How NP got a new definition:

1. How NP got a new definition: Probabilistically Checkable Proofs (PCPs) & Approximation Properties of NP-hard problems SANJEEV ARORA PRINCETON UNIVERSITY

2. Talk Overview • Recap of NP-completeness and its philosophical importance. • Definition of approximation. • How to prove approximation is NP-complete (new definition of NP; PCP Theorem) • Survey of approximation algorithms.

3. A central theme in modern TCS: Computational Complexity How much time (i.e., # of basic operations) are needed to solve an instance of the problem? Example: Traveling Salesperson Problem on n cities n =49 Number of all possible salesman tours = n! (> # of atoms in the universe for n =49) One key distinction: Polynomial time (n3, n7 etc.) versus Exponential time (2n, n!, etc.)

4. Is there an inherent difference between being creative / brilliant and being able to appreciate creativity / brilliance? • Writing the Moonlight Sonata • Proving Fermat’s Last Theorem • Coming up with a low-cost salesman tour • Appreciating/verifying any of the above When formulated as “computational effort”, just the P vs NP Question.

5. P vs. NP NPC “YES” answer has certificate of O(nc) size, verifiable in O(nc’) time. NP P Solvable in O(nc) time. NP-complete: Every NP problem is reducible to it in O(nc) time. (“Hardest”) e.g., 3SAT: Decide satisfiability of a boolean formula like

6. Practical Importance of P vs NP: 1000s of optimization problems are NP-complete/NP-hard. (Traveling Salesman,CLIQUE, COLORING, Scheduling, etc.) Pragmatic Researcher “Why the fuss? I am perfectly content with approximatelyoptimal solutions.” (e.g., cost within 10% of optimum) Good news: Possible for quite a few problems. Bad News: NP-hard for many problems.

7. Approximation Algorithms MAX-3SAT: Given 3-CNF formula , find assignment maximizing the number of satisfied clauses. An -approximation algorithm is one that for every formula, produces in polynomial time an assignment that satisfies at least OPT/ clauses. (¸ 1). Good News: [KZ’97] An 8/7-approximation algorithm exists. Bad News: [Hastad’97] If P  NP then for every  > 0, an(8/7 -)-approximation algorithm does not exist.

8. Observation (1960’s thru ~1990) NP-hard problems differ with respect to approximability [Johnson’74]: Provide explanation? Classification? Last 15 years: Avalanche of Good and Bad news

9. Next few slides: How to rule out existenceof good approximation algorithms(New definition of NP via PCP Theoremsand why it was needed)

10. Recall: “Reduction” “If you give me a place to stand, I will move the earth.” – Archimedes (~ 250BC) a 1.01-approximation for MAX-3SAT “If you give me a polynomial-time algorithm for 3SAT, I will give you a polynomial-time algorithm for every NP problem.” --- Cook, Levin (1971) [A., Safra] [A., Lund, Motwani, Sudan, Szegedy] 1992 “Every instance of an NP problem can be disguised as an instance of 3SAT.” MAX-3SAT

11. Desired Way to disguise instances of any NP problem as instances of MAX-3SAT s.t. “Yes” instances turn into satisfiable formulae“No” instances turn into formulae in which < 0.99fraction of clauses can be simultaneously satisfied “Gap”

12. Transcript of computation ? Transcript is “correct” if it satisfies all “local” constraints. Cook-Levin reduction doesn’t produce instanceswhere approximation is hard. Main point: Expressthese as boolean formula But, there always exists a transcript that satisfies almost alllocal constraints! (No “Gap”)

13. New definition of NP….

14. CERTIFICATE INPUT x n nc Recall: Usual definition of NP M x is a “YES” input there is a s.t. M accepts (x, ) x is a “NO” input M rejects (x, ) for every 

15. CERTIFICATE INPUT x n nc NP = PCP (log n, 1)[AS’92][ALMSS’92]; inspired by [BFL’90], [BFLS’91][FGLSS’91] M Reads Fixed number of bits(chosen in randomized fashion) Uses O(log n) random bits (Only 3 bits ! (Hastad 97)) Many other“PCP Theorems”known now. x is a “YES” input there is a  s.t. M accepts (x, ) x is a “NO” input for every , M rejects (x, ) Pr [ ] = 1 Pr [ ] > 1/2

16. Disguising an NP problem as MAX-3SAT ? INPUT x M O(lg n) random bits Note: 2O(lg n) = nO(1). ) M ≡ nO(1) constraints, each on O(1) bits x is YES instance )All are satisfiable x is NO instance ) · ½ fraction satisfiable “gap”

17. Of related interest…. Do you need to read a math proof completely to check it? Recall: Math can be axiomatized (e.g., Peano Arithmetic) Proof = Formal sequence of derivations from axioms

