Dinur s Proof of the PCP Theorem

Transcription

1 Dinur s Proof of the PCP Theorem Zampetakis Manolis School of Electrical and Computer Engineering National Technical University of Athens December 22, 2014

2 Table of contents 1 Introduction 2 PCP Theorem Proof Expaderize the Graph Alphabet Reduction Finishing the proof 3 Conclusions

3 Introduction Characterization of NP verifier accept reject input x proof y prover M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

4 Introduction Characterization of NP Prover The prover of a language L has to output a correct proof y with y poly( x ) when x L. When x / L prover could output anything. Verifier The verifier of a language L NP has to be efficient (polynomialy time bounded). M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

5 Introduction Characterization of NP NP prover-verifier For every language L NP there exists a prover P and an efficient verifier V, that reads : the input string x the proof string y and accepts or rejects such that : Completeness : If x L then P outputs a proof y such that V accepts. Soundness : If x / L then V rejects every proof y. M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

6 Introduction New characterization of NP [Arora, Safra] random-bits r verifier accept reject input x proof y prover M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

7 Introduction New characterization of NP [Arora, Safra] Probabilistically Checkable of Proofs Definition : PCP [p(n), q(n)] For every L PCP[p(n), q(n)] there exists a prover P and an efficient verifier V, that reads : an input string x with length n a random string r with length p(n) q(n) bits randomly from the proof y and accepts or rejects, such that : Completeness : If x L then P outputs a proof y such that V accepts with probability 1. Soundness : If x / L then V accepts with probability 1 2. M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

8 Introduction New characterization of NP [Arora, Safra] Probabilistically Checkable of Proofs Definition : PCP [O(log n), O(1)] For every L PCP[O(log n), O(1)] there exists a prover P and an efficient verifier V, that reads : an input string x with length n a random string r with length c log n k bits randomly from the proof y with c, k R + and accepts or rejects, such that : Completeness : If x L then P outputs a proof y such that V accepts with probability 1. Soundness : If x / L then V accepts with probability 1 2. M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

9 Introduction New characterization of NP [Arora, Safra] Probabilistically Checkable of Proofs PCP Theorem [Arora-Lund-Motwani-Sudan-Szegedy 92] NP = PCP[O(log n), O(1)] M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

10 Introduction Key Idea Graph Coloring Problem - Constraint Graph Problem M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

11 Introduction Key Idea Graph Coloring Problem - Constraint Graph Problem M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

12 Introduction Key Idea Graph Coloring Problem - Constraint Graph Problem Key Idea Add contraints for every path with length t in the graph G. M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

13 Introduction Good Idea but... M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

14 Introduction A Lemma on expanders... from Trevisan s lectures Lemma Let G = (V, E) be an expander and F E then for every path p in G with length t ( Pr[p completely misses F ] 1 t F ) E M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

15 Introduction Proof Overview Expanderize the Graph M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

16 Introduction Proof Overview Expanderize the Graph M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

17 Introduction to Expanders Definition The edge expansion of a graph G = (V, E), denoted by φ(g), is defined as E(S, φ(g) = min S) S V, S V /2 S M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

18 Introduction to Expanders Lemma There exists a constant φ 0 such that for every n N and d < n there is an efficient algorithm to construct a d regular graph G with φ(g) φ 0. M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

19 Introduction to Expanders Lemma There exists a constant φ 0 such that for every n N and d < n there is an efficient algorithm to construct a d regular graph G with φ(g) φ 0. Lemma Let G = (V, E) be an expander and F E then for every path p in G with length t ( Pr[p completely misses F ] 1 t F ) E M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

20 Introduction to Expanders Lemma There exists a constant φ 0 such that for every n N and d < n there is an efficient algorithm to construct a d regular graph G with φ(g) φ 0. Lemma Let G = (V, E) be an expander and F E then for every path p in G with length t ( Pr[p completely misses F ] 1 t F ) E useful property: If a we add edges to a graph which is expander then the graph remains expander. M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

21 Union with expander Expaderize the Graph G H d regular expander M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

22 Union with expander Expaderize the Graph G G H G is satifiable iff G H is satisfiable M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

23 Union with expander Expaderize the Graph G G H what about gap? M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

24 Union with expander Expaderize the Graph G d regular G H 2d regular G is satifiable iff G H is satisfiable M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

25 Union with expander Expaderize the Graph G d regular G H 2d regular gap(g H) 1/2gap(G) M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

26 Proof Overview Expaderize the Graph Make G d regular Union with expander M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

27 Proof Overview Expaderize the Graph Make G d regular Union with expander M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

28 Make G d regular Papadimitriou and Yannakakis Expaderize the Graph u M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

29 Make G d regular Papadimitriou and Yannakakis Expaderize the Graph u H u M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

