Linear-Degree Extractors and the Inapproximability of Max Clique and

Chromatic Number

David ZuckermanUniversity of Texas at Austin

Max Clique and Chromatic Number

• [FGLSS,…,Hastad]: Max Clique inapproximable to n1-, any >0, assuming NP ZPP.

• [LY,…,FK]: Same for Chromatic Number.• Can we assume just NP P?

Thm: Both inapproximable to n1-, any >0, assuming NP P.

Thm: Derandomized [Khot]: Both inapproximable to n/2(log n)1-, some >0, assuming NQ Q.

Derandomization tool: disperser.

Outline

• Extractors and Dispersers• Dispersers and Inapproximability• Extractor/Disperser Construction

– Additive Number Theory• Conclusion

Weak Random Sources

• Random element from set A, |A| 2k.

|A| 2k

{0,1}n

Weak Random Sources

• Can arise in different ways:– Physical source of randomness.– Condition on some information:

• Cryptography: bounded storage model.

• Pseudorandom generators for space-bounded machines.

• Convex combinations yield more general model, k-source: x Pr[X=x] 2-k.

Weak Random Sources

• Goal: Algorithms that work for any k-source.• Should not depend on knowledge of k-source.• First attempt: convert weak randomness to good

randomness.

Goal

Extvery long

weakly random

long

almost random

Should work for all k-sources.

Problem: impossible.

Solution: Extractor[Nisan-Z]

Extvery long

weakly random

long

almost random

short truly random

Extractor Parameters[NZ,…, Lu-Reingold-Vadhan-Wigderson]

Ext n bits

k-source

m=.99k bits

almost random

d=O(log n) truly random

Almost random in variation (statistical) distance.Error = arbitrary constant > 0.

(1- )M K=2k

Graph-Theoretical View of Extractor

D=2d

N=2n

M=2m

Think of K=N, M .Goal: D=O(log N).

output -uniform

x E(x,y1)

E(x,y2)

E(x,y3)

Disperser

Applications of Extractors

• PRGs for Space-Bounded Computation [Nisan-Z]

• PRGs for Random Sampling [Z]

• Cryptography [Lu, Vadhan, CDHKS, Dodis-Smith]

• Expander graphs and superconcentrators [Wigderson-Z]

• Coding theory [Ta-Shma- Z]

• Hardness of approximation [Z, Umans, Mossel-Umans]

• Efficient deterministic sorting [Pippenger]

• Time-space tradeoffs [Sipser]

• Data structures [Fiat-Naor, Z, BMRV, Ta-Shma]

Extractor Degree

• In many applications, left-degree D is relevant quantity:– Random sampling: D=# samples– Extractor codes: D=length of code– Inapproximability of Max Clique: size of graph

= large-case-clique-size cD (scaled-down).

Extractor/Disperser Constructions

• n=lg N=input length.• Previous typical good extractor: D=nO(1).• [Ta-Shma-Z-Safra]:

– D=O(n log* n), but M=Ko(1).– For K=N(1), D=n polylog(n), M=K(1).

New Extractor: For K=N(1), D=O(n) and M=K.99.New Disperser: Same, even D=O(n/log s-1),

s=1-=fraction hit on right side

Extractor Parameters[Nisan-Z,…, Lu-Reingold-Vadhan-Wigderson]

Ext n bits

k-source k=lg K

.99k bits

almost random

O(log n) random seed

Almost random in variation (statistical) distance.Error = arbitrary constant > 0.

•[TZS]: For k=(n), lg n + O(log log n) bit seed.•New theorem: For k=(n), lg n + O(1) bit seed.

(k)

Dispersers and Inapproximability

• Max Clique: can amplify success probability of PCP verifier using appropriate disperser [Z].

• Chromatic Number: derandomize Feige-Kilian reduction.– [FK]: randomized graph products [BS].– We use derandomized graph powering.

• Derandomized graph products of [Alon-Feige-Wigerson-Z] too weak.

Fractional Chromatic Number

• Chromatic number (G) N/(G), (G) = independence number.

• Fractional chromatic number f(G):

(G)/log N f(G) (G),

f(G) N/(G).

Overview of Feige-Kilian Reduction

• Poly-time reduction from NP-complete L to Gap-Chromatic Number:– x L f(G) b/c,– x L f(G) > b.

Overview of Feige-Kilian Reduction

• Poly-time reduction from NP-complete L to Gap-Chromatic Number:– x L f(G) b/c,– x L (G) < N/b, so f(G) > b

• Amplify: G GD, OR product.– (v1,…,vD) ~ (w1,…,wD) if i, vi ~ wi.– (GD) = (G)D.– f(GD) = f(G)D.– Gap c cD.

• Graph too large: take random subset of VD.

sM

|A| K

Disperser picks subset V’ of VD deterministically

D

V’

V

x (x1,x2,…,xD) y (y1,y2,…,yD)

x x1

x2

x3

y1

yy2

y3

Strong disperser: A i: i(A) sM

1

23

2

sM

|A| K

(G) < sM (G’) < KV’

V

x xi

yyi

If A independent in G’, |A| Kthen ( i) i(A) sM.

