the unique games conjecture and graph expansion school on approximability, bangalore, january 2011...

The Unique Games Conjecture and Graph Expansion

School on Approximability, Bangalore, January 2011

Joint work withS

Prasad Raghavendra

Georgia Institute of Technology

David Steurer

Microsoft Research New England

The Unique Games Conjecture

Graph Expansion

Reductions

Integrality Gaps

The Unique Games Conjecture

Graph Expansion

Reductions

Integrality Gaps

UNIQUE GAMESInput: list of constraints of form

Goal: satisfy as many constraints as possible

[𝑘 ][𝑘 ]

𝑥 𝑗𝑥𝑖

UNIQUE GAMESInput: list of constraints of form

Goal: satisfy as many constraints as possible

Input: UNIQUE GAMES instance with (say)

Goal: Distinguish two cases

YES: more than of constraints satisfiableNO: less than of constraints

satisfiable

Unique Games Conjecture (UGC) [Khot’02]

For every , the following is NP-hard:

Implications of UGCFor many basic optimization problems, it is NP-hard to beat current algorithms (based on simple LP or SDP relaxations)

Examples:

VERTEX COVER [Khot-Regev’03], MAX CUT [Khot-Kindler-Mossel-O’Donnell’04,

Mossel-O’Donnell-Oleszkiewicz’05],every MAX CSP [Raghavendra’08], …

What are hard instances for UNIQUE GAMES?

What are hard instances for UNIQUE GAMES?

Random instances

Random 3-SAT Hypothesis

[Appelbaum-Barak-Wigderson’10, Bhaskara--Charikar-Chlamtac-Feige-Vijayaraghavan’10]

Other problems:

Planted DENSEST k-SUBGRAPH

UNIQUE GAMES:

expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] few large eigenvalues [Kolla’10, Barak-Raghavendra-S.’11] strong small-set expanders [Arora-Impagliazzo-Matthews-S.’10]

[Feige’02, Schoenebeck’08]

Easy for random-like instances

What are hard instances for UNIQUE GAMES?

Random 3-SAT,Planted DENSEST k-SUBGRAPHUNIQUE GAMES:Easy for random-like instancesOther problems:

Random instances

Other problems:

CLIQUE on product graphs

PROJECTION GAMES from parallel repetition [Raz’98]

UNIQUE GAMES:

Easy for parallel-repeated instances of MAX CUT [Barak-Hardt-Haviv-Rao-Regev-S.’08] (based on counterexample for strong

parallel repetition [Raz’08])

Combinations of mildly hard instances

CLIQUE,UNIQUE GAMES:

PROJECTION GAMESEasy for parallel-repeated instances of MAX CUT

What are hard instances for UNIQUE GAMES?

Combinations of mildly hard instances

Random 3-SAT,Planted DENSEST k-SUBGRAPHUNIQUE GAMES:Easy for random-like instancesOther problems:

Random instances

Other problems:

Hard instances for UNIQUE GAMES from hard instances for SMALL SET EXPANSION[here]

natural generalization of SPARSEST CUT

The Unique Games Conjecture

Graph Expansion

Reductions

Integrality Gaps

d-regular graph Gd

vertex set S

Graph Expansion

expansion(S) = # edges leaving S

d |S|

volume(S) = |S||V|Important notion in many contexts:

derandomization, network routing, coding theory,Markov chains, differential geometry, group theory

S

expansion(S) = # edges leaving S

d |S|volume(S) =

|S||V|

S

expansion(S) = # edges leaving S

d |S|volume(S) =

|S||V|

SPARSEST CUT

Goal: find S with volume(S) < ½ so as tominimize expansion(S)

Input: graph G

eigenvalue gap: first non-trivial approximation [Cheeger’70]

semidefinite relaxation: -approximation [Arora-Rao-Vazirani’04]

linear relaxation: -approximation [Leighton-Rao’88]

vs. approximation

No strong hardness for approximating SPARSEST CUT known! (even assuming UGC)

Approximating SPARSEST CUT

S

expansion(S) = # edges leaving S

d |S|volume(S) =

|S||V|

SPARSEST CUT

Goal: find S with volume(S) < ½ so as tominimize expansion(S)

Input: graph G

For small (and constant ):

SMALL-SET EXPANSIONparameter > 0

𝜹

no poly-time algorithm with non-trivial approximation guarantee known

Input: graph G, parameter

Goal: Distinguish two cases

YES: exists set with volume and expansion NO: all sets with volume have

expansion

Small Set Expansion Hypothesis (SSE)

For every , the following is NP-hard:

S

expansion(S) = # edges leaving S

d |S|volume(S) =

|S||V|

SPARSEST CUT

Goal: find S with volume(S) < ½ so as tominimize expansion(S)

Input: graph G

For small (and constant ):

SMALL-SET EXPANSIONparameter > 0

𝜹

no poly-time algorithm with non-trivial approximation guarantee known

Input: graph G, parameter

Goal: Distinguish two cases

YES: exists set with volume and expansion NO: all sets with volume have

expansion

Small Set Expansion Hypothesis (SSE)

