Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several...

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Page 1: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity1

Hardness of Approximation

Page 2: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity2

Introduction

• Objectives:– To show several approximation

problems are NP-hard• Overview:

– Reminder: How to show inapproximability?

– Probabilistic Checkable Proofs– Hardness of approximation for clique

Page 3: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity3

Optimization Problems

Consider an optimization problem P:

instances: x1,x2,x3,…

optimization measure

feasible solutions

all graphs

Example:

all cliques in that graph

the clique’s size (max)

Page 4: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity4

Each Instance Has an Optimal Solution

OPTx1 x2 x3x4

Page 5: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity5

Approximation (Max Version)

OPTxi

Page 6: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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How To Show Hardness of Approximation?

Hardness of distinguishing far off instances Hardness of approximation

OPT

A B

gap

xi

Page 7: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Gap Problems (Max Version)

• Instance: …

• Problem: to distinguish between the following two cases:

The maximal solution B

The maximal solution ≤ A

Page 8: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Formally:

Claim: If the [A,B]-gap version of a problem

is NP-hard, then that problem is NP-hard to

approximate within factor B/A.

Page 9: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity9

Formally:

Proof: Suppose there is an approximation algorithm that outputs C so that C/C*≤B/A

A proper distinguisher:* If CB, return ‘YES’* Otherwise return ‘NO’

Page 10: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity10

Proof

Since C*≥AC/B, (1) If C>B (we answer ‘YES’), then

necessarily C*>A (the correct answer cannot be ‘NO’).

(2) If C*≤A (the correct answer is ‘NO’), then necessarily C≤B (we answer ‘NO’)

Page 11: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity11

Idea

• We’ve shown “standard” problems are NP-hard by reductions from 3SAT.

• We want to prove gap-problems are NP-hard,

• Why won’t we prove some canonical gap-problem is NP-hard and reduce from it?

• If a reduction reduces one gap-problem to another we refer to it as approximation-preserving

Page 12: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity12

Gap-3SAT[]

Instance: a set of clauses {c1,…,cm} over variables v1,…,vn.

Problem: to distinguish between the following two cases:

There exists an assignment which

satisfies all clauses.No assignment can satisfy more than 7/8+ of the clauses.

Page 13: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity13

Gap-3SAT: Example

( x1 x2 x3 )

( x1 x2 x2 )

( x1 x2 x3 )

( x1 x2 x2 )

(x1 x2 x3 )

( x3 x3 x3 )

= { x1 F ; x2 T ; x3 F }satisfies 5/6 of the clauses

Page 14: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity14

Why 7/8?

Claim: For any set of clauses with exactly three independent literals,

there always exists an assignment which satisfies at least 7/8.

Page 15: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity15

The Probabilistic Method

Proof: Consider a random assignment.

x1 x2 x3 xn

. . .

Page 16: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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1. Find the Expectation

Let Yi be the random variable indicating the outcome of the i-th clause.

For any 1im, E[Yi]=0·1/8+1·7/8=7/8

E[ Yi] = E[Yi] = 7/8m

Page 17: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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2. Conclude Existence

Expectedly, the number of clauses satisfied is 7/8m.

Thus, there exists an assignment which satisfies at least that many.

Page 18: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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PCP (Without Proof)

Theorem (PCP): For any >0,

Gap-3SAT[] is NP-hard.

This is tight! Gap-3SAT[0] is polynomial time

decidable

Page 19: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Approximation Preservation

A B

•YES

•don’t care

•NO

• YES

• don’t care

• NO

Page 20: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Hardness of Approximation

• Do the reductions we’ve seen also work for the gap versions?

• We’ll revisit the CLIQUE example.

Page 21: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity21

CLIQUE Construction

.

.

.

a part for each

clause

a vertex for each literal

edge indicates

consistency

Page 22: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Approximation Preservation

• If there is an assignment which satisfies all clauses, there is a clique of size m.

• If there is a clique of size (7/8+)m, there is an assignment which satisfies more than 7/8+ of the clauses.

Page 23: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Gap-CLIQUE (Ver1)

The following problem is NP-hard for any >0:

Instance: a graph G=(V,E) composed of m independent sets of size 3.

