Vol 1: Basics and Geometry

1
How to get
Hardness of approximation
results from
Integrality gaps
Guy KindlerWeizmann Institute
2
About this talk

• Well try to understand some notions, and their
  relations
• Combinatorial optimization problems
• Approximation relaxation by semi-definite
  programs.
• Integrality gaps
• Hardness of approximation
• Main example the Max-Cut problem

3
Combinatorial optimization problems

• Input, search space, objective function

Example MAX-CUT
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Example MAX-CUT

• input G (V,E)
• Search space Partition V(C, Cc)
• Objective function w(C) fraction
  of cut edges
• The MAX-CUT Problem Find mc(G)maxCw(C)
• Karp 72 MAX-CUT is NP-complete

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Example MAX-CUT

• input G (V,E)
• Search space Partition V(C, Cc)
• Objective function w(C) fraction
  of cut edges
• The MAX-CUT Problem Find mc(G)maxCw(C)
• ?-approximation Output S, s.t. mc(G) S
  ?mc(G).
• History ½-approximation easy, was best record
  for long time.

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Semi-definite Relaxation

• Introducing geometry into combinatorial
  optimization

GW 95
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Arithmetization
v
G(V,E)
u
xv
xu
Problem We cant maximize quadratic
functions,even over convex domains.
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Relaxation by geometric embedding
v
G(V,E)
u
xv
xu
Problem We cant maximize quadratic
functions,even over convex domains.
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Relaxation by geometric embedding
v
G(V,E)
u
xv
xu
Problem We cant maximize quadratic
functions,even over convex domains.
10
Relaxation by geometric embedding
v
G(V,E)
u
xv
xu
Problem We cant maximize quadratic
functions,even over convex domains.
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Relaxation by geometric embedding
Semi-definiterelaxation
v
G(V,E)
u
Now were maximizing a linear function over a
convex domain!
(unit sphere in Rn)
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Relaxation by geometric embedding
Is this really a relaxation?
(unit sphere in Rn)
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Relaxation by geometric embedding
Is this really a relaxation?
(unit sphere in Rn)
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Analysis by randomized rounding
xv
xu
xu
(unit sphere in Rn)
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Analysis by randomized rounding
arccos(ltxu,xvgt)

• So

(unit sphere in Rn)
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Analysis by randomized rounding

• So

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Analysis by randomized rounding

• So

mc(G) ?GWrmc(G) ?GWmc(G)
L.h.s. is tight, iff all inner products are ?
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An 0.879 approximation for Max-Cut
Is ?GW the best constant here?

• The GW 95 algorithm
• Given G, compute rmc(G)
• Let S?GWrmc(G)
• Output S.

mc(G) S ?GWmc(G)
mc(G) ?GWrmc(G) ?GWmc(G)
Is there a graph where this occurs?
L.h.s. is tight, iff all inner products are ?
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Integrality gap

• Measuring the quality of the geometric
  approximation

FS 96
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The integrality gap of Max-Cut
On instance G
w
rmc(G)
mc(G)
r-S(G)
S(G)
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The integrality gap of Max-Cut
?GW
?GW for some G
On instance G
w
rmc(G)
The GW 95 algorithm Given G, output IGrmc(G)
mc(G)
r-S(G)
S(G)
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The integrality gap of Max-Cut
?GW for some G
On instance G
w
rmc(G)
The GW 95 algorithm Given G, output IGrmc(G)
mc(G)
r-S(G)
S(G)

• Using ?rmc(G) to approximate mc(G), the factor
  ?IG cannot be improved!
• On G the algorithm computes mc(G) perfectly!

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The Feige-Schechtman graph, G
arccos(?)
Vertices Sn Edges uv iff ltu,vgt?

• rmc(F)(1-?)/2
• FS mc(G)arccos(?)/?
• so

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From IG to hardness

• A geometric trick may actually be inherent

KKMO 05
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Thoughts about integrality gaps
under some reasonable comlexity theoretic
assumptions

• The integrality gap is an artifact of the
  relaxation.
• The relaxation does great on an instance for
  which the integrality gap is achieved.
• And yet, sometimes the integrality gap is
  provably the best approximation factor
  achievable
• KKMO 04 under UGC, ?GW is optimal for
  max-cut.
• HK 03, HKKSW, KO 06, ABHS 05, AN
  02

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Thoughts about integrality gaps
And the IG instance G is used in the hardness
proof

• The integrality gap is an artifact of the
  relaxation.
• An algorithm does great on an instance for which
  the integrality gap is achieved.
• And yet, sometimes the integrality gap is
  provably the best approximation factor
  achievable
• KKMO 04 Under UGC, ?GW is optimal for
  max-cut.
• HK 03, HKKSW, KO 06, ABHS 05, AN
  02

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A recipe for proving hardness
Take the instance G
w
rmc(G)
mc(G)
r-S(G)
S(G)
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A recipe for proving hardness
Add teeth to S(G)which achieve rmc(G).
w
rmc(G)
mc(G)
r-S(G)
S(G)
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A recipe for proving hardness
Now combine several instances, such that finding
a point which belongs to all teeth becomes a
hard combinatorialoptimization problem.
w
rmc(G)
mc(G)
r-S(G)
S(G)
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A recipe for proving hardness
Now combine several instances, such that finding
a point which belongs to all teeth becomes a
hard combinatorialoptimization problem.
w
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A recipe for proving hardness
Determining whether mc(G)mc(G) or
whethermc(G)rmc(G) is intractable.
Factor of hardness mc(G)/rmc(G)IG !!
w
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Adding teeth to Feige-Schechtman
(unit sphere in Rq)
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Adding teeth to Feige-Schechtman
Vertices Sn Edges uv iff ltu,vgt?
Vertices -1,1n Edges uv iff ltu,vgt?

• a random edge (x,y) x-1,1n,
• Eltx,ygt ?

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Adding teeth to Feige-Schechtman

• mc arccos(?)/? ?
• For C(x) x7,
• w(C) Pedgex7 ? y7 (1-?)/2 !!

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Adding teeth to Feige-Schechtman

• mc arccos(?)/? ?
• For C(x) x7,
• w(C) Pedgex7 ? y7 (1-?)/2 !!
• For C(x) sign(?xi) Maj(x),
• w(C) PMaj(x)?Maj(y) (arccos ?)/p

no influential coordinates
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Adding teeth to Feige-Schechtman

• mc arccos(?)/? ?
• for axis parallel cut w(C)?(1-?)/2
• MOO 05 If ?i, Ii(C)lt?,
• w(C)?? (arccos ?)/p o?(1)
• Ratio between weight of teeth cuts and regular
  cuts is ?GW (for ? ?)

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Conclusion

• We tried to understand some notions, and their
  relations
• Combinatorial optimization problems
• Approximation relaxation by semi-definite
  programs.
• Integrality gaps
• Hardness of approximation