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Recent Progress in Approximability
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Administrivia
Most agreeable times:
Monday 2:30-4:00Wednesday 4:00-5:30Thursday 4:00-5:30Friday 1:00-2:30
Please Fill Up Survey: http://www.surveymonkey.com/s/9TSVQM7
Evaluation: 6-8 short homeworks and class participation.
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Max Cut
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Max CUTInput: A weighted graph G
Find:A Cut with maximum number/weight of crossing edges
Fraction of crossing edges
MaxCut is NP-complete
(Karp’s original list of 21 NP-complete problems (1971)
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An algorithm A is an α-approximation for a problem if for every instance I,
A(I) ≥ α OPT(I)∙
--Vast Literature--
Approximation Algorithms
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Max Cut
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Max CUTInput: A weighted graph G
Find:A Cut with maximum number/weight of crossing edges
Trivial ½ Approximation
Assign each vertex randomly to left or right side of the cutAnalysis
For every edge e,
Probability[edge is cut] = ½
Fraction of edges cut = ½
Optimum MaxCut < 1
So,
Solution returned = ½ > ½ *Optimum MaxCut
Till 1994, this was the state of the art.
Many linear programming techniques were known to NOT get any better approximation.
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The ToolsTill 1994,A majority of approximation algorithms directly or indirectly relied on Linear Programming.
In 1994,Semidefinite Programming based algorithm for Max Cut
[Goemans-Williamson]
Semidefinite Programming - A generalization of Linear Programming.
Semidefinite Programming is the one of the most powerful tools in approximation algorithms.
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Eji
jiij vvw),(
2||4
1
Semidefinite Program
Variables : v1 , v2 … vn
| vi |2 = 1
Maximize
Max Cut SDP
Quadratic Program
Variables : x1 , x2 … xn
xi = 1 or -1
Maximize
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1
1
1
-1
-1
-1
-1-1
-1
Eji
jiij xxw),(
2)(4
1
Relax all the xi to be unit vectors instead of {1,-1}. All products are replaced by inner products of vectors
1 -1
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Semidefinite Program:[Goemans-Williamson 94]
Embedd the graph on the N - dimensional unit ball, Maximizing
¼ (Average Squared Length
of the edges)
Eji
jiij vvw),(
2||4
1
Semidefinite Program[Goemans-Williamson 94]
Variables : v1 , v2 … vn
|vi|2 = 1
Maximize
MaxCut
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1
1
-1
-1
-1
-1-1
-11 -1
Max Cut ProblemGiven a graph G,Find a cut that maximizes the number of crossing edges
v1
v2
v3
v4
v5
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MaxCut Rounding
v1
v2
v3
v4
v5
Cut the sphere by a random hyperplane, and output the induced graph cut.
-A 0.878 approximation for the problem.
[Goemans-Williamson]
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Analysisv1
v2v3
v4
v5
SDP Optiumum
1015
3
711
OptimalMaxCut
v1
v2v3
v4
v5
Algorithm’sOutput
0 1
Rounding Ratio > 0.878
Integrality Gap
Algorithm Output > 0.878 X SDP Optimum > 0.878 X Optimum MaxCut
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minimumover all instances
=
value of rounded solution
value of SDP solution
rounding – ratioA
(approximation ratio)≤ integrality gap
=
value of optimal solution
value of SDP solution
minimumover all instances
For any rounding algorithm A, and a SDP relaxation ¦
v1v2 v3
v4v5
SDP Optiumum
10153
711OptimalMaxCut
v1v2 v3
v4v5
Algorithm’sOutput
0 1
Rounding Ratio > 0.878
Integrality Gap
=“algorithm achieves the gap’’
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InapproximabilityIs 0.878 the best possible approximation ratio for MaxCut?
Satisfiable
Unsatisfiable
MaxCut value = K
MaxCut value < K
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1
-1
-1
-1-1
3-SAT InstancePolynomial time
reduction
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What we need..
