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PCP and Inapproximability

- Irit Dinur
- NEC

Given any function.. How complicated is it to

compute it?

- Example the Minimum Vertex Cover function
- Facts 1. Best algorithm runs in time (1.21)n

Robson 86 - 2. VC is NP-hard. Karp 72
- What about approximation.. Output a vertex cover

thats nearly minimal!

Minimum Vertex Cover

Vertex-Cover Given a network of roads. Each road

requires a toll payment. Goal Put up the

smallest number of toll booths

What do we mean by approximation? Each instance

has many solutions, each has a value. In

optimization, we are seeking the minimal.

Approximation

An approximation algorithm finds a solution

within a certain neighborhood of MIN

- Example An algorithm for Approximating Vertex

Cover - Given G, find a maximal set of edges that do not

touch each

other. - Add both vertices of each edge to the vertex

cover.

MIN

Approximation

This is a solution all edges are covered

How big is it? No more than twice the minimum!

An approximation algorithm finds a solution

within a certain neighborhood of MIN

- Example An algorithm for Approximating Vertex

Cover - Given G, find a maximal set of edges that do not

touch each

other. - Add both vertices of each edge to the vertex

cover.

Approximation

How big is it? No more than twice the minimum!

An approximation algorithm finds a solution

within a certain neighborhood of MIN

- Example An algorithm for Approximating Vertex

Cover - Given G, find a maximal set of edges that do not

touch each

other. - Add both vertices of each edge to the vertex

cover.

Weve seen an approximation algorithm for

Vertex-Cover, with approximation factor 2.

Approximation

x 2

x 3/2

x 4/3

x 1.99

Weve seen a factor 2 algorithm. Q Is there a

factor 1.99 algorithm? 3/2 ? 4/3 ?

Not unless PNP, AS,ALMSS,BGS,Raz,Hastad,DS

Finding the Approximation Threshold

- Rich variety of approximation algorithms.
- Various problems with vastly differing

approximation ratios. - PCP theorem breakthrough, we can prove hardness

of approximation. - Moreover, its within reach to find the exact

threshold where a problem changes from being easy

to hard. (theoretic and practical importance) - Vertex Cover
- Upper bound 2-o(1) BYE, MS, Hal
- Best hardness result
- Thm DS02 NP-hard to approximate to within

1.36. - Conjecture NP-hard to within 2- ? ??gt0

How to show hardness?

Take any NP-hard problem, say SAT, and

reduce Translating a formula ? into a graph G,

s.t.

? is unSAT

VC(G) gt k

The gap-version of the problem

MIN k

- Given a 1c approximation algorithm,
- If MIN k then Approx lt (1c)?k.
- A 1c approximation algorithm could decide

whether the minimum is - below k or
- above (1c)?k

In other words, it would solve the gap

problem Given a graph G, decide if the minimum

VC is

How to show hardness of approximation?

Take any NP-hard problem, say SAT, and

reduce Translating a formula ? into a graph G,

s.t. Decades, complexity of approximations

remains mysterious Rich variety of algorithms,

hardly any lower-bounds Why?

?

? is unSAT

VC(G) gt k

VC(G) gt (1c)k

Why is it hard to prove?

?(x1,x2,,xn)

0 1 0

Simple Every SAT assignment translates to a

size-k VC for G.

Why is it hard to prove?

?(x1,x2,,xn)

0 1 0

?

? is unSAT

VC(G) gt k

VC(G) gt (1c)k

Simple Every SAT assignment translates to a

size-k VC for G. And Every size-k VC

translates to a SAT assignment. But, far from

easy Given a VC of size lt (1c)k, how to decode

it into a SAT assignment?

Combinatorial Decoding

Probabilistically Checkable Proofs

PCP a referees dream

- Background context Interactive Proofs
- Say you get a paper for refereeing.
- Select a few random lemmas and verify only them.
- Will this work? in general, of course not!
- PCP Theorem Every proof can be compiled into

PCP-language so that by reading only a few bits

of the new PCP proof, correct verification can be

achieved with high probability. - Why is this amazing?
- One tiny bug or hole or gap can cause much agony
- In a proof usually the correctness of each step

depends on much of what happened before it - It seems that if you only read a few bits, you

can easily be cheated!

