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Linear-Degree Extractors and the Inapproximability of Max Clique and Chromatic Number. David Zuckerman University of Texas at Austin. Max Clique and Chromatic Number. [FGLSS,…,Hastad]: Max Clique inapproximable to n 1- , any >0, assuming NP ZPP. - PowerPoint PPT Presentation
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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
• 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.
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.
– 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?