1 / 52

On the Unique Games Conjecture

On the Unique Games Conjecture. Subhash Khot NYU Courant CCC, June 10, 2010. Approximation Algorithms. A C-approximation algorithm for an NP-complete problem computes (C > 1), for problem instance I , solution A(I) s.t. Minimization problems :

merv
Download Presentation

On the Unique Games Conjecture

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. On the Unique Games Conjecture Subhash Khot NYU Courant CCC, June 10, 2010

  2. Approximation Algorithms A C-approximation algorithm for an NP-complete problem computes (C > 1), for problem instance I , solution A(I) s.t. Minimization problems : A(I)  C  OPT(I) Maximization problems : A(I)  OPT(I) / C

  3. PCP Theorem [B’85, GMR’89, BFL’91, LFKN’92, S’92,……] [PY’91] [FGLSS’91, AS’92 ALMSS’92] Theorem : It is NP-hard to tell whether a MAX-3SAT instance is * Satisfiable (i.e. OPT = 1) or * No assignment satisfies more than 99% clauses (i.e. OPT  0.99). i.e. MAX-3SAT is 1.01 hard to approximate.

  4. (In)approximability : Towards Tight Hardness Results • [Hastad’96]Clique n1- • [Hastad’97] MAX-3SAT 8/7 -  • [Feige’98] Set Cover (1- ) ln n [Dinur’05]Combinatorial Proof of PCP Theorem !

  5. Open Problems in (In) Approximability • Vertex Cover (1.36 vs. 2) [DinurSafra’02] • Coloring 3-colorable graphs (5 vs. n3/14) [KhannaLinialSafra’93, BlumKarger’97] • Sparsest Cut (1+ε vs. (logn)1/2) [AMS’07,AroraRaoVazirani’04] • Max Cut (17/16 vs 1/0.878… ) [Håstad’97, GoemansWilliamson’94] ………………………..

  6. Unique Games Conjecture [K’02] Supporting Evidence Connections to: • Inapproximability (UGC  several problems inapproximable) • Discrete Fourier Analysis (e.g. KKL, Majority is Stablest) • Geometry (Isoperimetry, Metric geometry, Integrality gaps) • Algorithms (Attempts to disprove UGC) • Parallel Repetition (Gap amplification, Foam construction)

  7. Example of Unique Game (2CSP) OPT = max fraction of equations that can be satisfied by any assignment. x1 - x3 = 2 (mod k) x5 -x2 = -1 (mod k) x2 - x1 = k-7 (mod k) …………. ………….

  8. Unique Game 2CSP w/ Permutation Constraints variable k labels Here k=4 constraints 

  9. Unique Game2CSP w/ Permutation Constraints variable k labels Here k=4 Permutations or matchings  : [k]  [k]

  10. Unique Game Find a labeling that satisfies max # constraints OPT(G) = 6/7

  11. Unique Games Considered before …… [Feige Lovasz’92] Parallel Repetition of UG reduces OPT(G). How hard is approximating OPT(G) ? Observation : Easy to decide whether OPT(G) = 1.

  12. Unique Games Conjecture For any , , there is integer k(, ), s.t. it is NP-hard to tell whether a Unique Game with k = k(, ) labels has OPT  1-  or OPT   i.e. Gap-Unique Game (1-  , ) is NP-hard. Gap Projection Game (1, ) is NP-hard. [ PCP Theorem + Raz’s Parallel Repetition Theorem ].

  13. Supporting Evidence [UGC] Gap-Unique Game (1-ε, ) is NP-hard. [Feige Reichman’04] Gap-Unique Game (C, ) is NP-hard. However C  --> 0 as C --> ∞. [KV’05] SDP relaxation for UG has “integrality gap” (1-, ). [KV’05] UGC based predictions were proven correct. Specifically, metric embedding lower bounds. [Wishful thinking] “There is structure in CS/math”.

  14. Small Set Expansion Conjecture [Raghavendra Steurer’ 10]Φ (S ) = Edge expansion of set S. For every ε > 0, there exists δ > 0, such that, it is NP-hard to tell whether in a graph G(V,E), • There is a set S, |S| = δ |V|,Φ (S) ≤ ε. • For every set S, |S| ≈ δ |V|,Φ (S) ≥ 1- ε. [Raghavendra Steurer’ 10] SSE Conjecture  Unique Games Conjecture.

