1 / 86

On the Unique Games Conjecture

On the Unique Games Conjecture. Subhash Khot Georgia Inst. Of Technology. At FOCS 2005. NP-hard Problems. Vertex Cover MAX-3SAT Bin-Packing Set Cover Clique MAX-CUT …………….. ……………. Approximability : Algorithms.

verity
Télécharger la présentation

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 Georgia Inst. Of Technology. At FOCS 2005

  2. NP-hard Problems • Vertex Cover • MAX-3SAT • Bin-Packing • Set Cover • Clique • MAX-CUT • …………….. • ……………..

  3. Approximability : Algorithms A C-approximation algorithm 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

  4. Some Known Approximation Algorithms • Vertex Cover2 - approx. • MAX-3SAT8/7 - approx. Random assignment. • Packing/Scheduling(1+) – approx.   > 0 (PTAS) • Set Coverln n approx. • Clique n/log n [Boppana Halldorsson’92] • Many more , ref. [Vazirani’01]

  5. 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 1/0.99 = 1.01 hard to approximate. i.e. MAX-3SAT and MAX-SNP-complete problems [PY’91] have no PTAS.

  6. 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 !

  7. Open Problems 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 vs. (logn)1/2) [AroraRaoVazirani’04] • Max Cut (17/16 vs 1/0.878… ) [Håstad’97, GoemansWilliamson’94] ………………………..

  8. Unique Games Conjecture [Khot’02] Implies these hardness results : • Vertex Cover 2-  [KR’03] • Coloring 3-colorable (1) [DMR’05] graphs (variant of UGC) • MAX-CUT 1/0.878.. -  [KKMO’04] • Sparsest Cut, Multi-cut [KV’05, (1) CKKRS’04] Min-2SAT-Deletion [K’02, CKKRS’04]

  9. Unique Games Conjecture Led to … [MOO’05] Majority Is Stablest Theorem [KV’05] “Negative type” metrics do not embed into L1 with O(1) “distortion”. Optimal “integrality gap” for MAX-CUT SDP with “Triangle Inequality”.

  10. Integrality Gap : Definition Given : Maximization Problem + Specific SDP relaxation. • For every problem instance G, SDP(G)  OPT(G) • Integrality Gap = Max G SDP(G) / OPT(G) • Constructing gap instance = negative result.

  11. Overview of the talk • The UGC • Hardness of Approximation Results • I hope UGC is true • Attempts to Disprove : Algorithms Connections/applications : • Fourier Analysis • Integrality Gaps • Metric Embeddings

  12. Unique Games Conjecture • A maximization problem called “Unique Game” is hard to approximate. • “Gap-preserving” reductions from Unique Game  Hardness results for Vertex Cover, MAX-CUT, Graph-Coloring, …..

  13. Example of Unique Game OPT = max fraction of equations that can be satisfied by any assignment. x1 + x3 = 2 (mod k) 3 x5 -x2 = -1 (mod k) x2 + 5x1 = 0 (mod k) UGC For large k, it is NP-hard to tell whether OPT 99% or OPT  1%

  14. 2-Prover-1-Round Game (Constraint Satisfaction Problem ) variables constraints 

  15. 2-Prover-1-Round Game (Constraint Satisfaction Problem ) variables k labels Here k=4 constraints 

  16. 2-Prover-1-Round Game (Constraint Satisfaction Problem ) variables k labels Here k=4 Constraints = Bipartite graphs or Relations   [k]  [k]

  17. 2-Prover-1-Round Game (Constraint Satisfaction Problem ) Find a labeling that satisfies max # constraints variables k labels Here k=4 OPT(G) = 7/7

  18. Hardness of Finding OPT(G) • Given a 2P1R game G, how hard is it to find OPT(G) ? • PCP Theorem + Raz’s Parallel Repetition Theorem : For every , there is integer k(), s.t. it is NP-hard to tell whether a 2P1R game with k = k() labels has OPT = 1 or OPT   In fact k = 1/poly()

  19. Reductions from 2P1R Game • Almost all known hardness results (e.g. Clique, MAX-3SAT, Set Cover, SVP, …. ) are reductions from 2P1R games. • Many special cases of 2P1R games are known to be hard, e.g. Multipartite graphs, Expander graphs, Smoothness property, …. What about unique games ?

  20. Unique Game = 2P1R Game with Permutations variable k labels Here k=4

  21. Unique Game = 2P1R Game with Permutations variable k labels Here k=4 Permutations or matchings  : [k]  [k]

  22. Unique Game = 2P1R Game with Permutations Find a labeling that satisfies max # constraints OPT(G) = 6/7

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

  24. MAX-CUT is Special Case of Unique Game • Vertices : Binary variables x, y, z, w, ……. • Edges : Equations x + y = 1 (mod 2) • [Hastad’97] NP-hard to tell whether OPT(MAX-CUT)  17/21 or OPT(MAX-CUT)  16/21

  25. 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.

  26. Overview of the talk • The UGC • Hardness of Approximation Results • I hope UGC is true • Attempts to Disprove : Algorithms Connections/applications : • Fourier Analysis • Integrality Gaps • Metric Embeddings

  27. Case Study : MAX-CUT • Given a graph, find a cut that maximizes fraction of edges cut. • Random cut : 2-approximation. • [GW’94] SDP-relaxation and rounding. min 0 <  < 1  / (arccos (1-2) /  ) = 1/0.878 … approximation. • [KKMO’04] Assuming UGC, MAX-CUT is 1/0.878… -  hard to approximate.