More Concretely

(x1vx3vx12)(x1vx2vx8)(x7vxnvx10)(x1vx15vx14)

1

0

1

1

1

0

0

0

1

0

1

1

0

0

1

0

0

1

0

1

- An NP statement can be written as a 3SAT formula
- A proof for its satisfiability is an assignment

of 0/1 - We can verify it clause by clause.

More Concretely

(x1vx3vx12)(x1vx2vx8)(x7vxnvx10)(x1vx15vx14)

x1

x5

xn-3

x9

x13

x2

x6

xn-2

x10

x14

x3

x7

xn-1

x11

x15

x4

x8

xn

x12

x16

- An NP statement can be written as a 3SAT formula
- A proof for its satisfiability is an assignment

of 0/1 - We can verify it clause by clause.
- Murphys law! we detect an error only on the

last clause

More Concretely

?(y1v y13v y2)(y15v y19v y29)(y22v y13v

y21)(y4v y31v y24)

- ?(x1vx3vx12)(x1vx15vx14)

PCP

- The PCP theorem, gives a compiler translating

? into ? s.t. ? admits super-efficient

verification. - If ? is satisfiable, then so is ?
- If ? is not satisfiable, then every assignment

satisfies at - most 99 of ?s clauses.

More Concretely

?(y1v y13v y2)(y15v y19v y29)(y22v y13v

y21)(y4v y31v y24)

- ?(x1vx3vx12)(x1vx15vx14)

PCP

- Conclusion AS,ALMSS gap-SAT is NP-hard
- Given a 3-SAT formula ? it is NP-hard to

decide whether - ? is satisfiable
- Every assignment satisfies at most 99 of ?s

clauses.

This is an Inapproximability Result!

PCP

- Hardness for gap-3SAT, shows that it is NP-hard

to approximate the following optimization

problem - Max-3-SAT Given a 3SAT formula,

?(y1vy3vy12)(y1vy15vy14), - find the maximal fraction of satisfiable

clauses. - A random assignment easily satisfies 7/8
- Håstad proved theres no better algorithm

Some Brief History

- Context Interactive proofs
- Connection to Approximation FGLSS 91
- PCP Theorem AS, ALMSS 92
- Immediately for many problems, e.g. vertex cover,

max-cut, metric-TSP, max-3SAT, bounded-degree-cliq

ue,

?

? is unSAT

? is lt 99 SAT

VC(G) gt (1c)k

Some Brief History

- Context Interactive proofs
- Connection to Approximation FGLSS 91
- PCP Theorem AS, ALMSS 92
- Immediately for many problems, e.g. vertex cover,

max-cut, metric-TSP, max-3SAT, bounded-degree-cliq

ue, PY 91 - Boom of hardness of approximation results
- Emphasis on
- Better PCP parameters
- Tight inapproximability results

Tighter Results

- ,BGLR,FK,BS Better reductions with explicit

constants - BGS 95 Introduced the Long-Code.
- e.g. for VertexCover 1.068, for Max-CUT 1.014
- Håstad 96-97 Clique is NP-hard to approximate

to within n1-? Optimal gap for 3-SAT and for

Linear equations. - Using Fourier analysis of the marvelous

Long-Code, and a Stronger PCP Raz 95 - Hardness factor for VertexCover 1.166, for

Max-CUT 1.062

My Work

- The Biased Long-Code a generalization of the

Long-Code. D.,Safra 02 - New perspectives on the Long Code yielding

powerful new techniques. - Analysis of influence of variables on Boolean

Functions - Extremal Set Combinatorics.
- New stronger enhancements of the PCP theorem,

e.g. the Layered PCP in DGKR 02, DRS 02 - Leading to best-known inapproximability results

for - Vertex-Cover Hardness for vertex cover 1.367.