18. Verification of math proofs PCP Theorem (spot-checking) n bits Theorem Proof M O(1) bits M runs in poly(n) time • Theorem correct  there is a proof that M accepts w. prob. 1 • Theorem incorrect  M rejects every claimed proof w. prob 1/2

19. Known Inapproximability ResultsThe tree of reductions [AL ‘96] MAX-3SAT [PY ’88] [FGLSS ’91, BS ‘89] [PY ’88];OTHERS MAX-3SAT(3) Metric TSP Vertex Cover MAX-CUT STEINER... Class I 1+ CLIQUE [LY ’93, ABSS ’93] [LY ’93] LABEL COVER COLORING [LY ’93] NEAREST VECTOR MIN-UNSATISFY QUADRATIC -PROGRAMMING LONGEST PATH ... INDEPENDENT SET BICLIQUE COVER FRACTIONAL COLORING MAX-PLANAR SUBGRAPH MAX-SET PACKING MAX-SATISFY Class IV n Class III 2(lg n)1- SET COVER Class II O(lg n) HITTING SET DOMINATING SET HYPERGRAPH - TRAVERSAL ...

20. Proof of PCP Theorems( Some ideas )

21. x x x 1 0 0 1 0 1 1 1 0 1 0 0 0 1 0 1 Need for “robust” representation  : O(lg n) random bits 3 bits Randomly corrupt 1% of  Correct proof still accepted with 0.97- probability! Original proof of PCP Thm used polynomial representations, Local “testing” algorithms for polynomials, etc. (~30-40 pages)

22. New Proof (Dinur’06); ~15-20 pages Repeated applications of two operations on the clauses: Globalize: Create new constraints using “walks” in the adjacency graph of the old constraints. Domain reduction: Change constraints so variables take values in a smaller domain (e.g., 0,1) (uses ideas from old proof)

23. Unique game conjecture and why it is useful Problem: Given system of equations modulo p (p is prime). 7x2 + 2x4 = 6 5x1 + 3x5= 2  7x5 + x2= 21 2 variables per equation • UGC (Khot03): Computationally intractable to distinguish between the cases: • 0.99 fraction of equations are simultaneously satisfiable • no more than 0.001 fraction of equations are simultaneously satisfiable. Implies hardness of approximating vertex cover, max-cut, etc. (K04), (KR05)(KKMO05)

24. Applications of PCP Techniques: Tour d’Horizon • Locally checkable / decodable codes. • List decoding of error-correcting codes. • Private Info Retrieval • Zero Knowledge arguments / CS proofs • Amplification of hardness / derandomization • Constructions of Extractors. • Property testing [Sudan ’96, Guruswami-Sudan] [Katz, Trevisan 2000] [Kilian ‘94] [Micali] [Lipton ‘88] [A., Sudan ’97] [Sudan, Trevisan, Vadhan] [Safra, Ta-shma, Zuckermann] [Shaltiel, Umans] [Goldreich, Goldwasser, Ron ‘97]

25. Approximation algorithms: Some major ideas How can you prove that the solution you found hascost at most 1.5 times (say) the optimum cost? • Relax, solve, and round : Represent problem using a linear or semidefinite program, solve to get fractional solution, and round to get an integer solution. (e.g., MAX-CUT, MAX-3SAT, SPARSEST CUT) • Primal-dual: “Grow” a solution edge by edge; prove its near optimality using LP duality. (Usually gives fasteralgorithms.) e.g., NETWORK DESIGN, SET COVER • Show existence of “easy to find” near-optimal solutions:e.g., Euclidean TSP and Steiner Tree

26. Next few slides: The semidefinite programming approach What is semidefinite programming? Ans. Generalization of linear programming; variables arevectors instead of fractions. “Nonlinear optimization.” [Groetschel, Lovasz, Schrijver ’81]; first used in approximation algorithms by [Goemans-Williamson’94]

27. v2 Rn v1 v3 vn n vertices n vectors, d(vi,vj) satisfy some constraints. Main Idea: G = (V,E) “Round” • Ex:1.13 ratio for MAX-CUT, MAX-2SAT [GW ’93] • O(lg n) ratio for min-multicut, sparsest cut. [LLR ’94, AR ’94] • n1/4-coloring of 3-colorable graphs. [KMS ’94] • (lg n)O(1) ratio for min-bandwidth and related problems [F ’98, BKRV ’98] • 8/7 ratio for MAX-3SAT [KZ ’97] • plog n-approximation for graph partitioning problems (ARV04) How do you analyze these vector programs? Ans. Geometric arguments; sometimes very complicated

28. G = (V,E) Find that maximizes capacity . Quadratic Programming Formulation Ratio 1.13.. for MAX-CUT [GW ’93] Semidefinite Relaxation [DP ’91, GW ’93]

29. vi ij vj Randomized Rounding [GW ’93] Rn v2 v1 v6 Form a cut by partitioning v1,v2,...,vn around a random hyperplane. v3 v5 SDPOPT Old math rides to the rescue...

