Vertex-Cover Inapproximability Irit Dinur NEC, Princeton Based on joint work with Muli Safra.

Vertex-Cover Inapproximability

Irit DinurNEC, Princeton

Based on joint work with Muli Safra

Talk OutlineTalk Outline

• Basic PCPBasic PCP• Best inapprox results: BGS-Hastad paradigm of Best inapprox results: BGS-Hastad paradigm of

composing the Raz-verifier with the long-code composing the Raz-verifier with the long-code • In this work:In this work:

• Same distinctive Outer/Inner composition structure. Same distinctive Outer/Inner composition structure. • New Outer PCP, New Outer PCP, • Inner = the biased Long Code, Inner = the biased Long Code,

• I will describe the Long-Code, then talk about I will describe the Long-Code, then talk about the properties needed from the outer verifier, for the properties needed from the outer verifier, for a construction to work with the LC. a construction to work with the LC.

Vertex CoverVertex Cover

• Vertex CoverVertex Cover: Given a graph G, find a : Given a graph G, find a smallest set of vertices that touches all smallest set of vertices that touches all edges.edges.

• The complement of a (minimum) vertex The complement of a (minimum) vertex cover is a (maximum) independent set. cover is a (maximum) independent set.

• Best algorithm, approximates VC within Best algorithm, approximates VC within 2-o(1) [BYE, MS, Hal]2-o(1) [BYE, MS, Hal]

• Best previous hardness, within 7/6 Best previous hardness, within 7/6 [Hastad] [Hastad]

• This work, we show hardness of 1.36This work, we show hardness of 1.36• How far can PCP techniques take us? How far can PCP techniques take us? (do (do

they always suffice for optimal hardness)they always suffice for optimal hardness)

Basic PCP Basic PCP (outer verifier)(outer verifier)

• There are all kinds of PCP theorems, with various There are all kinds of PCP theorems, with various properties of the local-tests, e.g. range of variables, properties of the local-tests, e.g. range of variables,

number of variables, size ofnumber of variables, size of , etc., etc.

yy11,y,y22,y,y33,y,y44,y,y55,y,y66,… ,y,… ,ymm

= = (y(y11vv yy1313vv yy22) (y) (y1515vv yy1919vv yy2929) () (yy2222vv yy1313vv yy2121) … () … (yy44vv y y3131v v yy2424))

Given Given , i, it is NP-hard to distinguish betweent is NP-hard to distinguish between

• 99 satisfying assignment satisfying assignment • No assignment satisfies more than 99%.No assignment satisfies more than 99%.

= = 11(y(y11,y,y44), ), 22(y(y11,y,y22), ), 33(y(y22,y,y44), ), 44(y(y33,y,y11), ), 55(y(y11,y,y66), ), 66(y(y66,y,y22))

Brief HistoryBrief History

• [AS, ALMSS]: [AS, ALMSS]: Basic PCP Thm – VertexCover, MaxCUT etc. Basic PCP Thm – VertexCover, MaxCUT etc. NP-hard to approx to within NP-hard to approx to within some some constant.constant.

• […,BGLR,FK,BS]: […,BGLR,FK,BS]: Better reductions with explicit constantsBetter reductions with explicit constants• [ BGS [ BGS ‘95‘95]:]: Introduced the Introduced the Long-CodeLong-Code. .

e.g. for VertexCover:e.g. for VertexCover: 1.0681.068, for Max-CUT: , for Max-CUT: 1.0141.014• [Håstad [Håstad 96-9796-97]:]: Clique is NP-hard to approximate to within n Clique is NP-hard to approximate to within n1-1-; ;

Optimal gap for 3-SAT and for Linear equations. Optimal gap for 3-SAT and for Linear equations. • Using Fourier analysis of the “marvelous” Using Fourier analysis of the “marvelous” Long-CodeLong-Code, and a, and a

Stronger PCP Stronger PCP [Raz [Raz ‘95‘95]]Hardness factor for VertexCover :Hardness factor for VertexCover : 1.1661.166, for Max-CUT:, for Max-CUT: 1.0621.062

• This work This work [DS [DS 0202]]: : hardness factor for VertexCover :hardness factor for VertexCover : 1.361.36

This WorkThis Work

1.1. The Biased Long-Code: a generalization The Biased Long-Code: a generalization of the Long-Code. of the Long-Code.

New techniques for analysis of the Long New techniques for analysis of the Long Code relying on tools from analysis and Code relying on tools from analysis and combinatorics.combinatorics.

