on the unique games conjecture … · implications of ugc ugc: true or false? the unique games...

Game, what game?The conjecture

Implications of UGCUGC: True or False?

On the Unique Games Conjecture

Fatima-Zahra Moataz

COATI, INRIA, I3S(CNRS/UNS), Sophia Antipolis, France

March 11,2015

COATI (INRIA/I3S) On the Unique Games Conjecture JCALM 2015 1/37

The Unique Games Conjecture (UGC)

Proposed by Subhash Khot in 2002 [Kho02]

It states that a problem called Unique Games (UG) is hard toapproximate

Gap-preserving reductions from UG → inapproximabilityresults for several other problems

The conjecture motivated work in the analysis of booleanfunctions, geometry . . .

Outline

1 Game, what game?Label coverWhy Label cover?

2 The conjecture

3 Implications of UGCAnalysis of boolean functionsMetric embeddingsInapproximability

MaxCutUGC and SDP

4 UGC: True or False?

Label coverWhy Label cover?

2 The conjecture

MaxCutUGC and SDP

Label Cover Problem (LC)

Input:

A bipartite graph ((V ,W ),E )

Two sets of labels M and N

∀ (v ,w) ∈ E , πv ,w : M → N

A labeling of the vertices of G :l : V → N and l : W → M.An edge (v ,w) is satisified ifπv ,w (l(v)) = l(w)

Output: An (optimal) labeling which maximizes the number ofsatisfied edgesFor an instance U of LC, OPT (U) = fraction of edges satisfied byan optimal labeling of U .

Input:

∀ (v ,w) ∈ E , πv ,w : M → N

Input:

∀ (v ,w) ∈ E , πv ,w : M → N

Input:

∀ (v ,w) ∈ E , πv ,w : M → N

Input:

∀ (v ,w) ∈ E , πv ,w : M → N

Input:

∀ (v ,w) ∈ E , πv ,w : M → N

Input:

∀ (v ,w) ∈ E , πv ,w : M → N

Where is the game?

Label Cover

2-Prover 1-Round Game

Unique Label Cover (ULC)

The label cover problem L = (G ,M,N, πvw ) is called unique if:

∀(v ,w) ∈ E , πv ,w is a bijection (permutation)

Unique Label Cover (ULC)

The label cover problem L = (G ,M,N, πvw ) is called unique if:

∀(v ,w) ∈ E , πv ,w is a bijection (permutation)

How is Label Cover (LC) useful?

It is all due to the following theorem:

Theorem

For ε > 0, it is NP-hard to decide whether a Label Cover problem:

satisfies all edges (OPT = 1)

satisfies at most a fraction ε of the edges (OPT ≤ ε)

Proved with PCP theorem [AS98] + Raz’s Parallel RepetitionLemma [Raz98]

Reductions from LC (1, ε) have allowed to proveinapproximability results for many problems.

Where does this problem fall short? → 2-CSPs

Mostly because of the ”many-to-one”ess of the constraints.

How about having a stronger result? the sameinapproximability theorem for Unique Label Cover?

2 The conjecture

MaxCutUGC and SDP

The Unique Games Conjecture[Kho02]

Conjecture

For ε, δ > 0, it is NP-hard to decide whether a Unique Label Coverproblem:

satisfies at least 1− ε fraction of the edges (OPT ≥ 1− ε)satisfies at most a fraction δ of the edges (OPT ≤ δ)

Conjecture

Why 1− ε and not 1?

Conjecture

Why 1− ε and not 1?→ Deciding if all edges can be satisfied is easy.

Analysis of boolean functionsMetric embeddingsInapproximability

2 The conjecture

MaxCutUGC and SDP

To the Unique Games Conjecture are associated:

Non-conditional results

Analysis of boolean functionsMetric embeddings

Conditional results: Inapproximability

Majority is Stablest [MOO05]

f : {0, 1}n → {0, 1} → voting scheme.

Dictatorship → f (x1, . . . , xn) = xi for some i

Influence of voter i in a scheme f :

Prx∈{1,−1}n(f (x1, . . . , xi , . . . , xn) 6= f (x1, . . . ,−xi , . . . , xn))

Noise stabilityρ of a scheme f : Probability that the result doesnot change if a random fraction ρ of voters flip their votes.

f : {0, 1}n → {0, 1} → voting scheme.