(Completeness)
Satisfiable
(Soundness)
Unsatisfiable
MaxCut value = K
MaxCut value < 0.9K
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1
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1
-1
-1
-1-1
3-SAT InstancePolynomial time
reduction
If we had a polytime 0.95 approximation algorithm for MaxCut
A polytime algorithm for 3-SAT
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A probabilistically checkable proof (PCP)
Goal: Alice wants to prove to Bob that 3-SAT instance A is satisfiable
3-SAT Instance A
Alex Bob (polytime machine)Satisfying assignment
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A probabilistically checkable proof (PCP)
Goal: Alice wants to prove to Bob that 3-SAT instance A is satisfiable
3-SAT Instance A
Alex Bob (polytime machine)
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-1-1
Polynomial
time
reduction
3-SAT Instance A
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Polynomial
time
reduction
Probabilistically Checkable ProofA cut of value > 0.9
Verifier (Bob):
Sample a random edge in graph,
Accept if edge is cut.
Prob[Bob Accepts] =
Value of the Cut
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Suppose,
(Completeness)
Satisfiable(Soundness)
Unsatisfiable
MaxCut value = 0.99
MaxCut value < 0.9
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3-SAT InstancePolynomial time
reduction
Completeness: There exists a ``proof” that Bob accepts with probability 0.99Soundness: No matter what Alex does, Bob accepts with probability < 0.9
Bob reads only 2 bits of the proof!!
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Analogy to Math Proofs
Could you check the proof of a theorem with any reasonable confidence by reading only 3 bits of the proof???
Guess: Probably Not..
Max-SNP complexity class was defined, because it was believable that
we will never be able to get a Gap Reduction aka Probabilistically Checkable Proof for NP.
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PCP Theorem: [Arora-Lund-Motwani-Sudan-Szegedy 1991]
Max-3-SAT is NP-hard to approximate better than 1- 10^{-100}.
Corollary:Max-Cut is NP-hard to approximate better than 1- 10^{-200}.
Long and very difficult proof, simplified over the years..(*Check out History of PCP Theorem:
http://www.cs.washington.edu/education/courses/cse533/05au/pcp-history.pdf)
Completely new proof by Irit Dinur in 2005.
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Hastad’s 3-Query PCP [Håstad STOC97]
For any ε > 0, NP has a 3-query probabilistically checkable proof system such that:
• Completeness = (1 – ε) • Soundness = 1/2 + ε
Verifier reads only 3-bits, and checks a linear equation on them!
Xi + Xj = Xk + c (mod p)
Alternately,
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Hastad’s 3-Query PCP [1997]
For any ε > 0, given a set of linear equations modulo 2 , it is NP-hard to distinguish between:
• (1 – ε) – fraction of the equations can be satisfied.• 1/2 + ε – fraction of the equations can be satisfied.
All equations are of the form Xi + Xj = Xk + c (mod p)
By Very Clever Gadget reductions, [Sudan-Sorkin-Trevisan-Williamson]
MaxCut is NP-hard to approximate beyond 0.94.
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ALGORITHMS[Charikar-Makarychev-Makarychev 06]
[Goemans-Williamson][Charikar-Wirth]
[Lewin-Livnat-Zwick][Charikar-Makarychev-Makarychev 07]
[Hast] [Charikar-Makarychev-Makarychev 07]
[Frieze-Jerrum][Karloff-Zwick]
[Zwick SODA 98][Zwick STOC 98]
[Zwick 99][Halperin-Zwick 01]
[Goemans-Williamson 01][Goemans 01]
[Feige-Goemans][Matuura-Matsui]
[Trevisan-Sudan-Sorkin-Williamson]
Approximability of CSPsGap for MaxCUTAlgorithm = 0.878Hardness = 0.941
MAX CUT
MAX 2-SAT
MAX 3-SAT
MAX 4-SAT
MAX DI CUT
MAX k-CUT
Unique GamesMAX k-CSP
MAX Horn SAT
MAX 3 DI-CUTMAX E2 LIN3
MAX 3-MAJ
MAX 3-CSPMAX 3-AND
0 1
NP HARD
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Given linear equations of the form:
Xi – Xk = cik mod p
Satisfy maximum number of equations.
x-y = 11 (mod 17)x-z = 13 (mod 17)
…….
z-w = 15(mod 17)
Unique Games Conjecture [Khot 02] [KKMO]
For every ε> 0, for large enough p,Given : 1-ε (99%) satisfiable system,
NP-hard to satisfyε (1%) fraction of equations.
Towards bridging this gap, In 2002, Subhash Khot introduced the
Unique Games Conjecture
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Unique Games Conjecture
A notorious open problem.