  15. Unique Game and Small Set Expansion |G’| = n k. S = Optimal labeling. |S| = 1/k |G ’|. Φ(S) = 1- OPT(G). Unique Game G with n variables, k labels

  16. Linear Equations Over Reals [K Moshkovitz’10] Homogeneous 3LIN(R): x1 – x3 + 2 x5 = 0. ∙∙∙∙∙∙∙∙ eq: xi + .5 x j -x k = 0. Theorem: It is NP-hard to tell if : • There is a “non-trivial” solution that satisfies 1-ε fraction of equations. • Any “non-trivial” solution fails on a constant fraction of equations with error Ω(√ε). 3LIN(R) to 2LIN(R) reduction ? 2LIN(R) ≈ Sparsest Cut

  17. Unique Games Conjecture [K’02] Connections to: • Inapproximability (UGC  several problems inapproximable) • Discrete Fourier Analysis (e.g. KKL, Majority is Stablest) • Geometry (Isoperimetry, Metric geometry, Integrality gaps) • Algorithms (Attempts to disprove UGC) • Parallel Repetition (Gap amplification, Foam construction)

  18. Generic Reduction from Unique Game[BGS’95 (Long Code), Hastad’97 (Fourier), UGC , ……]

  19. PCP Reduction Generic Reduction from Unique Game[BGS’95 (Long Code), Hastad’97 (Fourier), UGC , ……] MAX-CUT Instance Unique Game Instance k labels Gadget: {-1,1} k Match Goemans-Williamson’s SDP rounding Algorithm 1/0.878… Hardness OPT(UG) > 1-ε   sized cut. OPT(UG) < δ No cut with size arccos (1-2) / 

  20. x y {-1,1} k Gadget : Dictatorships (Long Codes) • Consider f: {-1, 1}k {-1,1}, i.e. Cuts. • Encode label i Є {1,2,…., k} by dictatorship function f(x) = xi. Weighted graph, total edge weight = 1. Picking random edge : x R{-1,1} k y <-- flip every co-ordinate of x with probability  (  0.8) Noise-sensitivity graph.

  21. xi = 1 xi = -1 Gadget: Cutthat“commits” toco-ordintae i Fraction of edges cut = Pr(x,y) [xi  yi ] =  Observation : These are the maximum cuts.

  22. Gadget : Cuts not committing to a co-ordinate Influence (i, f) = Prx [ f(x)  f(x+ei) ] How large can be cuts with no influential co-ordinate ? Random Cut : ½ Majority Cut :  > arccos (1-2) /  > ½ [KKMO’04, MOO’05]Majority Is Stablest (Under Noise) Any cut with no influential co-ordinate has size at most arccos (1-2) / .

  23. Integrality Gap Given : Maximization Problem + SDP relaxation. • For every problem instance G, SDP(G)  OPT(G) • Integrality Gap = Sup G SDP(G) / OPT(G)

  24. [Raghavendra’ 08] • Duality between Algorithms and Hardness. • For every CSP, write a natural SDP relaxation. • Integrality gap = β. Implies β-approximation. • Theorem: Every instance with gap β’ < β can be used to construct a gadget and prove UGC-based β’- hardness result ! • SDPs are optimal algorithm for CSPs.

  25. Unique Games Conjecture [K’02] Connections to: • Inapproximability (UGC  several problems inapproximable) • Discrete Fourier Analysis (e.g. KKL, Majority is Stablest) • Geometry (Isoperimetry, Metric geometry, Integrality gaps) • Algorithms (Attempts to disprove UGC) • Parallel Repetition (Gap amplification, Foam construction)

  26. Inapproximability and Fourier Analysis • f : {-1,+1} k  {-1,+1}, balanced. • Sparsest Cut[KV’05, CKKRS’05] [KKL’88] f has a co-ordinate with influence Ω(log k /k). [Bourgain’02] If NSε(f) << √ε, then f depends essentially on exp(1/ε2) co-ordinates. • MAX-CUT[KKMO’04]Majority Is Stablest [MOO’05] If f has no influential co-ordinate, then NS ε(f) ≥ NS ε(Majority) - o(1).

  27. Inapproximability and Fourier • f : {-1,+1}k  {-1,+1}, balanced. • Vertex Cover[DinurSafra’02, K Regev’03, K Bansal’09] [Friedgut’98] If total influence is k, then f depends essentially on exp(k) co-ordintaes. [MOO’05]It Ain’t Over Till It’s Over If f has no influential co-ordinate, then on almost every subcube of {-1, +1}k of dimension k/100, f = 1 and f = -1 with constant probability.