For every , the following is NP-hard:

The Unique Games Conjecture

Graph Expansion

Reductions

Integrality Gaps

label-extended graph constraint graph

cloud cloud

if a-b=c mod k

𝑖 𝑗

“Superficial” Connection of UNIQUE GAMES and SMALL-SET EXPANSION

assignment satisfying of constraints

vertex set of volume and expansion

SSE

UGC+ SSE

[Raghavendra-S.’10]

UGC

additional promise:small-set

expansion of constraint graph

MAX CUT

VERTEX COVER

Any MAX CSP…

SPARSEST CUT

BALANCED SEPARATOR

MINIMUM LINEAR

ARRANGEMENT

MIN k-CUT…

[Raghavendra- -Tulsiani-S.’10]

[Raghavendra- -Tulsiani-S.’10]

Small-Set Expansion Unique Games

Task: find non-expanding set of volume

graph G

A

B

Verifier

sample random edges M one vertex of each edgeother vertex of each edge

BA

Prover 1 Prover 2pick pick

ba

Verifieraccepts

if

To show:

expansion() no strategy has acceptanceprobability

expansion() strategy with acceptance probability

a

b

graph G

A

B

S

graph G

A

B

Completeness

Partial Strategy for Prover 1 (and 2):

pick if

(otherwise, refuse to answer)

With probability , no edge in M crosses

With constant probability,

acceptance probability conditioned on one prover answering

Suppose volume() = and expansion()

graph G

Soundness

edges such that conditioned on provers play small non-expanding set

But: analysis only works if graph contains -weight copy of complete graph, which we can arrange beforehand

S

The Unique Games Conjecture

Graph Expansion

Reductions

Integrality Gaps

Goal:rule out that certain (classes of) algorithms refute the UGC or the SSE hypothesis

Here: algorithms based on (hierarchies) of SDP relaxations

(capture current best approximation algorithms)

Basic SDP relaxation for SMALL-SET EXPANSION

minimize𝐄𝑖𝑗∈ 𝐸 (𝐺 )‖𝑣 𝑖−𝑣 𝑗‖

2

𝐄𝑖 ∈𝑉 (𝐺)‖𝑣 𝑖‖2

𝐄𝑖 , 𝑗 ∈𝑉 (𝐺)|⟨𝑣 𝑖 ,𝑣 𝑗 ⟩|≤𝛿𝐄𝑖∈𝑉 (𝐺 )‖𝑣 𝑖‖2

subject to

Basic SDP relaxation for SMALL-SET EXPANSION

minimize𝐄𝑖𝑗∈ 𝐸 (𝐺 )‖𝑣 𝑖−𝑣 𝑗‖

2

𝐄𝑖 ∈𝑉 (𝐺)‖𝑣 𝑖‖2

𝐄𝑖 , 𝑗 ∈𝑉 (𝐺)|⟨𝑣 𝑖 ,𝑣 𝑗 ⟩|≤𝛿𝐄𝑖∈𝑉 (𝐺 )‖𝑣 𝑖‖2

subject to

Integrality Gap Instance

vertices / vectors:

d-dimensional hypercube (with -length edges)

edges:

Note:

SDP solution with value for

vertices / vectors:

d-dimensional hypercube (with -length edges)

edges:

Integrality Gap Instance

SDP solution with value for

What is minimum expansion of sets of volume ?

𝑆= {𝑥∈ {±1 }𝑑 ∣ 𝑥1=…=𝑥𝑡=1 }volume (𝑆 )=2− 𝑡

expansion (𝑆 )≈ 𝜀𝑑⋅ 𝑡𝑑

=𝜀𝑡

sets of volume have expansion

vertices / vectors:

d-dimensional hypercube (with -length edges)

edges:

Integrality Gap Instance

SDP solution with value for

vertices / vectors:

d-dimensional hypercube (with -length edges)

edges:

sets of volume have expansion

Integrality Gap Instance

SDP solution with value for

Integrality Gap Instance

SDP solution with value for

vertices / vectors:

d-dimensional hypercube (with -length edges)

edges:

sets of volume have expansion

Integrality Gaps for UNIQUE GAMES via reductions

Basic SDP via SSEUG reduction [similar to Khot-Vishnoi’05]

-size SDP via alphabet reduction [Raghavendra-S.’09]

super-polynomial lower-bound for UNIQUE GAMES in a restricted computational model

Integrality Gap Instance

vertices / vectors:

d-dimensional hypercube (with -length edges)

edges:

SDP solution with value for

sets of volume have expansion

Integrality Gaps for UNIQUE GAMES via reductions

Basic SDP via SSEUG reduction [similar to Khot-Vishnoi’05]

-size SDP via alphabet reduction [Raghavendra-S.’09]

Open: -size SDP?

(contrast: solves [Barak-Raghavendra-S.’11] )

Thank you!Questions?