Approximation Problems

Goal:

• Discuss Optimization• Introduce Approximation• Problems/Algorithms

Plan:

• VERTEX-COVER, SET-COVER

• Greedy Algorithm• TSP

2

3

Network PartyP

lan

a

Part

ywith a set of invitees.Every pair of friends must have at least one invite!

Fin

d smallest subset of guests so as to cover all friendships.Can

you find

such a set?

• If (u,v)E, uC or vC

CV is a cover of G=(V, E)

• An undirected G=(V,E)

Instance:

• find a minimal cover C

Minimization Problem:

• An undirected G=(V,E), k

Instance:

• Is there a cover C, |C|=k?

Decision Problem:

VERTEX-COVER

• Min-V.C. is NP-hard

Theorem:

• For a cover C, V\C is an independent-set

Proof:

4

What to do if NP-

hard

• Best solution according to optimization parameter

Algorithm returns

Optimization

5

problem

CLIQUE

3SAT

V.C.

min/max

max

max

min

instance

graph

3CNF

graph

parameter

|clique|

#clause

|cover|

6

• An -approximation algorithm for a maximization/minimization problem with an optimum solution O, returns a solution that is O/O

Definition:

• and O may depend onthe size of the input

Note

Approximation

• |C| 2optimal

Theorem:

Proof:

• C 1• begin while C not a

cover2• Pick (u,v)E s.t. u,vC3• C C {u,v}4• end while5• return C7

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VC - Approximation AlgorithmCOR(B) 523-524

note

algorithm runs in poly-time

note

C is a V.C.

Edges algorithm picks:

Have no common vertexes

Optimal V.C. must contain 1 end of

each

8

You’d like to send some message to a large list of recipients (e.g. all campus)

Some mailing-lists are available, however, each list charges $1 for each message sent

You’d like to find the smallest set of lists that covers all recipients

Mass Mailing

9

• a finite set U and a family F of subsets of U, which covers U

Instance:

• find a smallest family CF that covers U

Minimization Problem:

SET-COVER

V.C.L

SET-COVER

SET-COVER NP-Hard

VCSC

• 1 element for each edgeU

• 1 set for each vertex, comprising adjacent edges

F10

Theorem:• SET-COVER is NP-Hard

Proof:

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The Greedy Algorithm• C , U’ U1

• begin while U’ 2

• Pick SF that maximizes |SU’|3

• C C {S}4

• U’ U’ – S5

• end while6

• return C7

COR(B) 530-533

note

algorithm runs in poly-time

note

C is a S.C.

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Just:

• Described the greedy algorithm for SET-COVER

Next:

• Analyze its approximation ratio in 3 distinct ways: lg2 n, ln n, even better

• compare the size of the cover returned by the greedy algorithm to optimal

Need to:

• The optimal is unknown

However

• k-cover any part of the universe has a k-cover!

Observation:

• Each step of the greedy algorithm removes 1/k of the remaining elements

Corollary:

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How to prove an Approximation Ratio?

• After k(=size of optimal S.C.) stages the algorithm covers at least ½ of U

Observe:

• By way of contradiction, assume <½ are coveredThe uncovered part of U intersects with a set in F in >n/2k elementsHence, all previous k stages have covered >n/2k elementsAnd must have covered >kn/2k=n/2

Proof:

14

lg2 n

It can be covered by k sets

ln n• S1, …, St be the sequence of sets

picked by the algorithm Let

• Ui be the set of elements not yet covered after i stages

Let

• |Ui+1|=|Ui-Si+1||Ui|(1-1/k)Note

• |Uik||Uo|(1-1/k)ik|U|e-iHence

• t ≤ k ln(n) + 1and

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Ui can be covered by k sets

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• Greedy algorithm approximates the optimal set-cover to within a factor H(max{ |S|: SF } )

Lemma:

• Split the “cost of $1”, for set Si picked ith by the greedy algorithm, among newly covered

• Now, bound the sum paid, over any SF, by H(|S|)

• “Imagine” the optimal solution, and bound the total paid

Proof:

Best Ratio-Bound

1lnn1dxx

11

k

1

k

1)(

n

1

n

2k

n

1k

nH

• Each segment of the tour problem now has a cost• find a least-costly tour

Generalized Tour Problem

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$2

$5

$4

$3$1

$4

$6

$5

$2

Approximate the optimal tour?

i.e. – find a tour that

costs, say, no more

than twice as much as

the least costly.

Next:

• NP-hard optimization problems• Approximation to within a certain

factor• NP-hard or in P?

Plan:

• Traveling Salesman Problem (TSP)• TSP is NP-hard• Approximation algorithm for special

cases• Inapproximability result

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• A complete weighted undirected G=(V,E) (non-negative weights)

Instance:

• find a Hamiltonian cycle (traversal) of minimal cost

Minimization Problem:

• TSP is NP-hard

Theorem:

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Traveling Salesperson Problem

2

1

34

3

4

L

By a simple reduction from Ham. cyc.

Next: show a 2-Approximation algorithm for TSP, in case the cost function

satisfies the triangle inequality

u,v,wV: c(u,v)+c(v,w)c(u,w)

viaMin

SpanningTree

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• Find a Minimum Spanning Tree (MST) T for G1

• Traverse along DFS of T --- jump over visited2

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Approximation AlgorithmCOR(B) 525-527

• MST weight Cheapest Traversal

Observation

• Algorithm’s traversal costs 2 weight MST

Observation

Next:

• Show it is NP-hard to approximate TSP –the general case– to within any factor h1

How?

• Introduce its gap version• Show it is NP-hard

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• a complete weighted undirected graph G=(V,E)

Instance:

• distinguish between the following cases:

GAP Problem: [|V|, h|V|]

• Hamiltonian cycle of cost < |V|

Yes

• The cost of Hamiltonian cycle > h|V|

No

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gap-TSP

• a complete weighted undirected graph G=(V,E)

Instance:

• distinguish between the following cases:

GAP Problem: [|V|, h|V|]

• Hamiltonian cycle of cost < |V|

Yes

• The cost of Hamiltonian cycle > h|V|

No

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gap-TSP

• Depending on the least costly traversal

Algorithm returns

Yes if

<|V|

No if

>h|V|

GAP

gap-TSP

• Whatever

GAP

• Efficient h-approximation for TSP resolves gap-TSP[C, hC]

Observation:

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• For any h1, HAM-CYCLE L gap-TSP[|V|, h|V|]

Theorem:

Proof:

• Approximating TSP to within any factor is NP-hard

Corollary:

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gap-TSP is NP-hard

L

Some NP-hard optimization problems can be efficiently approximated:• VERTEX-COVER (2)• SET-COVER (ln n)• TSP[triangle ineq.] (2)

For some factors it may still be NP-hardWe’ve introduced GAP problems for that purpose

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Synopsis

WWindex

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Optimization Vertex Cover

Greedy Algorithm

Spanning Tree

Set Cover

Minimum Spanning Tree

Independent Set

P NP-Hard

Hamiltonian Cycle

TSP

Clique

3SAT