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P robabilistically C heckable P roofs (and inapproximability)

P robabilistically C heckable P roofs (and inapproximability). Irit Dinur, Weizmann. open day, May 1 st 2009. P  NP (12 th Revision) By Ayror Sappen # Pages to follow: 15783. How Efficiently Can Proofs Be Checked ?. (slide by Madhu Sudan). our real interest: NP proofs.

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P robabilistically C heckable P roofs (and inapproximability)

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

  2. P  NP (12th Revision) By Ayror Sappen # Pages to follow: 15783 How Efficiently Can Proofs Be Checked? (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. Randomizing proof access 3-colorability • 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!)

  5. Prob.Checkable.Proof:  Add randomness, allow errors (ideas coming from interactive proofs and cryptography) Randomizing proof access • If x2 L then9proof, Pr[Ver accepts (x,)] = 1 • If x L then 8proof, Pr[Ver accepts (x,)] < s < 1 Possible gain: read fewer proof bits Verifier Input: x

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

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

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

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

  10. Approximation and Inapproximability

  11. 4 Optimization Problems – finding nearly optimal solutions • 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 Vertex-Cover: Given a graph find the smallest set of vertices that touch all edges.

  12. Approximation 4 5 7 What do we mean by approximation? Each instance has many solutions, each has a value. In optimization, we are seeking the minimal.

  13. Approx 4 5 7 Approximation An approximation algorithm: finds a solution within a certain neighborhood of MIN Example: An algorithm for Approximating Vertex Cover Given G, find a maximal set of edges that do not touch each other. Add both vertices of each edge to the vertex cover. MIN

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

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

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

  17. ma hakesher?

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

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

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

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

  22. Metric Embedding, Semi-definite programming Discrete Fourier Analysis Complexity of Boolean functions, Influences Extremal set theory, EKR intersection theorems Probability and Noise correlation, Invariance principles Getting tight results max-cut 3-SAT vertex-cover coloring

  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!

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