Download presentation

Presentation is loading. Please wait.

Published byCallie Gutierrez Modified over 3 years ago

1
Average-case Complexity Luca Trevisan UC Berkeley

2
Distributional Problem P computational problem – e.g. SAT D distribution over inputs – e.g. n vars 10n clauses

3
Positive Results: Algorithm that solves P efficiently on most inputs – Interesting when P useful problem, D distribution arising “in practice” Negative Results: If, then no such algorithm – P useful, D natural guide algorithm design – Manufactured P,D, still interesting for crypto, derandomization

4
Positive Results: Algorithm that solves P efficiently on most inputs – Interesting when P useful problem, D distribution arising “in practice” Negative Results: If, then no such algorithm – P useful, D natural guide algorithm design – Manufactured P,D, still interesting for crypto, derandomization

5
Holy Grail If there is algorithm A that solves P efficiently on most inputs from D Then there is an efficient worst-case algorithm for [the complexity class] P [belongs to]

6
Part (1) In which the Holy Grail proves elusive

7
The Permanent Perm (M) := i M(i, (i)) Perm() is #P-complete Lipton (1990): If there is algorithm that solves Perm() efficiently on most random matrices, Then there is an algorithm that solves it efficiently on all matrices (and BPP=#P)

8
Lipton’s Reduction Suppose operations are over finite field of size >n A is good-on-average algorithm (wrong on < 1/(10(n+1)) fraction of matrices) Given M, pick random X, compute A(M+X), A(M+2X),…,A(M+(n+1)X) Whp the same as Perm(M+X),Perm(M+2X),…,Perm(M+(n+1)X)

9
Lipton’s Reduction Given Perm(M+X),Perm(M+2X),…,Perm(M+(n+1)X) Find univariate degree-n polynomial p such that p(t) = Perm(M+tX) for all t Output p(0)

10
Improvements / Generalizations Can handle constant fraction of errors [Gemmel-Sudan] Works for PSPACE-complete, EXP-complete,… [Feigenbaum-Fortnow, Babai-Fortnow-Nisan-Wigderson] Encode the problem as a polynomial

11
Strong Average-Case Hardness [Impagliazzo, Impagliazzo-Wigderson] Manufacture problems in E, EXP, such that – Size-t circuit correct on ½ + 1/t inputs implies – Size poly(t) circuit correct on all inputs Motivation: [Nisan-Wigderson] P=BPP if there is problem in E of exponential average-case complexity

12
Strong Average-Case Hardness [Impagliazzo, Impagliazzo-Wigderson] Manufacture problems in E, EXP, such that – Size-t circuit correct on ½ + 1/t inputs implies – Size poly(t) circuit correct on all inputs Motivation: [Impagliazzo-Wigderson] P=BPP if there is problem in E of exponential average worst-case complexity

13
Open Question 1 Suppose there are worst-case intractable problems in NP Are there average-case intractable problems?

14
Strong Average-Case Hardness [Impagliazzo, Impagliazzo-Wigderson] Manufacture problems in E, EXP, such that – Size-t circuit correct on ½ + 1/t inputs implies – Size poly(t) circuit correct on all inputs [Sudan-T-Vadhan] – IW result can be seen as coding-theoretic – Simpler proof by explicitly coding-theoretic ideas

15
Encoding Approach Viola proves that an error-correcting code cannot be computed in AC0 The exponential-size error-correcting code computation not possible in PH

16
Problem-specific Approaches? [Ajtai] Proves that there is a lattice problem such that: – If there is efficient average-case algorithm – There is efficient worst-case approximation algorithm

17
Ajtai’s Reduction Lattice Problem – If there is efficient average-case algorithm – There is efficient worst-case approximation algorithm The approximation problem is in NP coNP Not NP-hard

18
Holy Grail Distributional Problem: – If there is efficient average-case algorithm – P=NP (or NP in BPP, or NP has poly-size circuits,…) Already seen: no “encoding” approach works Can extensions of Ajtai’s approach work?

19
A Class of Approaches L problem in NP, D distribution of inputs R reduction of SAT to : Given instance f of SAT, – R produces instances x 1,…,x k of L, each distributed according to D – Given L(x 1 ),…,L(x 1 ), R is able to decide f If there is good-on-average algorithn for, we solve SAT in polynomial time [cf. Lipton’s work on Permanent]

20
A Class of Approaches L,W problems in NP, D (samplable) distribution of inputs R reduction of W to Given instance w of W, – R produces instances x 1,…,x k of L, each distributed according to D – Given L(x 1 ),…,L(x 1 ), R is able to decide w If there is good-on-average algorithm for, we solve W in polynomial time; Can W be NP-complete?

