Two Query PCP with Subconstant Error Dana Moshkovitz Princeton University and The Institute for Advanced Study Ran Raz The Weizmann Institute 1

This Talk Hardness of Approximation & PCPs (Probabilistically Checkable Proofs) How we can construct PCPs that are useful for hardness of approximation. 2

Hardness of Approximation The 3SAT Maximization Problem: Given a 3CNF Á, how many clauses can be satisfied simultaneously? 3 Á = (x 7  : x 12  x 1 ) Æ … Æ ( : x 5  : x 9  x 28 )

Hardness of Approximating 3SAT Theorem (Håstad97): For any constant  >0, 3SAT is NP-hard to approximate within ⅞ + . 4 This work: Improving Håstad97 to  =o(1), and many more results!

The Bellare-Goldreich-Sudan Paradigm Projection Games Theorem (aka Hardness of Label-Cover, or two-query projection PCP) 5 Hardness of Approximating 3SAT Long-code based reduction

The Bellare-Goldreich-Sudan Paradigm Projection Games Theorem (aka Hardness of Label-Cover, or two-query projection PCP) 6 Hardness of Approximating Constraint Satisfaction Problems … and many more problems! Long-code based reduction e.g., Vertex-Cover [DS02] e.g., Set-Cover [Feige96]

Projection Games 7 A B Bipartite graph G=(A,B,E) Two sets of labels § A, § B Projections ¼ e : § A  § B Players A & B label vertices Verifier picks random e=(a,b) 2 E Verifier checks ¼ e (A(a) ) = B(b) Value of game = max A,B P(verifier accepts) ¼e¼e

Projection Games Theorem Projection Games Theorem There exists 0<c<1, s.t. For every ² ¸ 1/n c, 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. 8

How To Prove The Projection Games Theorem? ?? 9 Hardness of Approximation Projection Games Theorem [AS92,ALMSS92] PCP Theorem + [Raz94] Parallel Repetition

Caveat in Parallel Repetition Parallel repetition blows-up size to n θ( log 1=²  : – Proves Quasi-NP-hardness – NP-hardness only for constant ². [Feige, Kilian, 95]: No “de-randomization”! 10 Projection Games Theorem There exists 0<c<1, s.t. For every ² ¸ 1/n c, 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.

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. 11

Our Work 12 Hardness of Approximation Projection Games Theorem [AS92,ALMSS92] PCP Theorem + [Raz94]Parallel Repetition New construction with almost-linear size n 1+o(1) poly(1/ ² )

Caveat in Our Work Many labels: k=2 poly(1/ ² ) “Sliding-Scale Conjecture” [BGLR93]: k=poly(1/ ² ) k = poly(n) only for ² ¸ 1/(logn) ¯ for some ¯ >0 13 Projection Games Theorem There exists 0<c<1, s.t. For every ² ¸ 1/n c, 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.

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=n 1+o(1) ). Similarly, improvements to: 3LIN [Håstad,97], amortized query complexity and free bit complexity [Samorodnitsky-Trevisan,00], … 14

Starting Point 15 Projection Games Theorem with many labels For every ², there is k=k(n, ² )=2 poly(n o(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 n 1+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].

Composition Reduce the number of labels k=k(n, ² )=2 poly(n o(1),1/ ² ) k=k( ² )=2 poly(1/ ² ) by composition Previously: Either: 1)Increase in # queries [AS92...BGHSV04] 2)Two queries, but error ² ¼ 1 [DR04] Recently: Generalization by [Dinur-Harsha, 09] 16

Code Concatenation (Forney, 1966) Can iterate. When n  i · logn, can use Hadamard (exponential length). 17 poly(n ±,1/ ² ) poly(n ± 2,1/ ² ) n 1+o(1) poly(1/ ² ) Length: multiplies Distance: multiplies

Analogy to Codes 18 B A... labels to B  codeword A vertex  constraint: B neighborhood consistent with label to a value  max A E a [% consistent neighbors]

Composition As Concatenation? 19 B A... Main Issue: How to check the A constraints?

Perspective Change Perspective: Switch Sides! Associate each A vertex with its B neighborhood. View B vertices as posing constraints: consistency among containing neighborhoods. 20 B A

Sunflowers Label to B vertex = “sunflower” of labels to A neighbors log| § new | = Bdegree· log| § old | = poly(1/ ² )· log| § old | 21 B A Will reduce this!

The Key Idea Label to B vertex = a sunflower of sub-petals Encode A labels so can locally decode/reject center 22 B A

Local Decode/Reject (Strengthening of PCP) 23 A label center... B inner A inner

Composition Encode each neighborhood with LDRC 24 B A... B inner A inner... B inner A inner

Composition Encode each neighborhood with LDRC B inner...

Thank You! 26