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## PCPs in addition to Inapproximability: Recent Milestones in addition to Continuing ChallengesVenkatesa

Carter, Noelle, Contributor has reference to this Academic Journal, PHwiki organized this Journal PCPs in addition to Inapproximability: Recent Milestones in addition to Continuing ChallengesVenkatesan GuruswamiCarnegie Mellon University© V. Guruswami, 2011Intended to be a survey of some key developments of last 10 years Lets begin with a glimpse of where PCP theory was a decade backPCPs circa 2000

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PCP theoremPCP theorem: [Arora-Safra, Arora-Lund-Motwani-Sudan-Szegedy] Polynomial size witnesses as long as membership claims x L in NP languages L that are checkable by a r in addition to omized polytime verifier by just probing 3 bits with soundness < 0.99soundness = max. prob. of accepting some witness as long as strings x LProbabilistically Checkable ProofsPCP Theorem: characterization of NPDeterministicVerifierNP Proof yProbabilisticVerifierPCP Probabilistically Checkable ProofsPCP Theorem: characterization of NPDeterministicVerifierNP Proof yProbabilisticVerifierPCP PCP locally testable encoding of the NP proof y Probabilistically Checkable ProofsPCP Theorem: characterization of NPDeterministicVerifierNP Proof yProbabilisticVerifierPCP Completeness: If SAT, there exists a PCP such all local views are acceptingSoundness: If L, then as long as all PCPs at most 99% of the local views are accepting PCP theorem Gap3SAT1, is NP-hard as long as some < 1That is, polytime reduction mappingsatisfiable as long as mulae to satisfiable as long as mulae Unsatisfiable as long as mulae to as long as mulae that is at most -satisfiableImplies factor -approximation as long as Max 3SAT is NP-hardPCPs: early yearsPCP theorem implied hardness of approximation many problems (via their classic reductions from 3SAT).Generally very weak factors. Quest as long as better (optimal) factors followed2-prover -round proof systems emerged as the canonical PCP to reduce from:Constraint satisfaction problem called Label CoverExample early Label Cover-based success: (log n) hardness as long as set cover [Lund-Yannakakis] Label CoverArity 2 CSP over large domain [R] with projection constraints.Instance consists of:Bipartite graph G=(V,W,E)For each e =(v,w) E, a function e : [R] [R].Assignment (labeling) A: V W [R] satisfies an edge e=(v,w) if e(A(v)) = A(w)Value of instance = maximum fraction of edges satisfied by a labelingPCP Theorem + Razs parallel repetition theorem a strong (value 1 vs 1/R) gap hardness as long as Label CoverVWeStrong(er)/optimal PCPsImprovements in PCP parameters, aimed in part at better hardness resultsLabel Cover used as outer PCPComposed with inner PCPParadigm: Encode labels & test codewordsInner task: Check if (a)=b reading very few bits of Enc(a) in addition to Enc(b)Goal: trade off soundness as long as much smaller queriesWhich code to use Brilliant invention of [Bellare-Goldreich-Sudan]: LONG CODELong Code (aka Dictator functions)For a 2 {1,2, ,R}, LONG(a) : {0,1}R {0,1}LONG(a)(x) = xa as long as every x 2 {0,1}RVery redundant (encodes log R bits into 2R bits)has the value of every function [R] {0,1} at abut doesnt hurt to have around if R is constant. Surprisingly useful! The first optimal PCPs[Håstad96]: zero amortized free bit PCP factor n1- inapproximability as long as Clique[Håstad97]: Gap3LIN1-,½+ is NP-hard (3-query PCP with completeness 1- in addition to soundness ½+). Optimal! Also, similar result mod p, in addition to NP-hardness ofGap3SAT1,7/8+ & Gap-4-Set-Splitting1,7/8+ Several hardness results (currently best known, under NP P) via gadget reductions from Håstads results:MaxCut: 16/17 Max2SAT: 21/22 MaxDiCut: 11/12NAE-3SAT: 15/16 3-Set-Splitting: 19/20 3-coloring: 32/33 (these are with perfect completeness)Optimal!Optimal algorithmsIn parallel with hardness revolution, sophisticated SDP rounding methods developed. Eg.7/8 algo as long as Max3SAT [Karloff-Zwick]Factor 2/3 as long as Max 3MAJFactor ½ algo as long as Max3CSP in addition to factor 5/8 as long as satisfiable 3CSP.Optimal PCPs: queries vs soundness[G.