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Linear-Degree Extractors and the Inapproximability of Max Clique and Chromatic Number

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.

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Linear-Degree Extractors and the Inapproximability of Max Clique and Chromatic Number

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  1. Linear-Degree Extractors and the Inapproximability of Max Clique and Chromatic Number David Zuckerman University of Texas at Austin

  2. 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.

  3. Outline • Extractors and Dispersers • Dispersers and Inapproximability • Extractor/Disperser Construction • Additive Number Theory • Conclusion

  4. Weak Random Sources • Random element from set A, |A|  2k. {0,1}n |A|  2k

  5. 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.

  6. 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.

  7. Goal very long Ext long weakly random almost random Should work for all k-sources. Problem: impossible.

  8. Solution: Extractor[Nisan-Z] short truly random very long Ext long weakly random almost random

  9. Extractor Parameters[NZ,…, Lu-Reingold-Vadhan-Wigderson] d=O(log n) truly random n bits m=.99k bits Ext almost random k-source Almost random in variation (statistical) distance. Error  = arbitrary constant > 0.

  10. Graph-Theoretical View of Extractor N=2n output -uniform Disperser  K=2k M=2m D=2d x E(x,y1)  (1- )M E(x,y2) E(x,y3) Think of K=N, M . Goal: D=O(log N).

  11. 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]

  12. 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).

  13. 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

  14. Extractor Parameters[Nisan-Z,…, Lu-Reingold-Vadhan-Wigderson] • [TZS]: For k=(n), lg n + O(log log n) bit seed. • New theorem: For k=(n), lg n + O(1) bit seed. O(log n) random seed (k) n bits .99k bits Ext almost random k-source k=lg K Almost random in variation (statistical) distance. Error  = arbitrary constant > 0.

  15. 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.

  16. 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).

  17. 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.

  18. 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.

  19. Disperser picks subset V’ of VD deterministically V’ Strong disperser: A i: i(A)  sM V D |A|  K 1 x x1 y1 2 3 x2  sM y2 2 y x3 y3 x  (x1,x2,…,xD) y  (y1,y2,…,yD)

  20. (G) < sM (G’) < K V’ If A independent in G’, |A|  K then ( i) i(A)  sM. V |A|  K x xi yi  sM y Since OR graph product.

  21. Properties of Derandomized Powering • If (G) < s|V|, then (G’) < K. • f(G’) f(GD) = f(G)D.

  22. Properties of Derandomized Powering • If (G) < s|V|, then (G’) < K. • If x  L, then (G’) < K  N, so 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).

  23. Extractor/Disperser Outline  + O(1) random bits Condense: .9 Extract: + lg n+O(1) random bits  uniform

  24. 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]

  25. Condensing with O(1) bits • [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. • [Raz]: convert to condenser.

  26. Condensing via Incidence Graph lines • 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. points = Fp2 • (L,P) an edge iff P on L N3/2 edges P L

  27. 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.

  28. 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 L P P L few edges

  29. 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

  30. 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}

  31. 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.

  32. 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 |AA| < 2|A|. • Can’t have both simultaneously: • [ES,Elekes]: max(|A+A|,|AA|)  |A|5/4 • False in Fp: A=Fp

  33. 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 |AA| < 2|A|. • Can’t have both simultaneously: • [ES,Elekes]: max(|A+A|,|AA|)  |A|5/4 • [Bourgain-Katz-Tao, Konyagin]: similar bound over prime fields Fp:  |A|1+, assuming 1<|A|<p.9, some >0.

  34. Independent Sources • Corollary: if |A|  p.9, then |AA+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.

  35. 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?

  36. 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.

  37. Conclusions and Future Directions • NP-hard to approximate Max Clique and Chromatic Number to within n1-, any >0. • NQ-hard to within n/2(log n)1-some >0. • What is the right n1-o(1) factor? • 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?

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