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Subexponential Algorithms for Unique Games and Related Problems. Sanjeev Arora Princeton University & CCI. Boaz Barak MSR New England. David Steurer MSR New England. U. Approximation Algorithms, June 2011. Subexponential Algorithms for Unique Games and Related Problems. - PowerPoint PPT Presentation
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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 BarrierExample: -approximation for MAX CUT
at 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
Partitioningfind 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]