-Lewin-Sudan-Trevisan] 3-query adaptive PCP with perfect completeness & soundness ½[Samorodnitsky-Trevisan] amortized query complexity: k queries, soundness 2-k+o(k)Useful starting points in some reductions, eg. Clique, Low-congestion network routing Low-soundness Multiprover systems[Arora-Sudan; Raz-Safra] O() prover -round proof systems with exp(-(log n)()) soundnessNP-hardness of (log n) factor as long as set coverSimilar result as long as 2-prover case openWould have more applications, like hardness of lattice problems[Dinur-Fischer-Kindler-Raz-Safra] O() prover systems withexp(-(log n)0.99) soundness. Proof of BGLR sliding scale conjecture as long as up to (log n)0.99 bits readCovering PCPs [G.-Håstad-Sudan] Notion of soundness tailored to coloring problemsCovering soundness = minimum number of proofs that can cover all constraints ( as long as every check, at least one proof should cause acceptance)[G.-Håstad-Sudan] 4-query PCP with () covering soundnessSuper-constant hardness as long as coloring 2-colorable 4-uni as long as m hypergraphs.Later also as long as 3-uni as long as m hypergraphs [Khot] [Dinur-Regev-Smyth] Frontier: Rule out 5-coloring 3-colorable graphs in polytimePCPs till ~ 2000 summaryLabel Cover hardnessversatile starting point as long as inapproximability (continues to be prominent) Label Cover + Long Code + Fourier analysis paradigmTight hardness results as long as several CSPs of arity 3Arity 2 CSPs not well understood (results only via gadgets) PCPs in the last decadeNew proofs, notions: - Dinurs gap amplification - Robust PCPs - PCP of proximity (PCPP)Short PCPs (n1+o(1) size): - Best known n (log n)O(1) Low soundness error 2-query PCPsNew outer PCPs - multilayered, smooth, mixing, Dinur-Safra, etc. - Conjectural as long as ms: Unique Games, 2-to-1, Dictatorship tests in addition to new inner PCPs - New analytic machinery New Proof Composition methodsNew proofs in addition to notionsDinurs proofGap amplification: Reduce Gap-3Color, to Gap-3Color,2 provided 10-6Apply O(log n) times starting with =1/mShows that Gap-3Color, is NP-hard as long as some constant (this implies the PCP theorem)PCP via inapproximability instead of other way aroundRequires elements of old PCP in alphabet reductionA constant sized PCP (variant called PCPP or assignment tester) Robust PCPsPCP soundness: When SAT, as long as all proofs , with probability , check CI rejects I (I=r in addition to omly chosen query positions)Robust PCP: stronger soundness guaranteeI far from satisfying CI (with good prob.)Formally, ( = robust soundness)Check CI(¼I)= 1Proof IRobust PCPs & PCPPsCheck if I satisfies CI or is far from satisfying CI recursively, using another inner PCPInner primitive: PCP of proximity (PCPP)Input: circuit CProof: (purported) satisfying assignment x in addition to proof of proximity that x satisfies CVerifier (Assignment Tester): read few bits in both x in addition to ;For satisfying x, with acc. prob. If x is -far from satisfying C, then , rej. prob. Useful when robust distance of outer PCP > proximity parameter of inner PCPP Composition streamlined Robust PCPs compose syntactically with inner PCPs of proximity (when PCPP proximity parameter < robustness)[Ben-Sasson, Goldreich, Harsha, Sudan, Vadhan] Can check that original polynomial in addition to Hadamard based PCPs can be made robust PCPPs.Simplifies details of compositionUsed to give near-linear size PCPsPCPP also used in Dinurs alphabet reduction stepExplicit coding used to create distance between inconsistent assignments