Studying problems under UGC a distinction was made between:

Dictatorships

Schemes that are far from being dictatorships

A question has arised: between schemes that are far from beingdictatorships, what is the stablest scheme?

The answer: Majority is Stablest

The ”Majority is Stablest” (MIS) theorem [MOO05] states thatthe Majority function maximizes noise stability among balancedboolean functions on the discrete cube with ”small” influences.

Dictatorships

Metric embedding[KV05]

L× d(x , y) ≤ d ′(f (x), f (y)) ≤ C × L× d(x , y)

=⇒ f has distortion C

Metric Embedding: Goemans-Linial Conjecture

Goemans Linial Conjecture: Every negative type metricembeds into l1 with constant distortion.(d is a negative type metric if

√d is isometrically embeddable

in l2.)

Goemans Linial Conjecture: Every negative type metricembeds into l1 with constant distortion.(d is a negative type metric if

√d is isometrically embeddable

in l2.)

Insights from UGC have helped constructing a negative metricthat embeds in l1 with distortion at least log(log(n)) [KV05]→ The conjecture is false

Goemans-Linial Conjecture is true → O(1)-approximation for agraph partitioning problem P (sparsest cut) with some SDPrelaxation S.

Reduction from ULC to P to prove inapproximability for Punder UGC

Results on approximation ULC with some SDP relaxation

⇒ The ratio of the approximation of P with S is not constant

⇒ The ratio of the approximation of P with S is not constant⇒ The Goemans-Linial Conjecture is false

Some UGC inapproximability results [Kho10]

ProblemBestApprox.Known

Best In-approxknown

Best Inap-prox. knownunder UGC

Vertex Cover 2 1.36 2− εMaxCut 0.878 16

17 + ε 0.878 + ε

Max Acyclic Sub-graph

0.5 6566 + ε 0.5 + ε

Any CSP C withintegrality gap αC

αC αC + ε

Max-Cut: definition

Input:

A graph G = (V ,E )

Output:

MaxCut is NP-hard and hard to approximate within 1617 [Has01].

Max-Cut: definition

Input:

A graph G = (V ,E )

Output:

A partition (V1,V2) whichmaximizes the size of the set(V1,V2) ∩ E

Max-Cut: definition

Input:

A graph G = (V ,E )

Output:

A partition (V1,V2) whichmaximizes the size of the set(V1,V2) ∩ E

MaxCut: Goemans Williamson algorithm [GW95]

Quadratic Program

max:∑

(i ,j)∈E

1− xixj2

s.t.: x2i = 1 ∀i ∈ V

SDP relaxation

max:∑

(i ,j)∈E

1− vi .vj2

s.t: vi .vi = 1 ∀i ∈ V

vi ∈ Rn ∀i ∈ V

MaxCut: Goemans Williamson algorithm[GW95]

The algorithm

Solve the relaxed SDP

The algorithm

Cut the sphere with arandom hyperplane

The algorithm

The cut → the two sets ofvectors cut by thehyperplane {1, 4, 2} ∪ {3, 5}

The algorithm

The cut → the two sets ofvectors cut by thehyperplane {1, 4, 2} ∪ {3, 5}

What is the ratio achieved by the algorithm?

MaxCut: ratio of the Goemans Williamsonalgorithm[GW95]

OPT := value of the MaxCut

OPTSDP := value obtainedby the SDP relaxation

E(C ):= Expectation of thevalue of the cut obtained bythe algorithm

OPTSDP =∑

(i ,j)∈E

1−vi .vj2

OPTSDP =∑

(i ,j)∈E

1−vi .vj2

OPTSDP =∑

(i ,j)∈E

1−cos(θij )2

OPTSDP =∑

(i ,j)∈E

1−vi .vj2

OPTSDP =∑

(i ,j)∈E

1−cos(θij )2

E(C ) =∑

(i ,j)∈EPr(vi , vj separated)

OPTSDP =∑

(i ,j)∈E

1−vi .vj2

OPTSDP =∑

(i ,j)∈E

1−cos(θij )2

E(C ) =∑

(i ,j)∈E

θijπ

OPTSDP =∑

(i ,j)∈E

1−vi .vj2

OPTSDP =∑

(i ,j)∈E

1−cos(θij )2

E(C ) =∑

(i ,j)∈E

θijπ

E(C ) =∑

(i ,j)∈E

θijπ

21−cos(θij )