Since OR graph product.

Properties of Derandomized Powering

• If (G) < s|V|, then (G’) < K.

f(G’) f(GD) = f(G)D.

Properties of Derandomized Powering

• If (G) < s|V|, then (G’) < K.

– If x L, then (G’) K Nso f(G’) N1-

-Since disperser works for any entropy rate >0.

• f(G’) f(GD) = f(G)D.

– If x L, then f(G’) N.

– Uses D=O((log N)/log s-1).

Extractor/Disperser Outline

Condense:

Extract:

.9

uniform

+ lg n+O(1) random bits

+ O(1) random bits

Extractor for Entropy Rate .9(extension of [AKS])

• G=2c-regular expander on {0,1}m

• Weak source input: walk (v1,v2,…,vD) in G

– m + (D-1)c bits• Random seed: i [D]

• Output: vi.

• Proof: Chernoff bounds for random walks [Gil,Kah]

Condensing with O(1) bits

1. [BKSSW, Raz]: somewhere condenser

• Some choice of seed condenses.

• Uses additive number theory [BKT,BIW]

• 2-bit seed suffices to increase entropy rate.

• New result: 1-bit seed suffices. Simpler.

2. [Raz]: convert to condenser.

Condensing via Incidence Graph

• 1-Bit Somewhere Condenser:– Input: edge– Output: random endpoint

• Condenses rate to somewhere rate (1+), some > 0.– Distribution of (L,P) a somewhere rate (1+) source.

lines points = Fp2

LP (L,P) an edge

iff P on L

N3/2 edges

Somewhere r-source

• (X,Y) is an elementary somewhere r-source if either X or Y is an r-source.

• Somewhere r-source: convex combination of elementary somewhere r-sources.

Incidence Theorem [BKT]

P,L sets of points, lines in Fp2 with |P|, |L| M

p1.9.

# incidences I(P,L)=O(M3/2-), some >0.lines points = Fp2

LP

L P

few edges

Simple Statistical Lemma

• If distribution X is -far from an r-source, then S, |S|<2r: Pr[X S] .

• Proof: take S={x | Pr[X = x] > 2-r}.

2-r

S

Fp

Statistical Lemma for Condenser

• Lemma: If (X,Y) is -far from somewhere r-source, then S supp(X), T supp(Y), |S|,|T| < 2r, such that

Pr[X S and Y T] .

• Proof: S={s: Pr[X=s] > 2-r} T={t: Pr[Y=t] > 2-r}

Statistical Lemma for Condenser

• Lemma: If (X,Y) is -far from somewhere r-source, then S supp(X), T supp(Y), |S|,|T| < 2r, such that

Pr[X S and Y T] .

• Proof of Condenser: Suppose output -far from somewhere r-source. Get sets S and T.

I(S,T) 2r’, r’ = input min-entropy.Contradicts Incidence Theorem.

Additive Number Theory

• A=set of integers, A+A=set of pairwise sums.• Can have |A+A| < 2|A|, if A=arithmetic

progression, e.g. {1,2,…,100}.• Similarly can have |AA| < 2|A|.• Can’t have both simultaneously:

– [ES,Elekes]: max(|A+A|,|AA|) |A|5/4

– False in Fp: A=Fp

Additive Number Theory

• A=set of integers, A+A=set of pairwise sums.• Can have |A+A| < 2|A|, if A=arithmetic

progression, e.g. {1,2,…,100}.• Similarly can have |AA| < 2|A|.• Can’t have both simultaneously:

– [ES,Elekes]: max(|A+A|,|AA|) |A|5/4

– [Bourgain-Katz-Tao, Konyagin]: similar bound over prime fields Fp: |A|1+, assuming 1<|A|<p.9, some >0.

Independent Sources

• Corollary: if |A| p.9, then |AA+A| |A|1+.

• Can we get statistical version of corollary?

– If A,B,C independent k-sources, k .9n, is AB+C close to k’-source, k’=(1+)k? (n=log p)

– [Z]: under Generalized Paley Graph conjecture.

• [Barak-Impagliazzo-Wigderson] proved statistical version.

Simplifying and Slight Strengthening

• Strengthening: assume (A,C) a 2k-source and B an independent k-source.

• Use Incidence Theorem.• Relevance: lines of form ab+c.• How can we get statistical version?

Simplified Proof of BIW

• Suppose AB+C -far from k’-source.• Let S=set of size < 2k’ from simple stat lemma.• Let points P=supp(B) S.• Let lines L=supp((A,C)), where (a,c) line ax+c.• I(P,L) |L| |supp(B)| = 23k.• Contradicts Incidence Theorem.

Conclusions and Future Directions

1. NP-hard to approximate Max Clique and Chromatic Number to within n1-, any >0.

1. NQ-hard to within n/2(log n)1-some >0.2. What is the right n1-o(1) factor?

2. Extractor construction with linear degree for k=(n), m=.99k output bits.

• Linear degree for general k?• 1-bit somewhere-condenser.

• Also simplify/strengthen [BIW,BKSSW,Bo].• Other uses of additive number theory?