Hardness Results: No constant factor approximation for unique games. [Feige-Reichman]
Algorithm On (1-Є) satisfiable instances
[Khot 02]
[Trevisan]
[Gupta-Talwar] 1 – O(ε logn)
[Charikar-Makarychev-Makarychev]
[Chlamtac-Makarychev-Makarychev]
[Arora-Khot-Kolla-Steurer-Tulsiani-Vishnoi]
)2/( p)loglog(1 pnO
)log(1 3 nO
))/1log((1 5/12 pO
1log1
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Assuming UGCUGC Hardness
Results[Khot-Kindler-Mossel-O’donnell]
[Austrin 06][Austrin 07]
[Khot-Odonnell][Odonnell-Wu]
[Samorodnitsky-Trevisan]
NP HARDUGC HARD
0 1
MAX CUT
MAX 2-SAT
MAX 3-SAT
MAX 4-SAT
MAX DI CUT
MAX k-CUT
Unique GamesMAX k-CSP
MAX Horn SAT
MAX 3 DI-CUTMAX E2 LIN3
MAX 3-MAJ
MAX 3-CSPMAX 3-ANDFor MaxCut, Max-2-SAT,
Unique Games based hardness=
approximation obtained by Semidefinite programming!
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The Connection
MAX CUT
MAX 2-SAT
MAX 3-SAT
MAX 4-SAT
MAX DI CUT
MAX k-CUT
Unique GamesMAX k-CSP
MAX Horn SAT
MAX 3 DI-CUTMAX E2 LIN3
MAX 3-MAJ
MAX 3-CSPMAX 3-AND
UGC Hard
GENERICALGORITHM
Theorem:Assuming Unique Games Conjecture, For every CSP, “the simplest semidefinite programs give the best approximation computable efficiently.”
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Constraint Satisfaction Problems [Raghavendra`08][Austrin-Mossel]
MAX CUT [Khot-Kindler-Mossel-ODonnell][Odonnell-Wu]
MAX 2SAT [Austrin07][Austrin08]
Ordering CSPs [Charikar-Guruswami-Manokaran-Raghavendra-Hastad`08]
MAX ACYCLIC SUBGRAPH, BETWEENESS
Grothendieck Problems [Khot-Naor, Raghavendra-Steurer]
Metric Labeling Problems [Manokaran-Naor-Raghavendra-Schwartz`08]
MULTIWAY CUT, 0-EXTENSION
Kernel Clustering Problems [Khot-Naor`08,10]
Strict Monotone CSPs [Kumar-Manokaran-Tulsiani-Vishnoi`10]
VERTEX COVER [Khot-Regev], HYPERGRAPH VERTEX COVER
Assuming the Unique Games Conjecture,
A simple semidefinite program (Basic-SDP) yields the optimal approximation ratio for
Is the conjecture true?
Many many ways to disprove the conjecture! Find a better algorithm for any one of these problems.
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The UG Barrier
Constraint Satisfaction Problems
Graph Labelling Problems
Ordering CSPs
Kernel Clustering Problems
Monotone Min-One CSPs
UGC HARD
If UGC is true,
Then Simplest SDPs give the best approximation possible.
If UGC is false,
Hopefully, a new algorithmic technique will arise.
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What if UGC is false?
Could existing techniques ( LPs/SDPs) disprove the UGC?
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What if UGC is false?
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UGC is false New algorithms?
Unique Games
Constraint Satisfaction Problems [Raghavendra`08]MAX CUT, MAX 2SAT
Ordering CSPs [GMR`08]MAX ACYCLIC SUBGRAPH, BETWEENESS
Grothendieck Problems [KNS`08, RS`09]
Metric Labeling Problems [MNRS`08]MULTIWAY CUT, 0-EXTENSION
Kernel Clustering Problems [KN`08,10]
Strict Monotone CSPs [KMTV`10]VERTEX COVER, HYPERGRAPH VERTEX COVER
…
Problem X
UGC is false New algorithm for Problem X
Despite considerable efforts,No such reverse reduction known for any of the above problems
[Feige-Kindler-Odonnell,Raz’08, BHHRRS’08]
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Graph Expansion
d-regular graph G
d
expansion(S) = # edges leaving S
d |S|
vertex set S
A random neighbor of a random vertex in S is outside of S with probability expansion(S)
ФG = expansion(S)minimum|S| ≤ n/2
Conductance of Graph G
Uniform Sparsest Cut Problem Given a graph G compute ФG and find the set S achieving it.
