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Subexponential Algorithms for Unique Games and Related Problems
School on Approximability, Bangalore, January 2011
David SteurerMSR New England
Sanjeev Arora Princeton University & CCI
Boaz BarakMSR New England & Princeton
U
UNIQUE GAMESInput: list of constraints of form
Goal: satisfy as many constraints as possible
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, exp( no(1) )
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
Interlude: Graph Expansion
𝑉
𝑆
expansion() = # edges leaving
𝑑∨𝑆∨¿
normalized indicator vector
-regular graph
normalized adjacency matrix (stochastic)
expansion()
Subexponential Algorithm for Unique Games in time
Easy graphs for UNIQUE GAMES
Constraint Graph variable vertexconstraint edge 𝑖 𝑗
Expanding constraint graph
in time if eigenvalue gap
[Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08]
Expanding constraint graph
Constraint graph with few large eigenvalues
Easy graphs for UNIQUE GAMES
in time if at most eigenvalues
in time if eigenvalue gap
[Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08]
Subexponential Algorithm for Unique Games in time
Graph Eigenvalues
𝜆𝑚+1
1−100𝜀
𝜆1𝜆2
10
normalized adjacency matrix …
quasi-expander
[Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11]
Expanding constraint graph
Constraint graph with few large eigenvalues
Easy graphs for UNIQUE GAMES
in time if at most eigenvalues
in time if eigenvalue gap
[Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08]
[Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11]
Subspace Enumeration [Kolla-Tulsiani’07]
exhaustive search over certain subspace
Subexponential Algorithm for Unique Games in time
Question: Dimension of this subspace?
quasi-expander
[Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11]
quasi-expander
in time if at most eigenvalues
Expanding constraint graph
Constraint graph with few large eigenvalues
Easy graphs for UNIQUE GAMES
in time if eigenvalue gap
[Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08]
Subexponential Algorithm for Unique Games in time
Graph Decomposition
By removing 1% of edges, decompose constraint graphinto components for which is “easy”
Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10]
hardconstraint graph
remove 1% of edges
easy
easyeasy
easy easy
easy
easy
By removing 1% of edges, decompose constraint graphinto components for which is “easy”
Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10]
quasi-expander
in time if at most eigenvalues
Expanding constraint graph
Constraint graph with few large eigenvalues
Easy graphs for UNIQUE GAMES
in time if eigenvalue gap
[Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08]
[Kolla-Tulsiani’07, Kolla’10, here]
Subexponential Algorithm for Unique Games in time
Graph Decomposition: Given UG instance, distinguishYES: NO:
YES: opt>1-
opt>1- opt>1-
opt>1-
opt>1-
By removing 1% of edges, decompose constraint graphinto components for which is “easy”
Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10]
in time if at most eigenvalues
Expanding constraint graph
Constraint graph with few large eigenvalues
Easy graphs for UNIQUE GAMES
in time if eigenvalue gap
[Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08]
[Kolla-Tulsiani’07, Kolla’10, here]
Subexponential Algorithm for Unique Games in time
Graph Decomposition
YES:
?
YES?
YES YES
YES
YES
: Given UG instance, distinguishYES: NO:
By removing 1% of edges, decompose constraint graphinto components for which is “easy”
Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10]
in time if at most eigenvalues
Expanding constraint graph
Constraint graph with few large eigenvalues
Easy graphs for UNIQUE GAMES
in time if eigenvalue gap
[Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08]
[Kolla-Tulsiani’07, Kolla’10, here]
Subexponential Algorithm for Unique Games in time
Graph Decomposition: Given UG instance, distinguishYES: NO:
NO: opt<
opt< opt<
opt<
opt<
By removing 1% of edges, decompose constraint graphinto components for which is “easy”
Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10]
in time if at most eigenvalues
Expanding constraint graph
Constraint graph with few large eigenvalues
Easy graphs for UNIQUE GAMES
in time if eigenvalue gap
[Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08]
[Kolla-Tulsiani’07, Kolla’10, here]
Subexponential Algorithm for Unique Games in time
Graph Decomposition
NO:
NO
NO?
NO NO
NO
?