1−cos(θij )2

OPTSDP =∑

(i ,j)∈E

1−vi .vj2

OPTSDP =∑

(i ,j)∈E

1−cos(θij )2

E(C ) =∑

(i ,j)∈E

θijπ

E(C ) =∑

(i ,j)∈E

θijπ

21−cos(θij )

1−cos(θij )2

Let αGW = min0≤θ≤π

θij1−cos(θij ) ' 0.878

OPTSDP =∑

(i ,j)∈E

1−vi .vj2

OPTSDP =∑

(i ,j)∈E

1−cos(θij )2

E(C ) =∑

(i ,j)∈E

θijπ

E(C ) =∑

(i ,j)∈E

θijπ

21−cos(θij )

1−cos(θij )2

Let αGW = min0≤θ≤π

θij1−cos(θij ) ' 0.878 then

E(C ) ≥ αGWOPTSDP ≥ αGWOPT

From Unique Label Cover to Max-Cut [KKMO04]

ULC (δ)→ distinguishing between the cases:

OPT ≥ 1− δOPT ≤ δ

From Unique Label Cover to Max-Cut [KKMO04]

ULC (δ)→ distinguishing between the cases:

OPT ≥ 1− δOPT ≤ δ

Theorem

For every ε > 0 there exists δ such that there is a PCP for ULC (δ)in which the verifier reads two bits from the proof and accepts iffthey are unequal, and which has completeness c and soundness ssuch that s

c = αGW + ε

2-bit PCP for ULC(δ)

Completeness: If OPT (ULC ) ≥ 1− δ, then there is a proofthat the verifier accepts with probability ≥ c .

Soundness: If OPT (ULC ) ≤ δ, then all proofs are acceptedwith probability ≤ s.

Corollary

Assuming UGC, MaxCut is hard to approximate within αGW + ε.

Corollary

For ε > 0, let ULC (δ) be an instance with a 2-bit PCP.

Corollary

Probability of passing the test of a proof = fraction of edges of thecorresponding cut

Corollary

There is a proof that the verifier accepts with probability ≥ c →there is a cut with value ≥ c |E |

All proofs are accepted with probability ≤ s → all cuts have value≤ s|E |

Corollary

OPT (ULC ) ≥ 1− δ ⇒ MaxCut ≥ c |E |OPT (ULC ) ≤ δ ⇒ MaxCut ≤ s|E |

Corollary

UGC ⇒ MaxCut(s, c) is NP-hard

Corollary

UGC ⇒ MaxCut(s, c) is NP-hard

⇒ MaxCut is hard to approximate within sc = αGW + ε

How to build a 2-bit PCP for ULC?

The verifier expects codewords of the labels

The verifier has to have completeness c ans soundness s

Completeness: OPT (ULC ) ≥ 1− δ, then there is a proof thatthe verifier accepts with probability at least c

Completeness: OPT (ULC ) ≥ 1− δ, then there is a proof thatthe verifier accepts with probability at least c→ Code a good labeling into a proof that can be accepted withgood propability (c)

Completeness: OPT (ULC ) ≥ 1− δ, then there is a proof thatthe verifier accepts with probability at least c→ Code a good labeling into a proof that can be accepted withgood propability (c)→ Use ”Long code” to code labels

Definition

The long code of label i ∈ [1, n] is the truth table of the functionf : {0, 1}n → {0, 1} such that f (x) = xi .

Completeness: OPT (ULC ) ≥ 1− δ, then there is a proof thatthe verifier accepts with probability at least c→ Code a good labeling into a proof that can be accepted withgood propability (c)→ Use ”Long code” to code labels

Definition

The long code of label i ∈ [1, n] is the truth table of the functionf : {0, 1}n → {0, 1} such that f (x) = xi .

Soundness: If OPT (ULC ) ≤ δ, then all proofs are accepted withprobability at most s.

Soundness: If OPT (ULC ) ≤ δ, then all proofs are accepted withprobability at most s.→ If a proof is accepted with probability ≥ s then OPT (ULC ) > δ

Soundness: If OPT (ULC ) ≤ δ, then all proofs are accepted withprobability at most s.→ If a proof is accepted with probability ≥ s then OPT (ULC ) > δ→ Decode a proof into labels

Soundness: If OPT (ULC ) ≤ δ, then all proofs are accepted withprobability at most s.→ If a proof is accepted with probability ≥ s then OPT (ULC ) > δ→ Decode a proof into labels→ Distinguish dictatorships from functions far from beingdictatorships

A dictatorship depending on coordinate i can be decoded intolabel i .

