1. Probabilistically Checkable Proofs (and inapproximability) Irit Dinur, Weizmann open day, May 1 st 2009

2. How Efficiently Can Proofs Be Checked ? P  NP (12 th Revision) By Ayror Sappen # Pages to follow: 15783 (slide by Madhu Sudan)

3. our real interest: NP proofs – NP – class of problems with efficiently verifiable solutions Examples: 3-colorability, Satisfiability, Clique, etc. – Theory of NP-completeness provides enormous collection of new formats for writing proofs. – Strange, but just as valid (every thm has proof, but no false thm has one). Possibly new formats give more power? new features?

4. One proof for 3-colorability is a 3-coloring: We can verify it edge by edge Murphy’s law! we detect an “error” only on the last clause (no abundance of errors) How can we gain by randomizing? (ask for another proof!) 3-colorability Randomizing proof access

5. Prob.Checkable.Proof: Prob.Checkable.Proof:  Input: x Verifier – If x 2 L then 9 proof , Pr[Ver accepts (x,  )] = 1 – If x  L then 8 proof , Pr[Ver accepts (x,  )] < s < 1 Add randomness, allow errors (ideas coming from interactive proofs and cryptography) Possible gain: read fewer proof bits Randomizing proof access

6. Restricting proof access How much of the proof must the Verifier read? 1. stage 1: #proof-bit-queries = logarithmic in proof length 2. stage 2: #proof-bit-queries = absolute constant !!  “The PCP Theorem” [Arora-Safra, Arora-Lund-Motwani-Sudan-Szegedy `92] 3. stage 3: #proof-bit-queries = 3 [Hastad ‘97]

7. we want an “error”-amplifying reduction… How can this be done ??? G is not 3col H is <90% 3col G is 3col H is 3col every 3-col of H’s vertices violates > 10% edges G H

8. without looking… we want an “error”-amplifying reduction… How can this be done ??? (similar to error correcting codes)

9. Interactive Proofs, Cryptography approaches Expanders and pseudorandom objects Finite fields, Reed Muller & Reed Solomon codes, low degree curves

10. Approximation and Inapproximability

11. Example: the Minimum Vertex Cover problem Facts: 1. Best algorithm runs in time (1.21) n [Robson ‘86] 2. VC is NP-hard. [Karp ’72] What about approximation.. Output a vertex cover that’s “nearly” minimal! Minimum Vertex Cover Optimization Problems – finding nearly optimal solutions 4 Vertex-Cover: Given a graph find the smallest set of vertices that touch all edges.

12. 7 What do we mean by approximation? Each instance has many solutions, each has a value. In optimization, we are seeking the minimal. 45 Approximation

13. MIN An approximation algorithm: finds a solution within a certain neighborhood of MINApprox Example: An algorithm for Approximating Vertex Cover 1. Given G, find a maximal set of edges that do not touch each other. 2. Add both vertices of each edge to the vertex cover. 745

14. An approximation algorithm: finds a solution within a certain neighborhood of MIN Example: An algorithm for Approximating Vertex Cover 1. Given G, find a maximal set of edges that do not touch each other. 2. Add both vertices of each edge to the vertex cover. How big is it? No more than twice the minimum! Approximation This is a solution: all edges are covered This is a solution: all edges are covered

15. An approximation algorithm: finds a solution within a certain neighborhood of MIN We’ve seen an approximation algorithm for Vertex-Cover, with approximation factor 2. Example: An algorithm for Approximating Vertex Cover 1. Given G, find a maximal set of edges that do not touch each other. 2. Add both vertices of each edge to the vertex cover. Approximation How big is it? No more than twice the minimum!

16. MIN x 2Approx We’ve seen a factor 2 algorithm. Q: Is there a factor 1.99 algorithm? 3/2 ? 3/2 ? 4/3 ? 4/3 ? x 3/2 x 4/3 x 1.99 No, due to PCP thm (and more work) No, assuming very very strong PCP conjecture (“unique games”) Approximation

17. ma hakesher?

18. we want a “gap”-amplifying reduction… How does one prove inapproximability? VC(G) > k VC(H) > (2- ² ) k’ VC(G) = k VC(H) = k’ G H

19. we want a “gap”-amplifying reduction… How does one prove inapproximability? G is not 3col H is <90% 3col G is 3col H is 3col G H

20. The [FGLSS] connection “error”-amplifying reductions … are inapproximability results! & … are PCPs!

21. PCP & Inapprox Prob.Checkable.Proof x 2 ? L Verifier G is not 3col H is <90% 3col G is 3col H is 3col ( x  G  H ) [FGLSS, ALMSS] imability

22. Getting tight results Metric Embedding, Semi-definite programming Discrete Fourier Analysis Complexity of Boolean functions, Influences Probability and Noise correlation, Invariance principles Extremal set theory, EKR intersection theorems

23. summary Probabilistically Checkable Proofs – randomize proof access  gain locality – how? by amplifying “errors” in false proofs – like in error correcting codes Hardness of approximation – vertex cover – amplifying gaps – towards tight results Connections

24. thank you!