1

Two Query PCP with Subconstant Error

Dana MoshkovitzPrinceton University and

The Institute for Advanced Study

Ran RazThe Weizmann Institute

2

This Talk

Hardness of Approximation & PCPs (Probabilistically Checkable Proofs)

How we can construct PCPs that are useful for hardness of approximation.

3

Hardness of Approximation

The 3SAT Maximization Problem:Given a 3CNF Á, how many clauses can be

satisfied simultaneously?

x1 x2 x3 x4 x5 x6 xn

0 0 1 0 1 1 . . . 1

Á = (x7 : x12 x1) Æ … Æ (:x5 : x9 x28)

4

Hardness of Approximating 3SAT

Theorem (Håstad97): For any constant >0, 3SAT is NP-hard to

approximate within ⅞ + .

This work: Improving Håstad97 to =o(1), and many more results!

5

The Bellare-Goldreich-Sudan Paradigm

Projection Games Theorem(aka Hardness of Label-Cover, or

two-query projection PCP)

Hardness of Approximating 3SAT

Long-code based reduction

6

The Bellare-Goldreich-Sudan Paradigm

Projection Games Theorem(aka Hardness of Label-Cover, or

two-query projection PCP)

Hardness of Approximating Constraint Satisfaction Problems

… and many more problems!

Long-code based reduction e.g., Vertex-Cover [DS02]

e.g., Set-Cover [Feige96]

7

Projection Games

?

?

A

B• Bipartite graph G=(A,B,E) • Two sets of labels §A, §B

• Projections ¼e:§A§B

• Players A & B label vertices• Verifier picks random e=(a,b)2E• Verifier checks ¼e(A(a)) = B(b)

• Value of game = maxA,BP(verifier accepts)

¼e

8

Projection Games Theorem

Projection Games Theorem There exists 0<c<1, s.t.For every ²¸1/nc, there is k=k(²), such that it is NP-

hard to decide for a given projection game on k labels whether its value = 1 or < ².

9

How To Prove The Projection Games Theorem?

??

Hardness of Approximation

Projection Games Theorem

[AS92,ALMSS92] PCP Theorem + [Raz94] Parallel Repetition

10

Caveat in Parallel Repetition

• Parallel repetition blows-up size to nθ(log 1=²:– Proves Quasi-NP-hardness – NP-hardness only for constant ².

• [Feige, Kilian, 95]: No “de-randomization”!

Projection Games Theorem There exists 0<c<1, s.t.For every ²¸1/nc, there is k=k(²), such that it is NP-hard to decide for a given projection game on k labels whether its value = 1 or < ².

11

Subconstant Error for Projection Games?

• [RS97, AS97, DFKRS99, MR07]: subconstant error ²=²(n), as low as ²=2-(logn)1® for all ®>0.

• More than two queries! Not projection game! Much less useful for hardness of approx.

• Folklore: three queries for error ²=2-(logn)® for some ®>0.

12

Our Work

Hardness of Approximation

Projection Games Theorem

[AS92,ALMSS92] PCP Theorem + [Raz94]Parallel RepetitionNew construction with almost-linear size

n1+o(1)poly(1/²)

13

Caveat in Our Work

• Many labels: k=2poly(1/²)

• “Sliding-Scale Conjecture” [BGLR93]: k=poly(1/²) • k = poly(n) only for ²¸1/(logn)¯ for some ¯>0

Projection Games Theorem There exists 0<c<1, s.t.For every ²¸1/nc, there is k=k(²), such that it is NP-hard to decide for a given projection game on k labels whether its value = 1 or < ².

14

Implications

Improving Håstad: NP-hard to approximate 3SAT on inputs of size N within 7/8+1/(loglogN) for some constant >0 (blow-up N=n1+o(1)).

Similarly, improvements to: 3LIN [Håstad,97], amortized query complexity and free bit complexity [Samorodnitsky-Trevisan,00],

…

15

Starting Point

Projection Games Theorem with many labelsFor every ², there is k=k(n,²)=2poly(no(1),1/²), such that

it is NP-hard to decide for a given projection game on k labels whether its value = 1 or < ².

The reduction is almost-linear n1+o(1)poly(1/²).

• Construction is algebraic, based on low degree testing theorem with low error [AS97,RS97].

• Almost-linear size by [MR06,MR07].

16

Composition

Reduce the number of labelsk=k(n,²)=2poly(no(1),1/²) k=k(²)=2poly(1/²)

by composition

Previously: Either:1) Increase in # queries [AS92...BGHSV04]2) Two queries, but error ²¼1 [DR04]

Recently: Generalization by [Dinur-Harsha, 09]

17

Code Concatenation (Forney, 1966)

• Can iterate. When ni·logn, can use Hadamard (exponential length).

. . .

poly(n±,1/²)

. . .

poly(n±2,1/²)

n1+o(1)poly(1/²)

• Length: multiplies• Distance: multiplies

18

Analogy to Codes

. . .B

A . . .labels to B codewordA vertex constraint: B neighborhood

consistent with label to a

value maxA Ea[% consistent neighbors]

19

Composition As Concatenation?

. . .B

A . . .

. . .Main Issue: How to check the A constraints?

20

Change Perspective: Switch Sides!

• Associate each A vertex with its B neighborhood.• View B vertices as posing constraints: consistency

among containing neighborhoods.

. . .B

A. . . . . . . . . . .

. . .

. . . . . . . . . . .

21

Sunflowers• Label to B vertex = “sunflower” of labels to A neighbors• log|§new| = Bdegree· log|§old| = poly(1/²)· log|§old|

. . .B

A. . . . . . . . . . .

Will reduce this!

22

The Key Idea

Label to B vertex = a sunflower of sub-petalsEncode A labels so can locally decode/reject center

. . .B

A. . . . . . . . . . .

23

Local Decode/Reject (Strengthening of PCP)

. . .

...

A label

center

. . .

Binner

Ainner

CompositionEncode each neighborhood with LDRC

24

. . .B

A. . .

... .... . .Binner

Ainner. . . . . . . . . . . .

Binner

Ainner

CompositionEncode each neighborhood with LDRC

25

... .... . .Binner Binner

. . .

26

Thank You!