Approximation Algorithms:
•Cheeger’s Inequality [Alon][Alon-Milman] Given a graph G, if the normalized adjacency matrix has second eigen value λ2 then,
•A log n approximation algorithm [Leighton-Rao 98-99?].•A sqrt(log n) approximation algorithm using semidefinite programming [Arora-Rao-Vazirani 2004].
)1(22
)1(2
2
G
Extremely well-studied, many different contexts
pseudo-randomness, group theory, online routing,
Markov chains, metric embeddings, …
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A Reverse Reduction
Graph (Social Network)
Close-knitcommunity
Finding Small Non Expanding Sets
Suppose there exists is a small community say
(0.1% of the population)
99% of whose friends are within the community..
Find one such close-knit community.
Theorem [R-Steurer 10]UGC is false New algorithms to approximate expansion of small sets in graphs
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STILL OPEN:
Reverse reduction from Max Cut or Vertex Cover to Unique Games.
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What if UGC is false?
Could existing algorithmic techniques (LPs/SDPs) disprove the UGC?
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Could LPs/SDPs disprove the UGC?
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Question I:
Could some small LINEAR PROGRAM give a better approximation for MaxCut or Vertex Cover
thereby disproving the UGC?
Probably Not!
[Charikar-Makarychev-Makarychev][Schoenebeck-Tulsiani]
For MaxCut, for several classes of linear programs,
exponential sized linear programs are necessary to even beat the trivial ½ approximation!
Question II:
Could some small SEMIDEFINITE PROGRAM give a better approximation for MaxCut or Vertex Cover
thereby disproving the UGC?
We don’t know.
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v1
v2
v3
v4
v5
Max Cut SDP:
Embedd the graph on the N - dimensional unit ball,
Maximizing
¼ (Average squared length of
the edges)
In the integral solution, all the vectors vi are 1,-1. Thus they satisfy additional constraintsFor example : (vi – vj)2 + (vj – vk)2 ≥ (vi – vk)2
(the triangle inequality)
The Simplest Relaxation for
MaxCut
Does adding triangle inequalities improve approximation ratio?(and thereby disprove UGC!)
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[Arora-Rao-Vazirani 2002]
For SPARSEST CUT, SDP with triangle inequalities gives approximation.
An -approximation would disprove the UGC!
[Goemans-Linial Conjecture 1997] SDP with triangle inequalities would yield -approximation for SPARSEST CUT.
[Khot-Vishnoi 2005]
SDP with triangle inequalities DOES NOT give approximation for SPARSEST CUT
SDP with triangle inequalities DOES NOT beat the Goemans-Williamson 0.878 approximation for MAX CUT
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Until 2009:
Adding a simple constraint on every 5 vectorscould yield a better approximation for MaxCut, and disproves UGC!
Building on the work of [Khot-Vishnoi],
[Khot-Saket 2009][Raghavendra-Steurer 2009]
Adding all valid local constraints on at most vectors to the simple SDP DOES NOT improve the approximation ratio for MaxCut
[Barak-Gopalan-Hastad-Meka-Raghavendra-Steurer 2009]
Change to in the above result.As of Now:
A natural SDP of size (the round of Lasserre hierarchy) could disprove the UGC.
[Barak-Brandao-Harrow-Kelner-Steurer-Zhou 2012] round of Laserre hierarchy solves all known instances of Unique Games.
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Constraint Satisfaction Problems
Max 3 SAT
Find an assignment that satisfies the maximum number of clauses.
))()()(( 145532532321 xxxxxxxxxxxx
VariablesFinite Domain Constraints
{x1 ,x2 , x3 , x4 , x5}{0,1}Clauses
Kind of constraints permitted Different CSPs
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Deeper understanding of the UGC – why it should be true if it is.
Why play this game?
Connections between SDP hierarchies, Spectral Graph Theory and Graph Expansion.
New algorithms based on SDP hierarchies.
[Raghavendra-Tan] Improved approximation for MaxBisection using SDP hierarchies
[Barak-Raghavendra-Steurer]
Algorithms for 2-CSPs on low-rank graphs.
New Gadgets for Hardness Reductions:[Barak-Gopalan-Hastad-Meka-Raghavendra-Steurer]
A more efficient long code gadget.