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On the Unique Games Conjecture Subhash Khot Georgia Inst. Of Technology. At FOCS 2005

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

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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

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Some Known Approximation Algorithms Vertex Cover 2 - approx. MAX-3SAT 8/7 - approx. Random assignment. Packing/Scheduling (1+ ) – approx. > 0 (PTAS) Set Cover ln n approx. Clique n/log n [Boppana Halldorsson’92] Many more, ref. [Vazirani’01]

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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.

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

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Open Problems in Approximability –Vertex Cover (1.36 vs. 2) [DinurSafra’02] –Coloring 3-colorable graphs (5 vs. n 3/14 ) [ 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] ………………………..

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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/ [KKMO’04] Sparsest Cut, Multi-cut [KV’05, (1) CKKRS’04] Min-2SAT-Deletion [K’02, CKKRS’04]

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

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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.

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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

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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, …..

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

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

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

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

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

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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( )

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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 ?

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

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

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

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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.

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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

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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.

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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

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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.

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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])

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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

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

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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 {0,1} k y

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MAX-CUT Gadget : Co-ordinate Cut Along Dimension i Fraction of edges cut = Pr (x,y) [x i y i ] = Observation : These are the maximum cuts. x i = 0 x i = 1

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

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

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

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Reduction from Unique Game Variables k labels OPT 1 – o(1) or OPT o(1) Permutations : [k] [k]

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

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

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

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

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

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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) {0,1} k x y f(x 1 x 2 …….. x k ) = x i f(x 1 x 2 …….. x k ) = MAJORITY Influence (i, f) = Pr x [ f(x) f(x+e i ) ]

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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.

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

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Non-junta Cuts in Bipartite Gadget {0,1} k f : {0,1} k --> {0, 1} g : {0,1} k --> {0, 1} i Infl (i, f), Infl (i, g) > (1) cut > arccos (1-2 ) / +

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

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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 !

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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

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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.

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

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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

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I Hope UGC is True Implies all the “right” hardness results in a unifying way. Neat applications of Fourier theorems [Bourgain’02, KKL’88, Friedgut’98, MOO’05] Surprising application to theory of metric embeddings and SDP-relaxations [KV’05]. Mere coincidence ?

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Supporting Evidence [Feige Reichman’04] Gap-Unique Game (C , ) is NP-hard. i.e. For every constant C, there is s.t. it is NP-hard to tell if a UG has OPT > C or OPT < . However C --> 0 as --> 0.

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Supporting Evidence [Khot Vishnoi’05] SDP relaxation for Unique Game has integrality gap (1- , ).

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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

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Disproving UGC means.. For small enough (constant) , given a UG with optimum 1- , algorithm that finds a labeling satisfying (say) 50% constraints.

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Algorithmic Results Algorithm that finds a labeling satisfying f( , k, n) fraction of constraints. [Khot’02] 1- 1/5 k 2 [Trevisan’05] 1- 1/3 log 1/3 n [Gupta Talwar’05] 1- log n [CMM’05] 1/k , 1- 1/2 log 1/2 k None of these disproves UGC.

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Quadratic Integer Program For Unique Game [Feige Lovasz’92] variable k labels : [k] [k] u 1, u 2, …, u k {0,1} v 1, v 2, …, v k {0,1} u v v i = 1 if Label(v) = i = 0 otherwise

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Quadratic Program for Unique Games Constraints on edge-set E. Maximize u i v π(i) (u, v) E i=1,2,..,k u i [k], u i {0,1} u u i 2 = 1 i u i ≠ j, u i u j = 0

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SDP Relaxation for Unique Games Maximize u i, v π(i) (u, v) E i=1,2,..,k u i [k], u i is a vector. u || u i || 2 = 1 i=1,2,..,k u i≠j [k], u i, u j = 0

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[Feige Lovasz’92] OPT(G) SDP(G) 1. If OPT(G) < 1, then SDP(G) < 1. SDP(G m ) = (SDP(G)) m Parallel Repetition Theorem for UG : OPT(G) < 1 OPT(G m ) 0

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[Khot’02] Rounding Algorithm u1u1 ukuk u2u2 vkvk v2v2 v1v1 r r Label(u) = 2, Label(v) = 2 Pr [ Label(u) = Label(v) ] > 1 - 1/5 k 2 Labeling satisfies 1 - 1/5 k 2 fraction of constraints in expected sense. Random r u v

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[CMM’05] Algorithm Labeling that satisfies 1/k fraction of constraints. ( Optimal [KV’05]) vkvk v2v2 v1v1 r u1u1 ukuk u2u2 r All i s.t. u i is “close” to r are taken as candidate labels to u. Pick one of them at random.