  28. Reduction to MAX-CUT Unique Game Graph H • Completeness : OPT(UG) > 1-o(1)    - o(1) cut. • Soundness : OPT(UG) < o(1)  No cut with size arccos (1-2) /  + o(1) • Hardness factor =  / (arccos (1-2) /  ) - o(1) • Choose best  to get 1/0.878 … (= [GW’94])

  29. Reduction from Unique Game Gadget constructed via Fourier theorem + Connecting gadgets via Unique Game instance [DMR’05]“UGC reduces the analysis of the entire construction to the analysis of the gadget”. Gadget = Basic gadget ---> Bipartite gadget ---> Bipartite gadget with permutation

  30. Basic Gadget A graph on {0,1} k with specific properties (e.g. cuts, vertex covers, colorability) x = 011 k = # labels {0,1} k Y = 110

  31. {0,1} k y Basic Gadget : MAX-CUT Weighted graph, total edge weight = 1. Picking random edge : x R{0,1} k y <-- flip every co-ordinate of x with probability  (  0.8) x

  32. xi = 0 xi = 1 MAX-CUT Gadget : Co-ordinate CutAlong Dimension i Fraction of edges cut = Pr(x,y) [xi  yi ] =  Observation : These are the maximum cuts.

  33. Bipartite Gadget A graph on {0,1} k  {0,1} k (double cover of basic gadget) x = 011 y’ = 110

  34. Cuts in Bipartite Gadget {0,1} k {0,1} k Matching co-ordinate cuts have size = 

  35. x = 011 Y ’ = 110 Bipartite Gadget with Permutation  : [k] -> [k] Co-ordinates in second hypercube permuted via . Example :  = reversal of co-ordinates.  (y’) = 011

  36. OPT  1 – o(1) or OPT o(1) Variables k labels Permutations  : [k]  [k] Reduction from Unique Game

  37. {0,1} k  Vertices Edges Instance H of MAX-CUT Bipartite Gadget via 

  38. Proving Completeness (Completeness) : OPT(UG) > 1-o(1)  H has  - o(1) cut. Unique Game Graph H

  39. Completeness : OPT(UG)  1-o(1) label = 1 Labels = [1,2,3] label = 2 label = 3 label = 2 label = 1 label = 1 label = 3

  40. Completeness : OPT(UG)  1-o(1) {0,1} k Vertices Edges Hypercubes are cut along dimensions = labels. MAX-CUT   - o(1) 

  41. Proving Soundness Unique Game Graph H (Soundness) : OPT(UG) < o(1)  H has no cut of size arccos (1-2) /  + o(1)

  42. x {0,1} k y MAX-CUT Gadget Cuts = Boolean functions f : {0,1} k  {0,1} Compare boolean functions * that depend only on single co-ordinate vs * where every co-ordinate has negligible “influence” (i.e. “non-junta” functions) f(x1 x2 …….. xk) = xi Influence (i, f) = Prx [ f(x)  f(x+ei) ] f(x1 x2 …….. xk) = MAJORITY

  43. Gadget : “Non-junta” Cuts How large can non-junta cuts be ? i.e. cuts with all influences negligible ? Random Cut : ½ Majority Cut : arccos (1-2) /  > ½ • [MOO’05]Majority Is Stablest (Best) Any cut slightly better than Majority Cut must have “influential” co-ordinate.

  44. Non-junta Cuts in Bipartite Gadget {0,1} k {0,1} k [MOO’05] Any “special” cut with value arccos (1-2) /  +  must define a matching pair of influential co-ordinates.

  45. Non-junta Cuts in Bipartite Gadget {0,1} k {0,1} k f : {0,1} k --> {0, 1} g : {0,1} k --> {0, 1} cut > arccos (1-2) /  +   i Infl (i, f), Infl (i, g) > (1)

  46. {0,1} k  Vertices Edges Instance H of MAX-CUT Bipartite Gadget via 

  47. Proving Soundness • Assume arccos (1-2) /  +  cut exists. • On /2 fraction of constraints, the bipartite gadget has arccos (1-2) /  + /2 cut.  matching pair of labels on this constraint. This is impossible since OPT(UG) = o(1). Done !

  48. Other Hardness Results • Vertex Cover Friedgut’s Theorem Every boolean function with low “average sensitivity” is a junta. • Sparsest Cut, Min-2SAT Deletion KahnKalaiLinial Every balanced boolean function has a co-ordinate with influence log n/n. Bourgain’s Theorem(inspired by Hastad-Sudan’s 2-bit Long Code test) Every boolean function with low “noise sensitivity” is a junta. • Coloring 3-Colorable [MOO’05] inspired. Graphs

  49. Basic Paradigm by [BGS’95, Hastad’97] •  Hardness results for Clique, MAX-3SAT, ……. • Instead of Unique Games, use reduction from general 2P1R Games (PCP Theorem + Raz). • Hypercube = Bits in the Long Code [Bellare Goldreich Sudan’95] • PCPs with 3 or more queries (testing Long Code). • Not enough to construct 2-query PCPs.

  50. Why UGC and not 2P1R Games? Power in simplicity. “Obvious” way of encoding a permutation constraint. Basic Gadget ----> Bipartite Gadget with permutation.

More Related