D.,Safra 02 - Approximate Hypergraph Coloring Approximate

hypergraph coloring. D.,Regev,Smyth 02 - Hypergraph Vertex Cover D. D.,Guruswami,Khot

D.,Guruswami,Khot,Regev 02

Combinatorial Decoding

- Proving hardness for gap-VC, we translate ? into

G and then prove 2 things - I.
- II.
- The hard part of the proof is part II, showing

that - Every VC in G of size lt (1c)k can be decoded

into a satisfying assignment for ?. - In standard coding theory, we encode n bits by m

bits (mgtn), and are able to recover somewhat

corrupt codewords. - Every word, if close enough, we can decode
- In combinatorial decoding, we encode an

assignment for ? by a vertex cover in G and are

able to recover somewhat corrupt vertex covers.

- Every VC, if small enough, we can decode

? is lt 99 SAT

VC(G) gt (1c)k

The Underlying Structure

- Starting Point the PCP theorem
- Enhance it
- Apply the Long-Code on small sub-components.
- The hardest part of these works is the interplay

combining these two parts

PCP

Enhanced PCP

Long-Code

Vertex Cover

- A very loose outline of the construction
- A satisfying assignment can be encoded into a

vertex-cover. - A vertex-cover for the graph is a vertex cover in

each H . - Decode each small vertex cover in H into a

value for the underlying y variable. - Then, show consistency between these values.
- Combinatorial Question Construct such a graph H

.

? (y1v y13v y2) (y15v y19v y29) (y22v y13v y21)

(y4v y31v y24)

y1

y2

y3

ym

Sub-Goal Construct a graph H

- Such that,
- Each value in 1,2,..,R corresponds to a small

vertex cover for H (i.e. of size k ½V). - Every vertex cover for H , if smaller than

(2-?)k roughly corresponds to a single value in

1,2,..,R. - Technique
- Biased Long-Code,
- Analysis of influence of variables on Boolean

functions, - Erdös-Ko-Rado theorems on intersecting families

of subsets.

Long-Code of R

- R elements, can be most concisely
- encoded by log R bits.
- Seeking redundancy properties we use
- many more bits in the encoding.
- The Long-Code is the most redundant
- way, using 2R bits.

Long-Code of R, LCR?0,12R

- One bit for every subset of R

1

2

R

. . .

Long-Code of R, LCR?0,12R

- One bit for every subset of R
- How do we encode the element i?R?
- (Whats the value of LC(i)?)

1

2

R

. . .

1

0

0

1

1

The p-Biased Long-Code

- Endow the bits with the product distribution
- For each subset F, ?p(F) pF(1-p)R\F
- Roughly take only subsets whose size is p?R.

The Disjointness Graph of the Biased Long-Code

1

2

. . .

R

What is a codeword?

(No Transcript)

- A codeword is a vertex cover
- The complement of a vertex-cover is always an

independent set. - In this graph, an independent set is an

intersecting family of subsets. - Claim a long-code codeword, i.e. all subsets

containing i is a largest independent set, and

its complement, a smallest vertex cover. - Maximal Intersecting Families of Subsets

Erdös-Ko-Rado 61 - Lemma The ?p size of an intersecting family is ?

p (proof using shadows Kruskal 63, Katona

68) - Much more difficult to prove Any vertex cover

whose size is lt 1-p2 is decodable into a value

in 1,,R. (combinatorial decoding) - Using the complete characterization of maximal

intersecting families by Ahlswede and Khachatrian

97, and Friedguts Theorem on when Boolean

Functions are Juntas, etc.

(2-?)k ???

- We constructed a graph s.t.,
- Each value in 1,2,..,R corresponds to a small

vertex cover for H (i.e. of size k). - Every vertex cover for H , if smaller than

(4/3)k roughly corresponds to a single value in

1,2,..,R. - Now we can plug it into the whole construction

Future Directions

- Finding the true threshold (stronger

combinatorial decoding) - Factor 2 inapproximability for Vertex Cover
- Other problems approximate coloring, etc.
- Simplification of PCP, locally testable codes.
- Decoding in completely different contexts, with

applications for database privacy.

Thanks