Problem: to distinguish between:

There’s a clique of size m

Every clique is of size at most (7/8+)m

Page 24: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Corollary

Theorem: for any >0,CLIQUE is hard to approximate

within a factor of 1/(7/8+)

Page 25: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Amplification

• The bigger the gap is, the better the hardness result.

• We’ll see how a gap can be amplified.

Page 26: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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.

.

.

...

...

Amplification

A part for every k vertices

vertex for each Boolean

assignment

edge indicates

consistency

Given an instance of the Gap-CLIQUE problem and a constant k:

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Boolean assignments

• A Boolean assignment over k vertices {v1,…,vk} is a function A:{v1,…,vk}{0,1}.

• Think about it as if it indicates whether each vertex belongs to the clique.

Page 28: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Good Assignments

Page 29: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Consistency

• Two assignments are inconsistent, when they give the same vertex different truth-values.

. . .

n

. . .. . .

Page 30: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Consistency

• They are also inconsistent, if they both assign 1 to two vertices not connected by an edge.

non-edge

Page 31: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity31

Correctness

Page 32: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Chromatic Number

• Instance: a graph G=(V,E).• Problem: To minimize k, so that

there exists a function f:V{1,…,k}, for which

(u,v)E f(u)f(v)

Page 33: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity33

Chromatic Number

Page 34: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Chromatic NumberObservation: Each color group is an

independent set

Page 35: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Clique Cover Number (CCN)

• Instance: a graph G=(V,E).• Problem: To minimize k, so that

there exists a function f:V{1,…,k}, for which

(u,v)E f(u)=f(v)

Page 36: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity36

Clique Cover Number (CCN)

Page 37: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Reduction Idea

.

.

.

CLIQUE CCN

.

.

.

q

• cyclic shift-morphic

• clique preserving

m.

.

.

Page 38: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity38

Correctness

Page 39: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Transformation

T:V[q]

for any v1,v2,v3,v4,v5,v6,

T(v1)+T(v2)+T(v3) T(v4)+T(v5)+T(v6) (mod q)

{v1,v2,v3}={v4,v5,v6}T is unique for triplets

Page 40: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Observations

• Such T is unique for pairs and for single vertices as well:

• If T(x)+T(u)=T(v)+T(w), then {x,u}={v,w}

• If T(x)=T(y) (mod q), then x=y

Page 41: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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feasible values

Greedy Constructionv6

v6

v2

v2

v1

v1

v5

v5v

3

v3

v4

v4

vertices we determined

forbidden values

Page 42: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Greedy Construction - Analysis

At most values are ruled out totally, so for q=n5 the greedy construction works.

Corollary: There exists a polynomial time algorithm which constructs a triplet unique transformation with q=n5

5

n

Page 43: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity43

Using the Transformation

0 1 2 3 4 … (q-1)

vi

vj

T(vi)=1

T(vj)=4

CLIQUE

CCN

Page 44: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity44

Completing the CCN Graph Construction

T(s)

T(t)

(s,t)ECLIQUE

(T(s),T(t))ECCN

Page 45: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity45

Completing the CCN Graph Construction

T(s)

T(t)

Close the set of edges under shift:

For every (x,y)E,

if x’-y’=x-y (mod q), then (x’,y’)E

Page 46: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Max Clique of G-clique and G-ccn

• Lemma:Max-Clique(G-clique) = Max-Clique(G-CCN)

• Corollary: – MAX-clique(G-clique) = m CCN(G-ccn)=q– MAX-clqiue(G-clique) < m CCN(G-ccn)>

q

Page 47: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

Complexity47

Edge Origin Unique

T(s)

T(t)

First Observation: This edge comes

only from (s,t)

Page 48: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Triangle Consistency

Second Observation: A

triangle only come from a triangle

Page 49: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Clique Preservation

Corollary: {c1,…,ck} is a clique in the CCN graph

iff {T(c1),…,T(ck)} is a clique in the CLIQUE graph.

Page 50: Complexity 1 Hardness of Approximation. Complexity 2 Introduction Objectives: –To show several approximation problems are NP-hard Overview: –Reminder:

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Summary

• We’ve seen how to show hardness of approximation results in general,

• and even proven several such using the PCP theorem:– CLIQUE– CHROMATIC NUMBER