  28. Inapproximability and Fourier • f : {-1,+1}k  {-1,+1}, balanced. • MAX-k-CSP [Samorodintsky Trevisan ’06] If f has no influential co-ordinate, then f has low Gowers’ Uniformity norm. Open: f: [q] k  [q], q ≥ 3, no influential co-ordinate. • f balanced. Is Plurality Stablest ? • What is the maximum Fourier mass at the first level ?

  29. Unique Games Conjecture [K’02] Connections to: • Inapproximability (UGC  several problems inapproximable) • Discrete Fourier Analysis (e.g. KKL, Majority is Stablest) • Geometry (Isoperimetry, Metric geometry, Integrality gaps) • Algorithms (Attempts to disprove UGC) • Parallel Repetition (Gap amplification, Foam construction)

  30. Disproving UGC means .. For small enough (constant), given a UG with optimum 1- , algorithm that finds a labeling satisfying (say) 50% constraints, irrespective of k = #labels.

  31. Algorithmic Results Algorithm that finds a labeling satisfying f(, k, n) fraction of constraints. [K’02] 1- 1/5 k2 [Trevisan’05]1- 1/3 log1/3 n [Gupta Talwar’05] 1-  log n [CMM’05] 1/k , 1- 1/2 log1/2 k [CMM’06] 1-  log1/2 k log1/2 n [AKKSTV’08 , Kolla’10] UG on “mild” expander graphs. [ABS’10] Exp ( n ) time algorithm. None of these disproves UGC. However …

  32. If the UGC is true, then : • k >> 21/ε . • Graph of constraints cannot even be a “mild” expander. UG is easy on random graphs. • Reduction from 3SAT must blow up the size by n1/ε . • Conjecture does not hold for sub-constant ε, i.e. below 1/log n.

  33. Orthonormal Bases for Rk v1 , v2 , … , vk v variables k labels u u1 , u2 , … , uk Matchings [k]  [k] SDP Relaxation of Unique Games [FL’92] • OPT(G) = 1- εSDP(G) ≥ 1- ε . • For i = 1, …, k,  ui , vi  ≥ 1- ε , up to permutation of indices.

  34. vk v2 v1 uk u2 u1 [K’02, CMM’05] Rounding Algorithm r r Random r u v Pick the label closest to r. Label(u) = Label(v) = 2. Pr [ Label(u) = Label(v) ] > 1 - 1/5 k2[K’02]. Pr [ Label(u) = Label(v) ] > 1- 1/2 log1/2 k[CMM’05]. • Labeling satisfies 1- 1/2 log1/2 k fraction of constraints in expected sense.

  35. [Trevisan’05] Algorithm Graph of variables and constraints • [Leighton Rao’88] Delete 1% of edges so that all connected components have diameter O(log n). • Algorithm to solve UG on low diameter graph.

  36. [AKKSTV’08 Algorithm] Algorithm that works on a UG instance s.t. • 1-ε satisfiable and, • Every balanced cut in the graph cuts at least Ω ( √ε ) fraction of edges. • SDP-based. • “Mild” expansion Almost all SDP vector tuples are nearly identical  Yields a good labeling.

  37. Unique Game and Small Set Expansion Label extended Graph |G’| = n k. S = Optimal labeling. |S| = 1/k |G ’|. Φ(S) = 1- OPT(G). UG G with n variables, k labels

  38. [Arora Barak Steurer’10 Algorithm][Kolla’10, Naor’10] • Algo. runs in time exp(nε) on UG that is 1-ε satisfiable. • Good solution to UG  Small non-expanding set S in G’. • Small non-expanding set in label-extended graph G’ Either corresponds to a good UG solution (useful) Or is a non-expanding set in G (fake). • Iteratively remove all fake sets from G, sacrificing at most 1% edges.

  39. [Arora Barak Steurer’10 Algorithm][Kolla’10, Naor’10] Main Lemma (Algorithmic) : If every set of size n1-ε expands by Ω(ε2), then the number of eigenvalues exceeding 1-ε is nO(ε). • The UG solution is found in the linear span of eigenvectors with eigenvalues ≥ 1-ε. [Kolla’10] • Run-time exp ( nO(ε) ).

