NearOptimal Algorithms for Unique Games

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An algorithmic perspective on Unique Games
Moses Charikar
Joint work with
Konstantin Makarychev
Yury Makarychev
Princeton University
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Example

• Linear equations mod p, two vars per equation.
  Maximize of satisfied constraints.

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Unique Games
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Unique Games
Permutations
k labels
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Unique Games
Goal Satisfy as many constraints as possible.
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2 colors Max Cut
Two colors Red and Blue
Maximize the number of pairs of adjacent
vertices colored with distinct colors
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Greedy Algorithm
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Unique Games Conjecture

• Unique Games Conjecture Khot02
• Given a Unique Games instance where 1-? fraction
  of constraints is satisfiable, it is NP-hard to
  satisfy even ? fraction of all constraints
• (for every constant positive ? and ?
  and sufficiently large k).
• Used to prove (optimal ?) hardness of
  approximation results for several problems
• seem difficult to obtain by standard complexity
  assumptions.

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Hardness Results Assuming UGC
with UGC
without UGC
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Algorithmic Motivation

• Semidefinite programming techniques very useful
  for binary constraint satisfaction problems
• Seems difficult to extend techniques for problems
  over larger domains
• Unique Games is a good test case

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Approximation Algorithms
Assume 1-? fraction is satisfiable, k log n.

• Random Assignment 1/k.
• Andersson, Engebretsen, Hastad 01, slightly
  better than random for lin. eq.
• Khot 02, SDP based algorithm,
• Trevisan 05, SDP based algorithm,
• Gupta, Talwar 06, LP based,

GT
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Comparison
For what ? can we satisfy constant fraction of
constraints ? Assume k log n.
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Our results

• Given an instance where 1-? fraction of
  constraints is satisfiable, 1st algor. satisfies
• The 2nd algorithm satisfies

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Near Optimality

• Khot, Kindler, Mossel and O'Donnell showed that
  even a slight improvement of our results
• refutes the UGC.

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Matching upper and lower bounds ?
g
Gaussian random vector
v
u
u v 1 ? ?
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Roadmap

• SDP based algorithm for unique games
• Approaches to disproving UGC

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Semidefinite Program
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SDP Interpretation
Pr(u,v) constraint not satisfied
Prxu i
Prxu i and xvj
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Intuition
For each vertex, we have an orthogonal system of
vectors. For adjacent vertices the vectors are
close. Our goal is to pick one vector for each
vertex.
Green vectors correspond to vertex u. Red vectors
correspond to vertex v.
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Algorithm first attempt

• Pick value for xu based on ui
• Pick a random Gaussian vector g.
• Project g on ui .
• Pick i with largest projection Zi
• Works in uniform case

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Non-uniform Case

• Long vectors will be chosen with
  disproportionately high probability.
• Normalize all vectors
• To ensure Prxu i ? ui2, project on
  several Gaussians projections is proportional
  to ui2.

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Multiple Projections

• Let be the number of
  projections.
• Pick independent Gaussian random vectors g1, ,
  gk.
• Let
  .
• Pick i with largest projection Zi,r

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Max Cut vs. Unique Games
GVY 93
GT 06
GW 95
CMM 06
CMM 06
ACMM 05
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But UGC is just a conjecture

• UGC seems to predict limitations of SDPs
  correctly
• UGC based hardness for many problems matching
  best SDP based approximation
• UGC inspired constructions of gap examples for
  SDPs
• Disproof of Goemans-Linial conjecturel22 metrics
  do not embed into l1 with constant distortion.
  KV 05

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Is UGC true ?

• Points to limitations of current techniques
• Focuses attention on common hard core of several
  important optimization problems
• Motivates development of new techniques

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Approaches to disproving UGC

• Focus on possibly easier problems
• Max Cut
• OPT 1-?, beat 1-?1/2 GW 94
• Max k-CSP
• constraints are conjunctions of k literals
• maximize satisfied constraints
• Beat k/2k ST 06 CMM 07
• Distinguish between 1/k and 1/2k satisfiable

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Max k-CSP

• Constraints are conjunctions
• Maximize number of constraints satisfied
• Let us denote
• Then

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Integer Relaxation

• Introduce a -1, 1-variable for each
  Boolean variable .
• -1 encodes false, 1 encodes true.
• For each constraint C
• consider the term
• If the constraint is satisfied, then .

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Integer Relaxation
Note this integer program is a relaxation. Can
be solved using algorithm by Nesterov.
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Algorithm

• Solve the integer program, find .
• Let .
• For each i, let
• Flip the values of all with prob. 1/2.
• We show

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Bottleneck

• Objective function cannot distinguish between
  random instances and those where OPT1/k

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Approaches to disproving UGC

• Lifting procedures for SDPs
• Lovasz-Schrijver, Sherali-Adams, Lasserre
• Simulate products of k variables
• Can we use them ?
• Relaxation for k-CSP using sum of quartic terms ?
• Max Cut
• OPT 1-?, beat 1-?1/2 using f(1/?) rounds of
  lifting ?

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Moment matrices

• SDP solution gives covariance matrix M
• There exist normal random variables with
  covariances Mij
• Basis for SDP rounding algorithms
• There exist 1,-1 random variables with
  covariances Mij/log n
• Is something similar possible for higher order
  moment matrices ?

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Conclusion and Future Goals

• Precise tradeoff of parameters in Unique Games
  Conjecture
• Algorithms require geometric insights Is
  geometry intrinsic ?
• Prove or disprove the Unique Games Conjecture