2.2. New PCP constraint systemNew PCP constraint system

• Starting Point: the PCP theoremStarting Point: the PCP theorem1.1. Enhance itEnhance it

2.2. Compose with the Long-CodeCompose with the Long-Code

The hardest part of these works is the interplay The hardest part of these works is the interplay combining these two partscombining these two parts

PCPPCPEnhancedEnhanced

PCPPCPLong-CodeLong-Code

Common Composition Structure Common Composition Structure

Hardness for VCHardness for VC

We want to construct a graph G from We want to construct a graph G from , s.t., s.t.

by the PCP theorem it is NP-hard to distinguish between by the PCP theorem it is NP-hard to distinguish between the two cases, and to (1+c)-approximate Vertex Cover.the two cases, and to (1+c)-approximate Vertex Cover.

Given Given , i, it is NP-hard to distinguish betweent is NP-hard to distinguish between

• 99 satisfying assignment satisfying assignment • No assignment satisfies more than 99%.No assignment satisfies more than 99%.

VC(G) > (1+c)kVC(G) > (1+c)k

is SATis SAT VC(G)=kVC(G)=k

Is far from SATIs far from SAT

• Moreover, we encode a satisfying assignment Moreover, we encode a satisfying assignment for for into a small vertex-cover for G into a small vertex-cover for G

• And decode every “almost” small vertex-cover And decode every “almost” small vertex-cover for G into an “almost” satisfying assignment for for G into an “almost” satisfying assignment for ..

• In standard coding theory, we encode n bits by m bits In standard coding theory, we encode n bits by m bits (m>n), and are able to recover “somewhat corrupt” (m>n), and are able to recover “somewhat corrupt” codewords.codewords.

• In our setting, we can decode any “small enough” In our setting, we can decode any “small enough” vertex vertex cover in G into cover in G into an assignment for an assignment for . .

““Encoding / Decoding”Encoding / Decoding”

Construction: Loose OutlineConstruction: Loose Outline

• The H component is some “gadget”, constructed via the long-code (the The H component is some “gadget”, constructed via the long-code (the inner verifier)inner verifier)

• The The “skeleton”“skeleton” of the graph is a special PCP system (the outer verifier) of the graph is a special PCP system (the outer verifier)Properties we want from this graph G:Properties we want from this graph G:

• A satisfying assignment translates to a vertex cover of size kA satisfying assignment translates to a vertex cover of size k• A Vertex Cover for G of size <k(1+c) translates to an “almost” satisfying A Vertex Cover for G of size <k(1+c) translates to an “almost” satisfying

assignmentassignment

yy11 yy22 yy33 yymm

= = 11(y(y11,y,y44), ), 22(y(y11,y,y22), ), 33(y(y22,y,y44), ), 44(y(y33,y,y11), ), 55(y(y11,y,y66), ), 66(y(y66,y,y22))

Vertex CoverVertex Cover

• We want a (small) vertex cover in G to correspond to anWe want a (small) vertex cover in G to correspond to an(1) assignment(1) assignment

that isthat is(2) satisfying(2) satisfying

• A Vertex Cover for the graph is a vertex cover in each A Vertex Cover for the graph is a vertex cover in each HH . .• DecodeDecode each small vertex cover in each small vertex cover in H H into a value for the into a value for the

underlying y variable. underlying y variable. • Combinatorial Question:Combinatorial Question: Construct a graph Construct a graph HH s.t. any small s.t. any small

vertex cover for vertex cover for HH roughly correspondsroughly corresponds to a single value in the to a single value in the range range {1,2,..,R}{1,2,..,R}..

• Lemma:Lemma: Given Given {1,..,R}{1,..,R}, we can construct a graph , we can construct a graph H=H(H=H(RR)) s.t. s.t.

1.1. (encoding)(encoding) Each value in Each value in {1,2,..,R}{1,2,..,R} corresponds to corresponds to a vertex-cover for H, consisting of ~1/2 of the a vertex-cover for H, consisting of ~1/2 of the vertices. vertices.

2.2. (decoding)(decoding) Every Every vertex cover for vertex cover for HH of size <1- of size <1- still correspondsstill corresponds to some constant number of to some constant number of values in values in {1,2,..,R}.{1,2,..,R}.

• Technique:Technique: • Biased Long-Code, Biased Long-Code, • Analysis of influence of variables on Boolean functions, Analysis of influence of variables on Boolean functions, • Erdos-Ko-Rado theorems on intersecting families of subsets.Erdos-Ko-Rado theorems on intersecting families of subsets.