: Given UG instance, distinguishYES: NO:
[Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11]
quasi-expander
Expanding constraint graph
Constraint graph with few large eigenvalues
Easy graphs for UNIQUE GAMES
in time if at most eigenvalues
in time if eigenvalue gap
[Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08]
Subexponential Algorithm for Unique Games in time
Graph Decomposition
By removing 1% of edges, every graph can be decomposed into components with
Classical
at most eigenvalues
Here[Leighton-Rao’88, Goldreich-Ron’98, Spielman-Teng’04, Trevisan’05]
eigenvalue gap
Idea: quasi-expander small-set expander
Idea: quasi-expander small-set expander
Expanding constraint graph
Constraint graph with few large eigenvalues
Easy graphs for UNIQUE GAMES
in time if at most eigenvalues
in time if eigenvalue gap
[Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08]
Subexponential Algorithm for Unique Games in time
Graph Decomposition
By removing 1% of edges, every graph can be decomposed into components with
Classical
at most eigenvalues
Here
eigenvalue gap
quasi-expander
[Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11]
Constraint graph with few large eigenvalues
in time if at most eigenvalues
[Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11]
label-extended graph constraint graph
cloud cloud
if a-b=c mod k
assignment satisfying of constraints
vertex set of size and expansion
𝑖 𝑗
Assume: label-extended graph has at most eigenvalues
2
Constraint graph with few large eigenvalues
in time if at most eigenvalues
[Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11]
label-extended graph constraint graph
assignment satisfying 90%of
constraints
99% of indicator vector lies inspan of top m eigenvectors
Assume: label-extended graph has at most eigenvalues
enumerate subspace
2
vertex set of size and expansion
assignment satisfying of constraints
Constraint graph with few large eigenvalues
in time if at most eigenvalues
[Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11]
label-extended graph
assignment satisfying 90%of
constraints
99% of indicator vector lies inspan of top m eigenvectors
Assume: label-extended graph has at most eigenvalues
vertex set of size and expansion
enumerate subspace
assignment satisfying of constraints
Compare: Fourier-based learning
2
constraint graph normalized indicator vector
Suppose: > 1% of is orthogonal to span
⟨ 𝑥 , �̂� 𝑥 ⟩<0.99𝜆1+0.01 𝜆𝑚+1≤1−𝜀
expansion
Constraint graph with few large eigenvalues
in time if at most eigenvalues
[Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11]
Idea: quasi-expander small-set expander
Easy graphs for UNIQUE GAMES
Constraint graph with few large eigenvalues
in time if at most eigenvalues
[Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11]
Graph Decomposition
By removing 1% of edges, every graph can be decomposed into components with
Classical
at most eigenvalues
Here
eigenvalue gap
Subexponential Algorithm for Unique Games in time
Constraint graph with few large eigenvalues
in time if at most eigenvalues
[Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11]
Easy graphs for UNIQUE GAMES
Graph Decomposition
By removing 1% of edges, every graph can be decomposed into components with
Classical
at most eigenvalues
Here
eigenvalue gap
Subexponential Algorithm for Unique Games in time
Idea: quasi-expander small-set expander
non-expanding set
Graph DecompositionBy removing 1% of edges, every graph can be decomposed into components with at most eigenvalues
For ,
follows from Cheeger bound [Cheeger’70, Alon-Milman’85, Alon’86]
Veigenvalue gap
S
Assume graph is d-regular
expansion()
|𝑆|<|𝑉|/2
⇒
at most eigenvalues
For ,
Graph DecompositionBy removing 1% of edges, every graph can be decomposed into components with
Assume graph is d-regular
follows from Cheeger bound
V⇒S
“higher-order Cheeger bound” [here]
general
eigenvalues
small non-expanding set
¿𝑉∨¿𝑛
|𝑆|<|𝑉|/𝒏𝜷
expansion()
“higher-order Cheeger bound”
eigenvalues
¿𝑉∨¿𝑛
𝑛𝑂 ( 𝛽)
⇒
|𝑆|<|𝑉|/𝑛𝛽
expansion()
“higher-order Cheeger bound”
eigenvalues ¿𝑉∨¿𝑛
𝑛𝑂 ( 𝛽) ⇒|𝑆|<|𝑉|/𝑛𝛽
expansion()
“higher-order Cheeger bound”
eigenvalues ¿𝑉∨¿𝑛
𝑛𝑂 ( 𝛽)
|𝑆|<|𝑉|/𝑛𝛽
expansion()
Idea
⇒
“higher-order Cheeger bound”
eigenvalues ¿𝑉∨¿𝑛
𝑛𝑂 ( 𝛽)
|𝑆|<|𝑉|/𝑛𝛽
expansion()
volume growth in intermediate step
has expansion
( for degree )It would suffice to show:
vertex such that for
⇒
“higher-order Cheeger bound”
eigenvalues ¿𝑉∨¿𝑛
𝑛𝑂 ( 𝛽)
|𝑆|<|𝑉|/𝑛𝛽
expansion()
It would suffice to show:
vertex such that for
collision probability decay in intermediate step s < t
level set of has expansion and size
‖𝐺𝑡𝑒𝑖‖2>𝑛𝛽 /|𝑉|
Heuristic:collision probability
⇒collision probability of
t-step random walk from i
Suffices
local Cheeg
er bound
Heuristic:collision probability
collision probability decay in intermediate step s < t
“higher-order Cheeger bound”
eigenvalues ¿𝑉∨¿𝑛
𝑛𝑂 ( 𝛽)
|𝑆|<|𝑉|/𝑛𝛽
expansion()
It would suffice to show:
vertex such that for
level set of has expansion and size
‖𝐺𝑡𝑒𝑖‖2>𝑛𝛽 /|𝑉|
¿𝑛𝑂 (𝛽) (1−𝜀 )2 𝑡¿𝑛𝛽
“A Markov chain with many large eigenvalues cannot mix locally everywhere”
⇒collision probability of
t-step random walk from i
Suffices
Improved approximations for d-TO-1 GAMES ( Khot’s d-to-1 Conjecture) [S.’10]
Better approximations for MAX CUT and VERTEX COVER on small-set expanders
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?
More Subexponential Algorithms
Similar approximation for MULTI CUT and SMALL SET EXPANSION
What else can be done in subexponential time?
Towards refuting the Unique Games Conjecture
Better approximations for MAX CUT or VERTEX COVER on general instances?