21
A Class of Approaches Given instance w of W, – R produces instances x 1,…,x k of L, each distributed according to D – Given L(x 1 ),…,L(x 1 ), R is able to decide w Given good-on-average algorithm for, we solve W in polynomial time; If we have such reduction, and W is NP-complete, we have Holy Grail! Feigenbaum-Fortnow: W is in “coNP”

22
Feigenbaum-Fortnow Given instance w of W, – R produces instances x 1,…,x k of L, each distributed according to D – Given L(x 1 ),…,L(x 1 ), R is able to decide w Using R, Feigenbaum-Fortnow design a 2-round interactive proof with advice for coW Given w, Prover convinces Verifier that R rejects w after seeing L(x 1 ),…,L(x 1 )

23
Feigenbaum-Fortnow Given instance w of W, – R produces instances x of L distributed as in D – w in L iff x in L Suppose we know Pr D [ x in L]= ½ V P w R(w) = x 1 R(w) = x 2... R(w) = x m x 1, x 2,..., x m (Yes,w 1 ),No,..., (Yes, w m ) Accept iff all simulations of R reject and m/2 +/- sqrt(m) answers are certified Yes

24
Feigenbaum-Fortnow Given instance w of W, p:= Pr[ x i in L] – R produces instances x 1,…,x k of L, each distrib. according to D – Given L(x 1 ),…,L(x k ), R is able to decide w V w R(w) -> x 1 1,…,x k 1... R(w) -> x 1 m,…,x k m P x 1 1,…,x k m (Yes,w 1 1 ),…,NO Accept iff -pkm +/- sqrt(pkm) YES with certificates -R rejects in each case

25
Generalizations Bogdanov-Trevisan: arbitrary non-adaptive reductions Main Open Question: What happens with adaptive reductions?

26
Open Question 1 Prove the following: Suppose: W,L are in NP, D is samplable distribution, R is poly-time reduction such that – If A solves on 1-1/poly(n) frac of inputs – Then R with oracle A solves W on all inputs Then W is in “coNP”

27
By the Way Probably impossible by current techniques: If NP not contained in BPP There is a samplable distribution D and an NP problem L Such that is hard on average

28
By the Way Probably impossible by current techniques: If NP not contained in BPP There is a samplable distribution D and an NP problem L Such that for every efficient A A makes many mistakes solving L on D

29
By the Way Probably impossible by current techniques: If NP not contained in BPP There is a samplable distribution D and an NP problem L Such that for every efficient A A makes many mistakes solving L on D [Guttfreund-Shaltiel-TaShma] Prove: If NP not contained in BPP For every efficient A There is a samplable distribution D Such that A makes many mistakes solving SAT on D

30
Part (2) In which we amplify average-case complexity and we discuss a short paper

31
Revised Goal Proving “If NP contains worst-case intractable problems, then NP contains average-case intractable problems” Might be impossible Average-case intractability comes in different quantitative degrees Equivalence?

32
Average-Case Hardness What does it mean for to be hard-on-average? Suppose A is efficient algorithm Sample x ~ D Then A(x) is noticeably likely to be wrong How noticeably?

33
Average-Case Hardness Amplification Ideally: If there is, L in NP, such that every poly-time algorithm (poly-size circuit) makes > 1/poly(n) mistakes Then there is, L’ in NP, such that every poly-time algorithm (poly-size circuit) makes > ½ - 1/poly(n) mistakes

34
Amplification “Classical” approach: Yao’s XOR Lemma Suppose: for every efficient A Pr D [ A(x) = L(x) ] < 1- Then: for every efficient A’ Pr D [ A’(x 1,…,x k ) = L(x 1 ) xor … xor L(x k ) ] < ½ + (1 - 2 ) k + negligible

35
Yao’s XOR Lemma Suppose: for every efficient A Pr D [ A(x) = L(x) ] < 1- Then: for every efficient A’ Pr D [ A’(x 1,…,x k ) = L(x 1 ) xor … xor L(x k ) ] < ½ + (1 - 2 ) k + negligible Note: computing L(x 1 ) xor … xor L(x k ) need not be in NP, even if L is in NP

36
O’Donnell Approach Suppose: for every efficient A Pr D [ A(x) = L(x) ] < 1- Then: for every efficient A’ Pr D [ A’(x 1,…,x k ) = g(L(x 1 ), …, L(x k )) ] < ½ + small(k, ) For carefully chosen monotone function g Now computing g(L(x 1 ),…, L(x k )) is in NP, if L is in NP

37
Amplification (Circuits) Ideally: If there is, L in NP, such that every poly-time algorithm (poly-size circuit) makes > 1/poly(n) mistakes Then there is, L’ in NP, such that every poly-time algorithm (poly-size circuit) makes > ½ - 1/poly(n) mistakes Achieved by [O’Donnell, Healy-Vadhan-Viola] for poly-size circuits

38
Amplification (Algorithms) If there is, L in NP, such that every poly- time algorithm makes > 1/poly(n) mistakes Then there is, L’ in NP, such that every poly-time algorithm makes > ½ - 1/polylog(n) mistakes [T] [Impagliazzo-Jaiswal-Kabanets-Wigderson] ½ - 1/poly(n) but for P NP||

39
Open Question 2 Prove: If there is, L in NP, such that every poly-time algorithm makes > 1/poly(n) mistakes Then there is, L’ in NP, such that every poly-time algorithm makes > ½ - 1/poly(n) mistakes

40
Completeness Suppose we believe there is L in NP, D distribution, such that is hard Can we point to a specific problem C such that is also hard?