Encoding a value by Vertex CoversEncoding a value by Vertex Covers

2/32/3

8/98/9one value in {1,..,R}one value in {1,..,R}

Long-Code of Long-Code of RR

R elements, can be most concisely R elements, can be most concisely

encoded by log R bits.encoded by log R bits. Seeking redundancy properties: we use Seeking redundancy properties: we use

many more bits in the encoding.many more bits in the encoding. The Long-Code is the most redundantThe Long-Code is the most redundant

way, using 2way, using 2RR bits. bits.

• One bit for every subset of One bit for every subset of [R][R]

Long-Code of Long-Code of R, R, LC:[R]LC:[R]{0,1}{0,1}22RR

11 22 RR. . .. . .

• One bit for every subset of One bit for every subset of [R][R]

• How do we encode the element How do we encode the element ii[R]?[R]?

(What’s the value of LC(i)?)(What’s the value of LC(i)?)

Long-Code of Long-Code of R, R, LC:[R]LC:[R]{0,1}{0,1}22RR

0 0 1 1 1

11 22 RR. . .. . .

• Endow the bits with the product distribution:Endow the bits with the product distribution:

For each subset For each subset FF, , pp((FF) = p) = p||FF||(1-p)(1-p)|R\|R\FF||

(if p=0.5 this is the regular Long-Code; we take p<0.5)(if p=0.5 this is the regular Long-Code; we take p<0.5)

• Roughly: take only subsets whose size is pRoughly: take only subsets whose size is pR.R.

The p-The p-Biased Biased Long-CodeLong-Code

The Disjointness Graph of the The Disjointness Graph of the Biased Long-CodeBiased Long-Code

11 22 . . .. . . RR

What is a codeword?What is a codeword?

11 22 . . .. . . RR

• A codeword is a A codeword is a vertex coververtex cover• The complement of a vertex-cover is always an The complement of a vertex-cover is always an independent setindependent set. . • Minimum vertex-cover Minimum vertex-cover Maximum independent set Maximum independent set• Claim:Claim: a long-code codeword, i.e. {all subsets (not) containing i} a long-code codeword, i.e. {all subsets (not) containing i}

partitions H into a partitions H into a largest largest independent setindependent set, and its complement, a , and its complement, a smallestsmallest vertex cover vertex cover..

• Maximal Intersecting Families of Subsets: Maximal Intersecting Families of Subsets: [Erdös-Ko-Rado ’61][Erdös-Ko-Rado ’61]• Lemma: The Lemma: The pp size of an intersecting family is size of an intersecting family is p p (trivial for p=0.5, otherwise proven using “shadows” (trivial for p=0.5, otherwise proven using “shadows” [Kruskal `63, Katona [Kruskal `63, Katona

`68]`68]))

i i 22 R R

VC(H)=1-p = 1/2VC(H)=1-p = 1/2

ii11,..,i,..,ikk 22 R R

VC(H)< 1-

• Given some IS/VC partition… Given some IS/VC partition… • It can be viewed as a truth table of a Boolean function on R variables.It can be viewed as a truth table of a Boolean function on R variables.• We can prove, using Friedgut’s theorem, that this function is roughly a Junta We can prove, using Friedgut’s theorem, that this function is roughly a Junta (it (it

must have low average-sensitivity for a perturbed p value because it is monotone)must have low average-sensitivity for a perturbed p value because it is monotone)• This means that only const elements determine if a subset is in the VC or notThis means that only const elements determine if a subset is in the VC or not• Thus, whenever VC < 1-Thus, whenever VC < 1-, it is really a function of only a few elements i, it is really a function of only a few elements i11,..,i,..,ikk 22 [R], [R],

i.e. a combination (e.g. union) of their VC encodings.i.e. a combination (e.g. union) of their VC encodings.• Moreover, if p<1/3 and the VC has size < 1-pMoreover, if p<1/3 and the VC has size < 1-p22, the Junta has a special structure, , the Junta has a special structure,

highlighting one single value i highlighting one single value i 22 [R]. [R]. • Using: the complete characterization of maximal intersecting families by Using: the complete characterization of maximal intersecting families by Ahlswede Ahlswede

and Khachatrian ’97and Khachatrian ’97..

i i 22 R R

VC(H)=1-p = 1/2VC(H)=1-p = 1/2

ii11,..,i,..,iKK 22 R R

VC(H)< 1-

We constructed the disjointness graph of the biased long code, We constructed the disjointness graph of the biased long code, and “showed” that and “showed” that

1.1. Each Each value in value in {1,2,..,R} {1,2,..,R} correspondscorresponds to a small vertex to a small vertex cover for cover for HH (i.e. of size (i.e. of size kk).).

