Subexponential Algorithms for Unique Games and Related Problems

Approximation Algorithms, June 2011

David SteurerMSR New England

Sanjeev Arora Princeton University & CCI

Boaz BarakMSR New England

U

Subexponential Algorithms for Unique Games and Related Problems

David SteurerMSR New England

Sanjeev Arora Princeton University & CCI

Boaz BarakMSR New England

Rounding Semidefinite ProgrammingHierarchies via Global Correlation

David SteurerMSR New England

Prasad RaghavendraGeorgia Tech

Boaz BarakMSR New England

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:

UG (𝜀)

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], …

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

Unique Games Barrier

Example: -approximation for MAX CUTat least as hard as

UNIQUE GAMES is common barrier for improving current algorithms of

many basic problems

Goemans–Williamson

bound for MAX CUT

Subexponential Algorithm for Unique Games

Input: UNIQUE GAMES instance with alphabet size ksuch that of constraints are satisfiable,

Output:assignment satisfying of constraints Time:

in time

Time vs Approximation Trade-off

Analog of UGC with subconstant (say ) is false (*)(contrast: subconstant hardness for LABEL COVER [Moshkovitz-Raz’08])

NP-hardness reduction for must have blow-up (*)

rules out certain classes of reductions for proving UGC

(*) assuming 3-SAT does not have subexponential algorithms,

UGC-based hardness does not rule out subexponential algorithms, Possibility: -time algorithm for MAX CUT() ?

Subexponential Algorithm for Unique Games

in time

Consequences

poly (𝑛) exp (𝑛)

2-SAT

MAX 3-SAT()MAX CUT()

3-SAT (*)

FACTORING

exp (𝑛1 /2)exp (𝑛1 /3)exp (𝑛𝜀 1/3 )

UG (𝜀 )MAX 3-SAT()

LABEL COVER()

[Moshkovitz-Raz’08+ Håstad’97]MAX CUT()?

(*) assuming Exponential Time Hypothesis [Impagliazzo-Paturi-Zane’01]( 3-SAT has no algorithm )

Subexponential Algorithm for Unique Games

in time

GRAPH ISOMORPHISM

Subexponential Algorithm for Unique Games

in time

here: via semidefinite programming hierarchies

What we want:

jointly distributed random variables over

UNIQUE GAMESInput: list of constraints of form

Goal: satisfy as many constraints as possible

Pr (𝑋 𝑖− 𝑋 𝑗≡𝑐 )≥1−𝜀 for typical constraint

= indicator of

p.s.d. for all with -

same marginal for in distributions and

-local

Goal: produce global random variables

for most constraints

Here: iterative procedure for [Arora-Barak-S.’10+ Barak-Raghavendra-S.’11]

jointly distributed random variables over

Pr (𝑋 𝑖− 𝑋 𝑗≡𝑐 )≥1−𝜀 for typical constraint

distributions over for all with

time

consistency

positive semidefiniteness

𝑆𝑇

𝑆∩𝑇

-level SDP hierarchy:What we want:

Components of iterative procedure

Rounding

Conditioning

sample variables independently according to their marginals

pick a vertex and sample

condition on sample for

Partitioning

find vertex subset

break dependence between and

enough if constraint graph has few significant eigenvalues

[BRS’11]

general frameworkfor rounding

SDP hierarchies

Important fact:

can approximate by Gram matrix of unit vectors (tensoring trick)

Corr (𝑋 𝑖 , 𝑋 𝑗)(Pairwise) Correlation

max𝑐

∑𝑎

|Cov ( 𝑋 𝑖𝑎 , 𝑋 𝑗 (𝑎+𝑐 ) )|

∑𝑎

(𝑉𝑎𝑟 (𝑋 𝑖𝑎 )−𝑉𝑎𝑟 (𝑋 𝑖𝑎|𝑋 𝑗¿ )decrease in variance when conditioning on

statistical distance between and

Rounding Conditioning Partitioning

similar to mutual information

Rounding Conditioning Partitioning

sample variables independently according to their marginals

If then independent samplingsatisfies constraint with probability

Rounding fails 𝐄𝑖∼ 𝑗Corr (𝑋 𝑖 ,𝑋 𝑗 )>1−𝑂 (𝜀)

Local Correlation(over edges of constraint graph)

Rounding Conditioning Partitioning

statistical distance between independent and correlated

sampling

pick a vertex and sample

condition on sample for

computationally expensive(level level )

condition on vertex only if

can condition times on such vertices

Conditioning fails

Global Correlation(over random vertex pairs)

Rounding Conditioning Partitioning

Issue:

Idea:

measures decrease in variance when conditioning

on

find vertex subset

break dependence between and

destroy correlation for constraints between and

𝑉 ∖𝑆

𝑆

Rounding Conditioning Partitioning

Issue:

Wanted:set with small expansion & small cardinality

For endpoints of random path of length

𝐄𝑖∼𝑡 𝑗Corr (𝑋 𝑖 ,𝑋 𝑗 )>(1−𝑂 (𝜀) )𝑡≫𝑛−𝛽

random walk stuck for steps on fraction of vertices vertex set with and expansion

A vertex is cut in at most partitioning steps break only edges

Rounding Conditioning Partitioning

𝑉 ∖𝑆

𝑆

fails only if local correlation high

fails only if global correlation low

Correlation Propagation

is Grammatrix of unit

vectors

find vertex subset

break dependence between and

For general 2-CSP:

PTAS if constraint graph is random (degree alphabet)

QPTAS if constraint graph is hypercontractive very good expander

for small sets

[Barak-Raghavendra-S.’11]

Subsequent work: [Arora-Ge’11]

better 3-COLORING approximation on some graph families

Independent work:

approximation schemes for quadratic integer programming with p.s.d. objective & few relevant eigenvalues

More SDP-hierarchy algorithms

[Guruswami-Sinop’11]

Open Questions

Example: -approximation for SPARSEST CUT in time ?

How many large eigenvalues can a small-set expander have?

Is Boolean noise graph the worst case? (large eigenvalues)

Thank you!Questions?

What else can be done in subexponential time?

Towards settling the Unique Games Conjecture

Better approximations for MAX CUT or VERTEX COVER on general instances?

No: small set expander with large eigenvalues

[Barak-Gopalan-Håstad-Meka-Raghavendra-S.’11]