41
Completeness Suppose we believe there is L in NP, D distribution, such that is hard Can we point to a specific problem C such that is also hard? Must put restriction on D, otherwise assumption is the same as P != NP

42
Side Note Let K be distribution such that x has probability proportional to 2 -K(x) Suppose A solves on 1-1/poly(n) fraction of inputs of length n Then A solves L on all but finitely many inputs Exercise: prove it

43
Completeness Suppose we believe there is L in NP, D samplable distribution, such that is hard Can we point to a specific problem C such that is also hard?

44
Completeness Suppose we believe there is L in NP, D samplable distribution, such that is hard Can we point to a specific problem C such that is also hard? Yes we can! [Levin, Impagliazzo-Levin]

45
Levin’s Completeness Result There is an NP problem C, such that If there is L in NP, D computable distribution, such that is hard Then is also hard

47
Reduction Need to define reduction that preserves efficiency on average (Note: we haven’t yet defined efficiency on average) R is a (Karp) average-case reduction from to if 1.x in A iff R(x) in B 2.R(D A ) is “dominated” by D B : Pr[ R(D A )=y] < poly(n) * Pr [D B = y]

48
Reduction R is an average-case reduction from to if x in A iff R(x) in B R(D A ) is “dominated” by D B : Pr[ R(D A )=y] < poly(n) * Pr [D B = y] Suppose we have good algorithm for Then algorithm also good for Solving reduces to solving

49
Reduction If Pr[ Y=y] < poly(n) * Pr [D B = y] and we have good algorithm for Then algorithm also good for Reduction works for any notion of average-case tractability for which above is true.

50
Levin’s Completeness Result Follow presentation of [Goldreich] If is easy on average Then for every L in NP, every D computable distribution, is easy on average BH is non-deterministic Bounded Halting: given, does M(x) accept with t steps?

51
Levin’s Completeness Result BH, non-deterministic Bounded Halting: given, does M(x) accept with t steps? Suppose we have good-on-average alg A Want to solve, where L solvable by NDTM M First try: x ->

52
Levin’s Completeness Result First try: x -> Doesn’t work: x may have arbitrary distribution, we need target string to be nearly uniform (high entropy) Second try: x -> Where C() is near-optimal compression alg, M’ recover x from C(x), then runs M

53
Levin’s Completeness Result Second try: x -> Where C() is near-optimal compression alg, M’ recover x from C(x), then runs M Works! Provided C(x) has length at most O(log n) + log 1/Pr D [x] Possible if cumulative distribution function of D is computable.

54
Impagliazzo-Levin Do the same but for all samplable distribution Samplable distribution not necessarily efficiently compressible in coding theory sense. (E.g. output of PRG) Hashing provides “non-constructive” compression

55
Complete Problems BH with Uniform distribution Tiling problem with Uniform distribution [Levin] Generalized edge-coloring [Venkatesan-Levin] Matrix representability [Venkatesan-Rajagopalan] Matrix transformation [Gurevich]...

56
Open Question 3 L in NP, M NDTM for L is specified by k bits Levin’s reduction incurs 2 k bits in fraction of “problematic” inputs (comparable to having 2 k slowdown) Limited to problems having non-deterministic algorithm of 5 bytes Inherent?

57
More Reductions? Still relatively few complete problems Similar to study of inapproximability before Papadimitriou-Yannakakis and PCP Would be good, as in Papadimitriou-Yannakakis, to find reductions between problems that are not known to be complete but are plausibly hard

58
Open Question 4 (Heard from Russell Impagliazzo) Prove that If 3SAT is hard on instances with n variables and 10n clauses, Then it is also hard on instances with 12n clauses

59
See http://www.cs.berkeley.edu/~luca/average [slides, references, addendum to Bogdanov-T, coming soon] http://www.cs.berkeley.edu/~luca/average http://www.cs.uml.edu/~wang/acc-forum/ [average-case complexity forum] http://www.cs.uml.edu/~wang/acc-forum/ Impagliazzo A personal view of average-case complexity Structures’95 Goldreich Notes on Levin’s theory of average-case complexity ECCC TR-97-56 Bogdanov-T. Average case complexity F&TTCS 2(1): (2006)

Similar presentations

OK

Ragesh Jaiswal Indian Institute of Technology Delhi Threshold Direct Product Theorems: a survey.

Ragesh Jaiswal Indian Institute of Technology Delhi Threshold Direct Product Theorems: a survey.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on nationalism in india Ppt on albert einstein Download ppt on esterification Ppt on power generation by speed breaker road Ppt on indian union budget 2013-14 Class 7 science ppt on light Ppt on phonetic transcription dictionary Ppt on natural numbers meaning Ppt on job satisfaction of employees Ppt on glasgow coma scale