  40. Unique Games Conjecture [K’02] Connections to: • Inapproximability (UGC  several problems inapproximable) • Discrete Fourier Analysis (e.g. KKL, Majority is Stablest) • Geometry (Isoperimetry, Metric geometry, Integrality gaps) • Algorithms (Attempts to disprove UGC) • Parallel Repetition (Gap amplification, Foam construction)

  41. (Gaussian) Isoperimetry . [MOO’05]Majority Is Stablest reduces via Invariance Principle, to a geometric question: P: Rn  {-1,+1} be a partition of Gaussian space into two sets of equal measure. NSε(P) = Pr [ P(x) ≠ P(y) ], Cor (x,y) = 1-2ε. Which P minimizes the noise-sensitivity? [Borell’85] NSε(P) ≥ NSε( HALF-SPACE THRU ORIGIN ).

  42. (Gaussian) Isoperimetry Open: q ≥ 3. More Invariance. [IM’10] MAX-q-CUT Problem. Plurality is Stablest Conjecture. Partition Rn into q equal parts. (Geometric): Standard Simplex Conjecture. [K Naor’08]Kernel ClusteringProblem. Maximizing Fourier Mass at First Level. (Geometric): Propeller Conjecture.

  43. Integrality Gap • [Feige Schechtman’01] [Goemans Williamson’92] 1/0.878.. Integrality gap for MAX-CUT. • SDP with “triangle inequality constraints” ? • ω(1) Integrality gap for Sparsest Cut? • UGC  NP-hardness  These integrality gaps exist.

  44. Orthonormal Bases for Rk v1 , v2 , … , vk v variables k labels u u1 , u2 , … , uk Matchings [k]  [k] [KV’05] Integrality Gap for Unique Games SDP SDP(G) = 1-o(1) Unique Game G with OPT(G) = o(1)

  45. u1 , u2 , … , uk Integrality Gap for MAX-CUT with Triangle Inequality OPT(G) = o(1) PCP Reduction No large cut Good MAX-CUT SDP solution  u1  u2  u3 ……… uk-1  uk {-1,1}k

  46. MAX-CUT and Sparsest Cut I.G. • [KV’05] MAX-CUT gap matching Goemans-Williamson even with triangle inequality constraints. • [KV’05, KrauthgamerRabani’05, DKSV’06] (loglog n) integrality gap for Sparsest Cut SDP.  An n-point “negative type” metric that needs distortion (loglog n) to embed into L1.  Refutation of [Goemans Linial’97, ARV’04] conjectures. • [KS’09, RS’09] Similar gaps for SDP + Sherali-Adams LP. Negative type metric that is L1 embeddable locally but not globally.

  47. Open Problems • Integrality gaps for the Lasserre SDP Relaxation? Lasserre Relaxation could potentially disprove UGC. • Sparsest Cut (NEG versus L1 Metrics) : [ARV’04, AroraLeeNaor’05] O(√log n). [LeeNaor’06, CheegerKleiner’06, CKNaor’09]. Ω(logc n), c = ½?

  48. Unique Games Conjecture [K’02] Connections to: • Inapproximability (UGC  several problems inapproximable) • Discrete Fourier Analysis (e.g. KKL, Majority is Stablest) • Geometry (Isoperimetry, Metric geometry, Integrality gaps) • Algorithms (Attempts to disprove UGC) • Parallel Repetition (Gap amplification, Foam construction)

  49. Gap Amplification Prove UGC in two steps (?): • Prove “mild” hardness, i.e. GapUG (1-ε’ , 1-ε’’ ) is hard. • Amplify gap via parallel repetition to GapUG (1-ε , δ). Note however that even proving “mild” hardness is a huge challenge.

  50. Strong Parallel Repetition ? OPT(G) = 1-ε. [Raz’98] OPT(Gm) ≤ (1-ε32 )m/log k . 2P1R Games [Holenstein’07] OPT(Gm) ≤ (1-ε3 )m/log k . 2P1R Games [Rao’08] OPT(Gm) ≤ (1-ε2)m . Projection Games (UG). GapUG (1-ε , δ ) is NP-hard iff GapUG (1-ε , 1 - √ε C(ε) ) is NP-hard where C(ε) –> ∞ as ε –> 0. [Raz’08] The rate (1-ε2)m cannot be improved further.

More Related