30. S E(S, S) Input: A graph G=(V,E). sparsest cut: edge expansion For a cut (S,S) let E(S,S) denote the edges crossing the cut. The sparsity of S is the value The SPARSEST CUT problem is to find the cut which minimizes (S). SDPs used to give plog n -approximation involves proving a nontrivial fact about high-dimensional geometry [ARV04]

31. Arora, Rao, and Vazirani showed how the SDP could be rounded to obtain an approximation to Sparsest Cut (2004) ARV structure theorem After we have such A and B, it is easy to extend them to a good sparse cut in G. ARV structure theorem: If the points xu2Rn are well-spread, e.g. u,v (xu-xv)2¸ 0.1 and xu2· 10 for u 2 V and d(u,v) = (xu-xv)2 is a metric, then… A There exist two large, well-separated sets A, B µ {x1, x2, …, xn} with |A|,|B| ¸ 0.1 n and B

32. Unexpected progress inother disciplines… ARV structure theorem led to new understanding ofthe interrelationship between l1 and l2 norms (resolved open question in math) l1 distances among n points can be realized as l2 distances among some other set of n points, andthe distortion incurred is only plog n [A., Lee, Naor’05], building upon [Chawla Gupta Raecke’05]

33. factor () factor () Approx. upto () All interesting problems . . . . . . . Theory of Approximability? • Desired Ingredients: • Definition of approximation-preserving reduction. • Use reductions and algorithms to show: Partial Progress Max-SNP: Problems similar to MAX-3SAT. [PY ’88] RMAX(2): Problems similar to CLIQUE. [PR ‘90] F+2(1): Problems similar to SET COVER. [KT ’91]] MAX-ONES CSP, MIN-CSP,etc. (KST97, KSM96)

34. Further Directions • Investigate alternatives to approximation • Average case analysis • Slightly subexponential algorithms (e.g. 2o(n) algorithm for CLIQUE??) • Resolve the approximability of graph partitioning problems. (BISECTION, SPARSEST CUT, plog n vs loglog n??) and Graph Coloring 3. Complete the classification of problems w.r.t. approximability. 4. Simplify proofs of PCP Thms even further. 5. Resolve “unique games”conjecture. 6. Fast solutions to SDPs? Limitations of SDPs?

35. [Fortnow, Rompel, Sipser ’88] [Feige, Goldwasser, Lovász, Safra, Szegedy ’91] [Arora, Safra ’92] Definition of PCP PCP Hardness of Approx. Polynomial Encoding Method Verifier Composition Fourier Transform Technique Attributions [FGLSS ’91] [ALMSS ’92] [Arora, Safra ’92] [Lund, Fortnow, Karloff, Nisan ’90] [Shamir ’90] [Babai, Fortnow ’90] [Babai, Fortnow, Levin, Szegedy ’91] [Håstad ’96, ’97]

36. {CSP(F) : F is finite} Iff F is 0-valid, 1-valid, weakly positive or negative, affine, or 2CNF P NP Complete Constraint Satisfaction Problems [Schaefer ’78] Let F = a finite family of boolean constraints. An instance of CSP(F): x1 x2 . . . . . . . . . . . . xn variables functions from F g1 g2 . . . . . . . . . . . . gm Ex: Dichotomy Thm:

37. MAX-CSP [Creignou ‘96] [Khanna, Sudan, Williamson ‘97] MAX-SNP-hard (1+) ratio is NP-hard P Iff F is 0-valid, 1-valid, or 2-monotone (Supercedes MAXSNP work) MAX-ONES-CSP [KSW ‘97] Ex: Feasibility is undecidable P Feasibilty NP-hard n 1+ MIN-ONES-CSP [KST ‘97] Ex: Feasibilty NP-hard MIN-HORN-DELETION-complete P n 1+ NEAREST-CODEWORD-complete

38. v2 Rn v1 G = (V,E) v3 vn n vertices n vectors, d(vi,vj) satisfy some constraints. • Ex:1.13 ratio for MAX-CUT, MAX-2SAT [GW ’93] • O(lg n) ratio for min-multicut, sparsest cut. [LLR ’94, AR ’94] • n1/4-coloring of 3-colorable graphs. [KMS ’94] • (lg n)O(1) ratio for min-bandwidth and related problems [F ’98, BKRV ’98] • 8/7 ratio for MAX-3SAT [KZ ’97] • plog n-approximation for graph partitioning problems (ARV04) Geometric Embeddings of Graphs