2.2. EveryEvery vertex cover for vertex cover for HH , if smaller than , if smaller than (4/3)(4/3)¢¢kk roughly roughly correspondscorresponds to a single value in to a single value in {1,2,..,R}{1,2,..,R}..

Now we can plug it into the whole construction… Now we can plug it into the whole construction…

i i 22 R R

VC(H)=1-p = 2/3VC(H)=1-p = 2/3

i i 22 R R

VC(H)< (1-p2) = 8/9

Vertex CoverVertex Cover• We want a (small) vertex cover in G to correspond to anWe want a (small) vertex cover in G to correspond to an

(1) assignment(1) assignment

that isthat is

(2) satisfying(2) satisfying

• (1) is accomplished by a lemma like the above(1) is accomplished by a lemma like the above• Achieving (2) entails representing the PCP constraints by Achieving (2) entails representing the PCP constraints by redred

graph edges (between H components).graph edges (between H components).

• The lemma gives a correspondence The lemma gives a correspondence between values in [R] and small vertex between values in [R] and small vertex covers in H.covers in H.

• Next step: consider two copies of H, Next step: consider two copies of H, representing two y variables with a representing two y variables with a constraint between them.constraint between them.

Expressing a local-constraintExpressing a local-constraint

Expressing a local-constraintExpressing a local-constraint

• Two copies of H, representing variables yTwo copies of H, representing variables y11 and y and y22

• Add red edges so that “consistent” pairs of VCs will always Add red edges so that “consistent” pairs of VCs will always cover the red edges too. cover the red edges too. (thus there is no freedom in (thus there is no freedom in choosing the red edges)choosing the red edges)

• Limitations- the red edges can themselves be covered Limitations- the red edges can themselves be covered “cheaply”… “cheaply”…

• We want to ensure that a semi-small VC corresponds to an We want to ensure that a semi-small VC corresponds to an assignment to yassignment to y11,y,y22 that satisfies the constraint. that satisfies the constraint.

• For this, For this, • either the VCs are required to be rather large (no gap)either the VCs are required to be rather large (no gap)• or the constraint must have high “uniqueness”or the constraint must have high “uniqueness”

UniquenessUniqueness

yy11yy22

11

22

33

RR

11

22

33

RR

UniquenessUniqueness

yy11yy22

11

22

33

RR

11

22

33

RR

UniquenessUniqueness

yy11yy22

11

22

33

RR

11

22

33

RR

““High Uniqueness” PCPHigh Uniqueness” PCP

For every For every h>1h>1,, there is a system of tests there is a system of tests with “high uniqueness” s.t. it is NP- with “high uniqueness” s.t. it is NP-hard to distinguish betweenhard to distinguish between

1.1. There is an assignment satisfying 1-There is an assignment satisfying 1- of the constraintsof the constraints

2.2. Every assignment to at least Every assignment to at least of the of the variables, must contain variables, must contain hh variables that variables that are pairwise inconsistent.are pairwise inconsistent.

Going Back to the ConstructionGoing Back to the Construction

yy11 yy22 yy33 yymm

= = 11(y(y11,y,y44), ), 22(y(y11,y,y22), ), 33(y(y22,y,y44), ), 44(y(y33,y,y11), ), 55(y(y11,y,y66), ), 66(y(y66,y,y22))

A satisfying assignment can be encoded into a A satisfying assignment can be encoded into a small vertex coversmall vertex coverA semi-small vertex cover can be decoded into a A semi-small vertex cover can be decoded into a satisfying-assignment. satisfying-assignment. (an assignment to (an assignment to variables without an h- variables without an h-sized clique of inconsistency)sized clique of inconsistency)

2½ parts2½ parts

1.1. Construct HConstruct H

2.2. Construct PCP with high “uniqueness”Construct PCP with high “uniqueness”

3.3. Combine the twoCombine the two

How to get factor 2?How to get factor 2?

1.1. Get a better PCP system with higher Get a better PCP system with higher uniquenessuniqueness

2.2. New ways of combining the Long-New ways of combining the Long-Code into the soundness proof.Code into the soundness proof.