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