30 Make G d regular Papadimitriou and Yannakakis Expaderize the Graph u H u? What kind of constraints in H u? M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

31 Make G d regular Papadimitriou and Yannakakis Expaderize the Graph u H u? What kind of constraints in H u? EQUALITY M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

32 Make G d regular Papadimitriou and Yannakakis Expaderize the Graph u H u G is satifiable iff G is satisfiable M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

33 Make G d regular Papadimitriou and Yannakakis Expaderize the Graph u H u What about gap? M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

34 Make G d regular Papadimitriou and Yannakakis Expaderize the Graph u (d 1) regular expander H u gap(g ) 1/O(1)gap(G) M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

35 Proof Overview Expaderize the Graph Make G d regular Union with expander M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

36 Proof Overview Expaderize the Graph Make G d regular Union with expander M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

37 Key Idea Add contraints for every path with length t in the graph G. M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

38 Key Idea Add contraints for every path with length t in the graph G. for t = 2 Our initial alphabet is Σ = {red, green, blue}. Now every vertex u has an opinion about the color of the vertices in N(u) and therefore the alphabet becomes Σ = Σ Σ d. For every path {u, w, v} with length 2 we add an edge {u, v} with constraint : c {u,v} is true if the opinion u has about w is the same as the opinion v has about w. M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

39 Key Idea Add contraints for every path with length t in the graph G. for every t Our initial alphabet is Σ = {red, green, blue}. Now every vertex u has an opinion about the color of the vertices in N(u) and therefore the alphabet becomes Σ = Σ Σ d Σ dt. For every path {u, w,..., v} with length t we add an edge {u, v} with constraint : c {u,v} is true if the opinion u has about every internal vertex w is the same as the opinion v has about w, and every internal edge of G is valid using this opinion. M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

40 Key Idea Add contraints for every path with length t in the graph G. for every t Our initial alphabet is Σ = {red, green, blue}. Now every vertex u has an opinion about the color of the vertices in N(u) and therefore the alphabet becomes Σ = Σ Σ d Σ dt. For every path {u, w,..., v} with length t we add an edge {u, v} with constraint : c {u,v} is true if the opinion u has about every internal vertex w is the same as the opinion v has about w, and every internal edge of G is valid using this opinion. G is satifiable iff G is satisfiable M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

41 Computational Complexity Oded Goldreich M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

42 We have now to use the following : Lemma Let G = (V, E) be an expander and F E then for every path p in G with length t ( Pr[p completely misses F ] 1 t F ) E Where we set F the set of unsatisfied constraints in G. M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

43 Lemma gap(g ) t O(1) gap(g) Proof Sketch gap(g ) 1/3 Pr e [e passes through F ] 1/3(1 Pr e [e completely misses F ]) 1/3(1 (t gap(g))) = t O(1) gap(g) M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

44 If we set t = O(n) we have finished! M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

45 If we set t = O(n) we have finished! But we cannot do this because then size(g ) = O(d t ) = O(d O(n) ) which is inefficient!!! M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

46 If we set t = O(n) we have finished! But we cannot do this because then size(g ) = O(d t ) = O(d O(n) ) which is inefficient!!! Therefore t must be a constant! M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

47 Proof Overview Make G d regular Union with expander Double gap M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

48 Proof Overview Make G d regular Union with expander Double gap Alphabet Reduction M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

49 Proof Overview Make G d regular Union with expander Double gap Alphabet Reduction 1.. log n M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

50 Proof Overview Make G d regular Union with expander Double gap Alphabet Reduction 1.. log n M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

51 Alphabet Reduction Alphabet Reduction Effects of the Reduction Size increases a constant factor Gap decreases a constant factor Alphabet size redused to 16 M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

52 Alphabet Reduction Alphabet Reduction Effects of the Reduction Size increases a constant factor Gap decreases a constant factor Alphabet size redused to 16 Proof Techniques Hadamard Codes Linearity Testing Fourier Analysis M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

53 Finishing the proof Finishing the proof random-bits r Reduction 3-Coloring Dinur s Process verifier accept reject input x proof y prover M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

54 Conclusions Let s flip the coin Coin For every language L in NP there is a way to write proofs such that for every instance x : If x L then there is a correct proof If x / L then every proof has a lot of errors M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

55 Conclusions Let s flip the coin Coin For every language L in NP there is a way to write proofs such that for every instance x : If x L then there is a correct proof If x / L then every proof has a lot of errors One side PCP Theorem M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

56 Conclusions Let s flip the coin Coin For every language L in NP there is a way to write proofs such that for every instance x : If x L then there is a correct proof If x / L then every proof has a lot of errors One side PCP Theorem The other side Hardness of Approximation M. Zampetakis (NTUA) Dinur s Proof December 22, / 42

57 Thanks! Thank you! :)