-approximation algorithm is the optimal

solution.solution. The goal is to find a polynomial-time The goal is to find a polynomial-time

approximation algorithm with small approximation algorithm with small constant approximation ratios.constant approximation ratios.


Approximation schemeApproximation scheme Approximation schemeApproximation scheme is an is an

approximation algorithm that takes Є>0 approximation algorithm that takes Є>0 as an input such that for any fixed Є>0 as an input such that for any fixed Є>0 the scheme is (1+Є)-approximation the scheme is (1+Є)-approximation algorithm.algorithm.

Polynomial-time approximation Polynomial-time approximation scheme scheme is such algorithm that runs in is such algorithm that runs in time polynomial in the size of input.time polynomial in the size of input. As the Є decreases the running time of the As the Є decreases the running time of the

algorithm can increase rapidly:algorithm can increase rapidly: For example it might be O(nFor example it might be O(n2/Є2/Є))


Approximation schemeApproximation scheme We haveWe have Fully Polynomial-time Fully Polynomial-time

approximation scheme approximation scheme when its when its running time is polynomial not only running time is polynomial not only in n but also in in n but also in 11//ЄЄ

For example it could be O((For example it could be O((11//ЄЄ))33nn22))


Some examples:Some examples:

Vertex cover problem.Vertex cover problem. Traveling salesman problem.Traveling salesman problem. Set cover problem.Set cover problem.


The vertex-cover problemThe vertex-cover problem A vertex-cover of an undirected A vertex-cover of an undirected

graph G is a subset of its vertices graph G is a subset of its vertices such that it includes at least one such that it includes at least one end of each edge.end of each edge.

The problem is to find minimum size The problem is to find minimum size of vertex-cover of the given graph.of vertex-cover of the given graph.

This problem is an NP-Complete This problem is an NP-Complete problem.problem.


The vertex-cover problemThe vertex-cover problem ……

Finding the optimal solution is hard Finding the optimal solution is hard (its NP!) but finding a near-optimal (its NP!) but finding a near-optimal solution is easy.solution is easy.

There is an 2-approximation There is an 2-approximation algorithm:algorithm: It returns a vertex-cover not more It returns a vertex-cover not more

than twice of the size optimal than twice of the size optimal solution.solution.



APPROX-VERTEX-COVER(G)APPROX-VERTEX-COVER(G)

1 1 C ← ØC ← Ø2 2 E′ ← E[G]E′ ← E[G]3 3 while while E′ ≠ ØE′ ≠ Ø4 4 do do let (u, v) be an arbitrary edge of E′let (u, v) be an arbitrary edge of E′5 5 C ← C U {u, v}C ← C U {u, v}6 6 remove every edge in E′ incident on u or remove every edge in E′ incident on u or

vv7 7 return return CC



OptimalSize=3

Near Optimalsize=6



This is a polynomial-time This is a polynomial-time

2-aproximation algorithm. (Why?)2-aproximation algorithm. (Why?) Because:Because:

APPROX-VERTEX-COVER is O(V+E)APPROX-VERTEX-COVER is O(V+E) |C*| ≥ |A||C*| ≥ |A|

|C| = 2|A||C| = 2|A|

|C| ≤ 2|C*||C| ≤ 2|C*|Optimal

Selected Edges

Selected Vertices


Traveling salesman problemTraveling salesman problem Given an undirected weighted Given an undirected weighted

Graph G we are to find a minimum Graph G we are to find a minimum cost Hamiltonian cycle.cost Hamiltonian cycle.

Satisfying triangle inequality or not Satisfying triangle inequality or not this problem is NP-Complete.this problem is NP-Complete.

We can solve Hamiltonian path.We can solve Hamiltonian path.