Talk PlanNew proofs in addition to notionsRobust PCPs in addition to PCPs as long as proximityShort PCPsLow-soundness error Label CoverUnique Games, Dictatorship tests, etc.NP-hardness via structured outer PCPsSome challengesQuasi-linear PCPs[BGHSV] PCPs of length n 2(log n)² (O() queries)[Ben-Sasson, Sudan] Univariate polynomial based PCPProof of proximity as long as Reed-Solomon codes which makes it locally testablen (log n)O() sized PCP with (log n)O() queriesO(log log n) steps of Dinurs gap amplification gives n (log n)O() sized PCP with O(1) queriesImplication as long as approximation: APX-hardness via reduction from 3SAT hold as long as time algos, under the ETHNew proofs in addition to notionsRobust PCPs in addition to PCPs as long as proximityShort PCPsLow-soundness error Label CoverUnique Games, Dictatorship tests, etc.NP-hardness via structured outer PCPsSome challenges

Label cover with o(1) soundnessHåstads 7/8+ hardness as long as 3SAT requires soundness error of Label Cover to be Getting this via parallel repetition makes Label Cover instance size n(log (/²)) (with large hidden constant factor)Can we get such soundness with LC size say O(n3) Answer: [Moshkovitz-Raz] YES. In fact, with near-linear size ! (Was very surprising to me)For soundness , Label Cover with n1+o(1) poly(1/) vertices. Worse dependence on domain size: R = exp(poly(1/)) instead of poly(1/) in Raz. Inapproximability consequence7/8+ approx. as long as 3SAT requires exp(²(n1-o(1))) time (under Exponential Time Hypothesis)Similar claims as long as other hardness results based on Label cover + Long code testing Sharp complexity dichotomy at the approximation threshold 7/8: polytime vs. exponential time.[Moshkovitz-Raz] approachStart with Label Cover of low soundness error but large alphabet = (log n)-0.01, = poly(n) (based on low-degree testing)reduce alphabet size via composition New composition method to reduce alphabet size that does not increase proversNext: Few words on an alternate approach (giving polynomial instead of near-linear size)

Wrap-upPCPs remarkably successful in showing inapproximability (even beyond initial expectations)Breadth of problems. I find it amazing what all Label Cover can be reduced to!Many tight resultsSome notorious problems have withstood resolutionDensest subgraph, minimum linear arrangement, bipartite clique, sparsest cut, graph bisection, etc.Known algorithms have superconstant approx. ratio, but even APX-hardness not known Cut challengesEg. Uni as long as m Sparsest Cut, Minimum Graph Bisection: Best approximation (log n)). Hardness evidence:Refuting r in addition to om 3SAT is hard Factor 1.1 hardness [Feige]Polytime ( approximation NP has 2n² time algorithms [Khot, quasi-r in addition to om PCPs]Superconstant hardness under SSE hypothesis (stronger than Unique Games conjecture) [Raghavendra-Steurer-Tulsiani]Easiness evidence [G.-Sinop]Factor (1r approximation in time where r is the rth smallest eigenvalue of normalized Laplacian.ChallengesCan PCP machinery (even assuming UGC) give strong hardness results as long as Steiner Tree, TSP, Asymmetric TSP Lasserre integrality gaps beyond known hardness bound as long as Vertex Cover, Max Cut (or Unique Games)Just 4 rounds could improve [GW] in addition to refute UGC !Unique-Games-completenessBypass UGC as long as some other consequencesUnchartered terrain as long as inapproximability:eg. , nearest codeword in algebraic codes, bin packing,

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