39. Example: Low Degree Test F =GF(q) Is f a degree d polynomial ? Doesfagree with a degree d polynomial in 90% of the points? f : Fm!F Easy:f is a degree d polynomial iff its restriction on every line is a univariate degree d polynomial. [Line ≡ 1 dimensional affine subspace] ≡ q points. Theorem: Iff on ~ 90% of lines, f has agreement ~90% with a univariate degree d polynomial. Weaker results: [Babai, Fortnow, Lund ‘90] [Rubinfeld Sudan ‘92] [Feige, Goldwasser, Lovász, Szegedy ‘91] Stronger results: [A. Sudan ‘96]; [Raz, Safra ‘96]

40. The results described in this paper indicate a possible classification of optimization problems as to the behavior of their approximation algorithms. Such a classification must remain tentative, at least until the existence of polynomial-time algorithms for finding optimal solutions has been proved or disproved. Are there indeed O(log n) coloring algorithms? Are there any clique finding algorithms better than O(ne) for all e>0? Where do other optimization problems fit into the scheme of things? What is it that makes algorithms for different problems behave the same way? Is there some stronger kind of reducibility than the simple polynomial reducibility that will explain these results, or are they due to some structural similarity between te problems as we define them? And what other types of behavior and ways of analyzing and measuring it are possible? David Johnson, 1974

41. NP-hard Optimization Problems MAX-3SAT: Given 3-CNF formula , find assignment maximizing the number of satisfied clauses. MAX-LIN(3): Given a linear system over GF(2) of the form find its largest feasible subsystem.

42. Theorem: If P  NP, (2-)-approximation does not exists. Approximation Algorithms Defn: An -approximation for MAX-LIN(3) is a polynomial-time algorithm that computes, for each system, a feasible subsystem of size ¸ . (¸ 1) Easy Fact: 2-approximation exists.

43. Common Approx. Ratios

44. Early History • Graham’s algorithm for multiprocessor scheduling [approx. ratio = 2] 1971,72 NP-completeness • Sahni and Gonzalez: Approximating TSP is NP-hard 1975 FPTAS for Knapsack [IK] 1976 Christofides heuristic for metric TSP 1977 Karp’s probabilistic analysis of Euclidean TSP 1980 PTAS for Bin Packing [FL; KK] 1980-82 PTAS’s for planar graph problems [LT, B]

45. Subsequent Developments 1988 MAX-SNP: MAX-3SAT is complete problem [PY] 1990 IP=PSPACE, MIP=NEXPTIME 1991 First results on PCPs [BFLS, FGLSS] 1992 NP=PCP(log n,1) [AS,ALMSS] 1992-95 Better algorithms for scheduling, MAX- CUT [GW], MAX-3SAT,... 1995-98 Tight Lowerbounds (H97); (1+ )- approximation for Euclidean TSP, Steiner Tree... 1999-now Many new algorithms and hardness results. 2005 New simpler proof of NP=PCP(log n,1) (Dinur)

46. SOME NP-COMPLETE PROBLEMS 3SAT: Given a 3-CNF formula, like decide if it has a satisfying assignment. THEOREMS: Given decide if T has a proof of length · n in Axiomatic Mathematics Philosophical meaning of P vs NP: Is there an inherent difference between being creative / brilliant and being able to appreciate creativity / brilliance?

47. “Feasible” computations: those that run in polynomial (i.e.,O(nc)) time (central tenet of theoretical computer science) e.g., time is “infeasible”

48. CERTIFICATE INPUT x n nc Reads 3 bits; Computes sum mod 2 O(1) bits [A., Safra ‘92] [A., Lund, Motwani, Sudan, Szegedy ’92] NP=PCP(log n, 1) Håstad’s 3-bit PCP Theorem (1997) M O(lg n) random bits Accept / Reject > 1 -  > ½ +  x is a “YES” input  there is s.t. M accepts x is a “NO” input  for every  M rejects Pr[ ] Pr[ ]

49. 1- _ ½ + (2-)-approx. to MAX-LIN(3)) P=NP ? INPUT x M O(lg n) random bits Note: 2O(lg n) = nO(1). ) M ≡ nO(1) linear constraints x is YES instance )> 1- fraction satisfiable x is NO instance ) · ½+ fraction satisfiable

50. Idea 1 Polynomial Encoding [LFKN ‘90] [BFL ’90] Sequence of bits / numbers 2 4 5 7 Represent as m-variate degree d polynomial: 2x1x2 + 4x1(1-x2) + 5x2(1-x1) + 7(1-x1)(1-x2) Evaluate at all points in GF(q)m Note: 2 different polynomials differ in (1-d/q) fraction of points.