Traveling salesman problemTraveling salesman problem Exact exponential solution:Exact exponential solution:

Branch and boundBranch and bound Lower bound:Lower bound:

(sum of two lower degree of vertices)/2(sum of two lower degree of vertices)/2


Traveling salesman problemTraveling salesman problem

A: 2+3A: 2+3B: 3+3B: 3+3C: 4+4C: 4+4D: 2+5D: 2+5E: 3+6E: 3+6= 35= 35

Bound: 17,5Bound: 17,5d

ce

ba

7 4

3

3

56

2

64


Traveling salesman problemTraveling salesman problem

Ab,~ac,ad

Ab,~ac,ad~ac,bc Ab,~ac,ad,~ac,~bc

All Terms

ab ~ab

~ab,~ac ab,acab,ac => ~ac,ad ab

17.5

Ab,~ac,~ad23

2123

18

17.5

17.5 18.5

20.5 21 18.5


Traveling salesman problemTraveling salesman problem Near Optimal solutionNear Optimal solution

FasterFaster More easy to impliment.More easy to impliment.


Traveling salesman problem Traveling salesman problem with triangle inequality.with triangle inequality.

APPROX-TSP-TOUR(G, c)APPROX-TSP-TOUR(G, c)

1 1 select a vertex select a vertex rr ЄЄ V[V[GG] to be root.] to be root.

2 2 compute a compute a MSTMST for for GG from root r using from root r using Prim Alg.Prim Alg.

3 3 LL=list of vertices in preorder walk of that =list of vertices in preorder walk of that MSTMST..

4 4 returnreturn the Hamiltonian cycle the Hamiltonian cycle HH in the order in the order LL..


Traveling salesman problem Traveling salesman problem with triangle inequality.with triangle inequality.

root

MST

Pre-Order walk

Hamiltonian Cycle


Traveling salesman problemTraveling salesman problem This is polynomial-time 2-This is polynomial-time 2-

approximation algorithm. (Why?)approximation algorithm. (Why?) Because:Because:

APPROX-TSP-TOUR is O(VAPPROX-TSP-TOUR is O(V22)) C(MST) C(MST) ≤ C(H*)≤ C(H*)

C(W)=2C(MST)C(W)=2C(MST)

C(W)≤2C(H*)C(W)≤2C(H*)

C(H)≤C(W)C(H)≤C(W)

C(H)≤2C(H*)C(H)≤2C(H*)

Optimal

Pre-order

Solution


Traveling salesman problem In Traveling salesman problem In GeneralGeneral

Theorem:Theorem:If P ≠ NP, then for any constant ρ≥1, If P ≠ NP, then for any constant ρ≥1,

there is no polynomial time ρ-there is no polynomial time ρ-approximation algorithm. approximation algorithm.

c(u,w) ={uEw ? 1 : ρ|V|+1}c(u,w) ={uEw ? 1 : ρ|V|+1}

ρ|V|+1ρ|V|+1++|V|-1|V|-1>ρ|V|>ρ|V|

Selected edge not in

E

Rest of edges


The set-CoverThe set-Cover Generalization of vertex-cover Generalization of vertex-cover

problem.problem. We have given (X,F) :We have given (X,F) :

X : a finite set of elements.X : a finite set of elements. F : family of subsets of X such that F : family of subsets of X such that

every element of X belongs to at least every element of X belongs to at least one subset in F.one subset in F.

Solution C : subset of F that Includes Solution C : subset of F that Includes all the members of X.all the members of X.


The set-Cover …The set-Cover …

Minimal Covering

setsize=3


The set-Cover …The set-Cover …GREEDY-SET-COVER(X,F)GREEDY-SET-COVER(X,F)

1 M ← X1 M ← X

2 C ← Ø2 C ← Ø

3 3 while while M ≠ Ø M ≠ Ø dodo

44 select an SЄF that maximizes |S select an SЄF that maximizes |S |M|M חח

55 M ← M – SM ← M – S

66 C ← C U {S}C ← C U {S}

7 7 return return CC



3rd chose

1st chose

2nd chose

4th chose

GreedyCovering

setsize=4


This greedy algorithm is polynomial-This greedy algorithm is polynomial-time ρ(n)-approximation algorithmtime ρ(n)-approximation algorithm

ρ(n)=H(max{|S| : S Є F})ρ(n)=H(max{|S| : S Є F}) HHdd

The proof is beyond of scope of this The proof is beyond of scope of this presentation.presentation.


=∑=∑i=1i=1dd


Any Question?Any Question?

Thank you Thank you for your attendancefor your attendance

and attention.and attention.

an introduction to approximation algorithms presented by iman sadeghi

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