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[Trevisan’05] Algorithm Given a unique game with optimum 1- 1/log n, algorithm finds a labeling that satisfies 50% of constraints. Limit on hardness factors achievable via UGC (e.g. loglog n for Sparsest Cut).

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[Trevisan’05] Algorithm [Leighton Rao’88] Delete a few constraints and remaining graph has connected components of low diameter. Variables and constraints

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[Trevisan’05] Algorithm A good algorithm for graphs with low diameter.

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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

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Already Covered Let’s move on ….

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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

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[KV’05] Integrality Gaps for SDP-relaxations MAX-CUT Sparsest Cut Unique Game Gaps hold for SDPs with “Triangle Inequality”.

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Integer Program for MAX-CUT Given G(V,E) Maximize ¼ |v i - v j | 2 (i, j) E i, v i {-1,1} Triangle Inequality (Optional) : i, j, k, |v i - v j | 2 + |v j - v k | 2 |v i - v k | 2

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Goemans-Williamson’s SDP Relaxation for MAX-CUT Maximize ¼ || v i - v j || 2 (i, j) E i, v i R n, || v i || = 1 Triangle Inequality (Optional) : i, j, k, || v i - v j || 2 + || v j - v k || 2 || v i - v k || 2

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Integrality Gap for MAX-CUT [Goemans Williamson’94] Integrality gap 1/ [Karloff’99] [Feige Schetchman ’01] Integrality gap 1/ SDP solution does not satisfy Triangle Inequality. Does Triangle Inequality make the SDP tighter ? NO if Unique Games Conj. is true !

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Integrality Gap for Unique Games SDP Unique Game G with OPT(G) = o(1) SDP(G) = 1-o(1) Orthonormal Bases for R k u 1, u 2, …, u k v 1, v 2, …, v k variables k labels Matchings [k] [k] u v

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Integrality Gap for MAX-CUT with Triangle Inequality {-1,1} k u 1, u 2, …, u k u 1 u 2 u 3 ……… u k-1 u k PCP Reduction OPT(G) = o(1) No large cut Good SDP solution

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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

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Metrics and Embeddings Metric is a distance function on [n] such that d(i, j) + d(j, k) d(i, k). Metric d embeds into metric with distortion 1 if i, j d(i, j) (i, j) d(i, j).

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Negative Type Metrics Given a set of vectors satisfying Triangle Inequality : i, j, k, || v i - v j || 2 + || v j - v k || 2 || v i - v k || 2 d(i, j) = || v i - v j || 2 defines a metric. These are called “negative type metrics”. L 1 NEG METRICS

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NEG vs L 1 Question [Goemans, Linial’ 95] Conjecture : NEG metrics embed into L 1 with O(1) distortion. Sparsest Cut O(1) Integrality Gap O(1) Approximation [Linial London Rabinovich’94] [Aumann Rabani’98] Unique Games Conjecture [Chawla Krauthgamer Kumar Rabani Sivakumar ’05] [KV’05] (1) hardness result

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NEG vs L 1 Lower Bound ( loglog n) integrality gap for Sparsest Cut SDP. [KhotVishnoi’05, KrauthgamerRabani’05] A negative type metric that needs distortion ( loglog n) to embed into L 1.

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Open Problems (Dis)Prove Unique Games Conjecture. Prove hardness results bypassing UGC. NEG vs L 1, Close the gap. (log log n) vs ( log n loglog n) [Arora Lee Naor’04]

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Open Problems Prove hardness of Min-Deletion version of Unique Games. (log n approx. [GT’05]) Integrality gaps with “k-gonal” inequalities. Is hypercube (Long Code) necessary ?

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Open Problems More hardness results, integrality gaps, embedding lower bounds, Fourier Analysis, …… [Samorodnitsky Trevisan’05] “Gowers Uniformity, Influence of Variables, and PCPs”. UGC Boolean k-CSP is hard to approximate within 2 k- log k Independent Set on degree D graphs is hard to approximate within D/poly(log D).

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Open Problems in Approximability Traveling Salesperson Steiner Tree Max Acyclic Subgraph, Feedback Arc Set Bin-packing (additive approximation) …………………… Recent progress on Edge Disjoint Paths Network Congestion Shortest Vector Problem Asymmetric k-center (log * n) Group Steiner Tree (log 2 n) Hypergraph Vertex Cover ………………

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Linear Unique Games System of linear equations mod k. x 1 + x 3 = 2 3 x 5 - x 2 = -1 x x 1 = 0 [KKMO’04] UGC UGC in the special case of linear equations mod k.

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Variations of Conjecture 2-to-1 Conjecture [K’02] -Conjecture [DMR’05] NP-hard to color 3-colorable graphs with O(1) colors. [k] [k]

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