Functions far from dictatorships cannot be decoded

The PCP with only two bits can be designed thanks to:

Unique games (permutations)

Majority is stablest

UGC and SDP

Under UGC, the SDP-based algorithm provides the bestapproximation for MaxCut.

UGC and SDP

Under UGC, the SDP-based algorithm provides the bestapproximation for MaxCut.

A stronger result [Rag08]: UGC → for every MAX-CSP, thesimplest SDP relaxation is the best possible poly-timeapproximation.

UGC and SDP [Rag08]

For every MAX-CSP there is a semi-definite programmingrelaxation SAssuming UGC, no other polynomial time algorithm canprovide a better approximation than S

2 The conjecture

MaxCutUGC and SDP

UGC: True or False?

Validates the quality of SDPrelaxations

It provides very ”neat”inapproximability results

There is no algorithm torefute it

GapULCC(δ)δ,δ is NP-hard[FR04]

False?

The results can still holdeven if the conjecture is false

A sub-exponential timealgorithm has been designed[ABS10]

C (δ)δ → 0 as δ → 0

UGC: True or False?

False?

C (δ)δ → 0 as δ → 0

UGC: True or False?

False?

C (δ)δ → 0 as δ → 0

UGC: True or False?

False?

C (δ)δ → 0 as δ → 0

Thank you

Sanjeev Arora, Boaz Barak, and David Steurer.Subexponential algorithms for unique games and relatedproblems.In In 51 st IEEE FOCS, 2010.

Sanjeev Arora and Shmuel Safra.Probabilistic checking of proofs: A new characterization of np.J. ACM, 45(1):70–122, January 1998.

Uriel Feige and Daniel Reichman.On systems of linear equations with two variables perequation.In Approximation, Randomization, and CombinatorialOptimization. Algorithms and Techniques, volume 3122 ofLecture Notes in Computer Science, pages 117–127. SpringerBerlin Heidelberg, 2004.

Michel X. Goemans and David P. Williamson.

Improved approximation algorithms for maximum cut andsatisfiability problems using semidefinite programming.J. ACM, 42(6):1115–1145, November 1995.

Johan Hastad.Some optimal inapproximability results.J. ACM, 48(4):798–859, July 2001.

Subhash Khot.On the power of unique 2-prover 1-round games.In Proceedings of the Thiry-fourth Annual ACM Symposiumon Theory of Computing, STOC ’02, pages 767–775, NewYork, NY, USA, 2002. ACM.

Subhash Khot.On the unique games conjecture.Technical report, Courant Institute of Mathematical Sciences,NYU, 2010.

S. Khot, G. Kindler, E. Mossel, and R. O’Donnell.Optimal inapproximability results for max-cut and other2-variable csps?In Foundations of Computer Science, 2004. Proceedings. 45thAnnual IEEE Symposium on, pages 146–154, Oct 2004.

Subhash A. Khot and Nisheeth K. Vishnoi.The unique games conjecture, integrality gap for cut problemsand embeddability of negative type metrics into l 1.In Proceedings of the 46th Annual IEEE Symposium onFoundations of Computer Science, FOCS ’05, pages 53–62,Washington, DC, USA, 2005. IEEE Computer Society.

Elchanan Mossel, Ryan O’Donnell, and Krzysztof Oleszkiewicz.

Noise stability of functions with low influences: Invariance andoptimality.

In Proceedings of the 46th Annual IEEE Symposium onFoundations of Computer Science, FOCS ’05, pages 21–30,Washington, DC, USA, 2005. IEEE Computer Society.

Limits of approximation algorithms: Pcps and unique games(dimacs tutorial lecture notes).

Prasad Raghavendra.Optimal algorithms and inapproximability results for every csp.In In Proc. 40 th ACM STOC, pages 245–254, 2008.

Ran Raz.A parallel repetition theorem.SIAM Journal on Computing, 27(3):763–803, 1998.

Luca Trevisan.On khots unique games conjecture, 2011.

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