How NP got a new definition: P vs. NP

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How NP got a new definition: P vs. NP

Cubrilovic, Nic, Contributor has reference to this Academic Journal, PHwiki organized this Journal How NP got a new definition: Probabilistically Checkable Proofs (PCPs) & Approximation Properties of NP-hard problems SANJEEV ARORA PRINCETON UNIVERSITY Recap of NP-completeness in addition to its philosophical importance. Definition of approximation. How to prove approximation is NP-complete (new definition of NP; PCP Theorem) Survey of approximation algorithms. Talk Overview 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 Number of all possible salesman tours = n! (> of atoms in the universe as long as n =49) One key distinction: Polynomial time (n3, n7 etc.) versus Exponential time (2n, n!, etc.) n =49

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Is there an inherent difference between being creative / brilliant in addition to 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 as long as mulated as “computational ef as long as t”, just the P vs NP Question. P vs. NP NP P NPC “YES” answer has certificate of O(nc) size, verifiable in O(nc’) time. 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 as long as mula like Pragmatic Researcher Practical Importance of P vs NP: 1000s of optimization problems are NP-complete/NP-hard. (Traveling Salesman, CLIQUE, COLORING, Scheduling, etc.) “Why the fuss I am perfectly content with approximately optimal solutions.” (e.g., cost within 10% of optimum) Bad News: NP-hard as long as many problems. Good news: Possible as long as quite a few problems.

Approximation Algorithms MAX-3SAT: Given 3-CNF as long as mula , find assignment maximizing the number of satisfied clauses. An -approximation algorithm is one that as long as every as long as mula, 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 as long as every > 0, an (8/7 -)-approximation algorithm does not exist. Observation (1960’s thru ~1990) NP-hard problems differ with respect to approximability [Johnson’74]: Provide explanation Classification Last 15 years: Avalanche of Good in addition to Bad news Next few slides: How to rule out existence of good approximation algorithms (New definition of NP via PCP Theorems in addition to why it was needed)

Recall: “Reduction” “If you give me a place to st in addition to , I will move the earth.” – Archimedes (~ 250BC) “If you give me a polynomial-time algorithm as long as 3SAT, I will give you a polynomial-time algorithm as long as every NP problem.” — Cook, Levin (1971) “Every instance of an NP problem can be disguised as an instance of 3SAT.” a 1.01-approximation as long as MAX-3SAT [A., Safra] [A., Lund, Motwani, Sudan, Szegedy] 1992 MAX-3SAT Desired Way to disguise instances of any NP problem as instances of MAX-3SAT s.t. “Yes” instances turn into satisfiable as long as mulae “No” instances turn into as long as mulae in which < 0.99 fraction of clauses can be simultaneously satisfied “Gap” Cook-Levin reduction doesn’t produce instances where approximation is hard. Main point: Express these as boolean as long as mula But, there always exists a transcript that satisfies almost all local constraints! (No “Gap”) New definition of NP . 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, ) as long as every NP = PCP (log n, 1) [AS’92][ALMSS’92]; inspired by [BFL’90], [BFLS’91][FGLSS’91] x is a “YES” input there is a s.t. M accepts (x, ) x is a “NO” input as long as every , M rejects (x, ) M Reads Fixed number of bits (chosen in r in addition to omized fashion) Pr [ ] = 1 Pr [ ] > 1/2 Uses O(log n) r in addition to om bits (Only 3 bits ! (Hastad 97)) Many other “PCP Theorems” known now.

Disguising an NP problem as MAX-3SAT INPUT x M O(lg n) r in addition to om 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” 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 Verification of math proofs Theorem Proof M M runs in poly(n) time n bits (spot-checking) O(1) bits PCP Theorem Theorem correct there is a proof that M accepts w. prob. 1 Theorem incorrect M rejects every claimed proof w. prob 1/2

Known Inapproximability Results The tree of reductions [AL ‘96] HITTING SET DOMINATING SET HYPERGRAPH – TRAVERSAL [PY ’88]; OTHERS [LY ’93] [LY ’93, ABSS ’93] MAX-3SAT MAX-3SAT(3) CLIQUE LABEL COVER SET COVER COLORING [PY ’88] [LY ’93] [FGLSS ’91, BS ‘89] Metric TSP Vertex Cover MAX-CUT STEINER NEAREST VECTOR MIN-UNSATISFY QUADRATIC -PROGRAMMING LONGEST PATH INDEPENDENT SET BICLIQUE COVER FRACTIONAL COLORING MAX-PLANAR SUBGRAPH MAX-SET PACKING MAX-SATISFY Class II O(lg n) Class I 1+ Class III 2(lg n)1- Class IV n Proof of PCP Theorems ( Some ideas ) Need as long as “robust” representation O(lg n) r in addition to om bits 3 bits : R in addition to omly corrupt 1% of Correct proof still accepted with 0.97- probability! Original proof of PCP Thm used polynomial representations, Local “testing” algorithms as long as polynomials, etc. (~30-40 pages)

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) Unique game conjecture in addition to why it is useful Problem: Given system of equations modulo p (p is prime). 7×2 + 2×4 = 6 5×1 + 3×5 = 2 7×5 + 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) 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 / der in addition to omization 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]

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Approximation algorithms: Some major ideas Relax, solve, in addition to round : Represent problem using a linear or semidefinite program, solve to get fractional solution, in addition to 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 faster algorithms.) e.g., NETWORK DESIGN, SET COVER How can you prove that the solution you found has cost at most 1.5 times (say) the optimum cost Show existence of “easy to find” near-optimal solutions: e.g., Euclidean TSP in addition to Steiner Tree What is semidefinite programming Ans. Generalization of linear programming; variables are vectors instead of fractions. “Nonlinear optimization.” [Groetschel, Lovasz, Schrijver ’81]; first used in approximation algorithms by [Goemans-Williamson’94] Next few slides: The semidefinite programming approach G = (V,E) Ex: 1.13 ratio as long as MAX-CUT, MAX-2SAT [GW ’93] O(lg n) ratio as long as min-multicut, sparsest cut. [LLR ’94, AR ’94] n1/4-coloring of 3-colorable graphs. [KMS ’94] (lg n)O(1) ratio as long as min-b in addition to width in addition to related problems [F ’98, BKRV ’98] 8/7 ratio as long as MAX-3SAT [KZ ’97] plog n-approximation as long as graph partitioning problems (ARV04) Main Idea: “Round” How do you analyze these vector programs Ans. Geometric arguments; sometimes very complicated

Ratio 1.13 as long as MAX-CUT [GW ’93] Semidefinite Relaxation [DP ’91, GW ’93] R in addition to omized Rounding [GW ’93] v6 v2 v3 v5 Rn v1 Form a cut by partitioning v1,v2, ,vn around a r in addition to om hyperplane. SDPOPT Old math rides to the rescue sparsest cut: edge expansion Input: A graph G=(V,E). 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]

c-balanced separator S Assign {+1, -1} to v1, v2, , vn to minimize (i, j) 2 E vi –vj2/4 Subject to i < j vi –vj2/4 ¸ c(1-c)n2 +1 -1 vi –vj2/4 =1 Semidefinite relaxation as long as Find unit vectors in Cubrilovic, Nic Contributor

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