On the Unique Games Conjecture

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.