Introduction to PCP and Hardness of Approximation Dana Moshkovitz Princeton University and The Institute for Advanced Study 1

This Talk A Groundbreaking Discovery! 2 (From ) The PCP Theorem and Hardness of Approximation

A Canonical Optimization Problem MAX-3SAT: Given a 3CNF Á, what fraction of the clauses can be satisfied simultaneously? 3 Á = (x 7  : x 12  x 1 ) Æ … Æ ( : x 5  : x 9  x 28 ) x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 x n-3 x n-2 x n-1 xnxn...

4 Good Assignment Exists Claim: There must exist an assignment that satisfies at least 7/8 fraction of clauses. Proof: Consider a random assignment. x1x1 x2x2 x3x3 xnxn...

5 1. Find the Expectation Let Y i be the random variable indicating whether the i-th clause is satisfied. For any 1  i  m,  FFFF FFTT FTFT FTTT TFFT TFTT TTFT TTTT

6 1. Find the Expectation The number of clauses satisfied is a random variable Y=  Y i. By the linearity of the expectation: E[Y] = E[  Y i ] =  E[Y i ] = 7/8m

7 2. Conclude Existence Thus, there exists an assignment which satisfies at least the expected fraction (7/8) of clauses.

8 ® -Approximation (Max Version) OPT OPT(x) For every input x, computed value C(x): ® ¢ OPT(x) · C(x) · OPT(x) Corollary: There is an efficient ⅞-approximation algorithm for MAX-3SAT.

Better Approximation? Fact: An efficient tighter than ⅞-approximation algorithm is not known. Our Question: Can we prove that if P≠NP such algorithm does not exist? 9

10 Computation  Decision Hardness of distinguishing far off instances  Hardness of approximation AB gap OPT(x) OPT

11 Gap Problems (Max Version) Instance: … Problem: to distinguish between the following two cases: The maximal solution ≥ B The maximal solution < A

12 Gap NP-Hard  Approximation NP-hard Claim: If the [A,B]-gap version of a problem is NP-hard, then that problem is NP-hard to approximate to within factor A/B.

13 Gap NP-Hard  Approximation NP-hard Proof (for maximization): Suppose there is an approximation algorithm that, for every x, outputs C(x) ≤ OPT so that C(x) ≥ A/B ¢ OPT. Distinguisher(x): * If C(x) ≥ A, return ‘YES’ * Otherwise return ‘NO’ AB

14 (1) If OPT(x) ≥ B (the correct answer is ‘YES’), then necessarily, C(x) ≥ A/B ¢ OPT(x) ≥ A/B·B = A (we answer ‘YES’) (2) If OPT(x)<A (the correct answer is ‘NO’), then necessarily, C(x) ≤ OPT(x) < A (we answer ‘NO’). Gap NP-Hard  Approximation NP-hard

New Focus: Gap Problems Can we prove that gap-MAX-3SAT is NP-hard? 15

Connection to Probabilistic Checking of Proofs [FGLSS91,AS92,ALMSS92] Claim: If [A,1]-gap-MAX-3SAT is NP-hard, then every NP language L has a probabilistically checkable proof (PCP): There is an efficient randomized verifier that queries 3 proof symbols: x  L: There exists a proof that is always accepted. x  L: For any proof, the probability to err and accept is ≤A. Note: Can get error probability ² by making O(log1/ ² ) queries. 16

Probabilistic Checking of x  L? 17 If yes, all of Á clauses are satisfied. If no, fraction ≤A of Á clauses can be satisfied. x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 x n-3 x n-2 x n-1 xnxn... Prove x  L! This assignment satisfies Á ! Enough to check a random clause!

Other Direction: PCP  Gap-MAX-3SAT NP-Hard Note: Every predicate on O(1) Boolean variables can be written as a conjunction of O(1) 3-clauses on the same variables, as well as, perhaps, O(1) more variables. – If the predicate is satisfied, then there exists an assignment for the additional variables, so that all 3-clauses are satisfied. – If the predicate is not satisfied, then for any assignment to the additional variables, at least one 3-clause is not satisfied. 18

The PCP Theorem Theorem […,AS92,ALMSS92]: Every NP language L has a probabilistically checkable proof (PCP): There is an efficient randomized verifier that queries O(1) proof symbols: x  L: There exists a proof that is always accepted. x  L: For any proof, the probability to accept is ≤½. Remark: Elegant combinatorial proof by Dinur,

Conclusion Probabilistic Checking of Proofs (PCP) 20 Hardness of Approximation

Tight Inapproximability? Corollary: NP-hard to approximate MAX-3SAT to within some constant factor. Question: Can we get tight ⅞-hardness? 21

The Bellare-Goldreich-Sudan Paradigm, 1995 Projection Games Theorem (aka Hardness of Label-Cover, or low error two-query projection PCP) 22 Tight Hardness of Approximating 3SAT [Håstad97] Long-code based reduction

The Bellare-Goldreich-Sudan Paradigm, 1995 Projection Games Theorem (aka Hardness of Label-Cover, or low error two-query projection PCP) 23 Tight Hardness of Approximation for Many Problems Long-code based reduction e.g., Set-Cover [Feige96]

Projection Games & Label-Cover 24 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 = max A,B P(verifier accepts) ¼e¼e Label-Cover: given projection game, compute value.

Equivalent Formulation of PCP Thm Theorem […,AS92,ALMSS92]: NP-hard to approximate Label-Cover within some constant. Proof: by reduction to Label- Cover (see picture). 25 Verifier randomness Proof entries Verifier queries… Accepting verifier view Projection = consistency check symbol

Projection Games Theorem: Low Error PCP Theorem Claim: There is an efficient 1/k-approximation algorithm for projection games on k labels (i.e., | § A |,| § B | · k). 26 Projection Games Theorem For every ² >0, 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.

The Bellare-Goldreich-Sudan Paradigm Projection Games Theorem (aka Hardness of Label-Cover, or low error two-query projection PCP) 27 Tight Hardness of Approximation for Many Problems

?? How To Prove The Projection Games Theorem? 28 Hardness of Approximation Projection Games Theorem [AS92,ALMSS92] PCP Theorem Parallel repetition Theorem [Raz94] [M-Raz08] Construction

The Khot Paradigm, 2002 Unique Games Conjecture 29 Tight Hardness of Approximation for More Problems e.g., Vertex-Cover [DS02,KR03] e.g., Max-Cut [KKMO05] Long-code based reduction Constraint Satisfaction Problems [Raghavendra08]

Thank You! 30