1. CSP
(STOC 2011, to appear)
( )

2. Max CSP

3. CSP ( )
•
• : SAT
: (¬x1 ∨ x2 ∨ x3), (x1 ∨ ¬x2), (x1 ∨ ¬x3)
: Yes ( β = (1,1,0))
• : k , Fq
• NP

4. Max CSP
•
• : Max Cut
I: (v1 ⊕ v2), (v1 ⊕ v3), (v1 ⊕ v4), (v2 ⊕ v3), (v2 ⊕ v4)
v1 v2 β = (1,1,0,0)
opt(I) = val(I, β) = 4 / 5
v3 v4 ( )
• NP

5. CSP Λ
• CSP Λ = ([q], s, t, P)
[q] = {1,...,q}:
s=
t=
P=
• Max SAT = ({0, 1}, ∞, ∞, {∨})
• Max Cut = ({0,1}, 2, ∞, {⊕})

6. CSP Λ I
• CSP Λ I = (V, P)
V:
P:
• Max Cut I = (V, P)
v1 v2
V = {v1,v2,v3,v4}
P = {(v1⊕v2), (v1⊕v3),
v3 v4
(v1⊕v4), (v2⊕v3), (v2⊕v4)}

7. Max CSP
• Max CSP NP
• x x* α : αx* ≦ x ≦ x*
•β I α :
val(I, β) opt(I) α
• CSP Λ α :
I α β

8. Max CSP
• Random Assignment
• SDP
• PCP/Unique Games Conjecture
CSP RA SDP UGC-Hard
Max SAT 0.5 0.787[ABZ06] ?
Max Cut 0.5 0.878[GW95] 0.878+ε[KKMO04]
Max Dicut 0.25 0.874[LLZ06] ?
Max k-CSP 1/2k poly(k)/2k[CMM09] poly(k)/2k[ST06]

9. Max CSP
• [Rag08] (informal) CSP Λ UGC
”BasicSDP”+
UGC-Hard
CSP RA SDP UGC-Hard
Max SAT 0.5
Max Cut 0.5 0.878 0.878+ε
Max Dicut 0.25
Max k-CSP 1/2k

10. Max CSP

11. Max CSP
•
(o(n) )
• x x* (α, ε) : αx* - ε ≦ x ≦ x*
•β I (α,ε) :
val(I, β) opt(I) (α, ε) (0 ≦ opt(I) ≦ 1 )
• CSP Λ (α, ε) :
I 2/3 (α,ε)
β
v ( 1 )βv

12. Max CSP
• Ω(n)
• CSP( )
1. Dense model
2. Bounded-degree model ( )
•
(Dense model CSP (1,ε) )

13. • t
• I = (V, P) OI: V × [t] → P
OI(v, i) = v i
• OI (
)

14. t=3 v1 v2
V = {v1,v2,v3,v4}
P = {(v1 ⊕ v2), (v1 ⊕ v3), v3 v4
(v1 ⊕ v4), (v2 ⊕ v3), (v2 ⊕ v4)}
OI(v1, 1) = (v1⊕v3)
OI(v3, 2) = (v2⊕v3)
OI(v3, 3) = ⊥

15. (α, ε)
[Rag08]

16. • (informal) CSP Λ ”BasicLP”+
CSP RA O(1) (LP) Ω(√n)
Max SAT 0.5 0.75-ε 0.75+ε
Max Cut 0.5 0.5 0.5+ε
Max Dicut 0.25 0.5-ε 0.5+ε
Max k-CSP 1/2k 2/2k-ε 2/2k+ε

17. Integrality Gap
• lp(I): I BasicLP
opt(I)
• Integrality Gap: αΛ = inf
I∈Λ lp(I)
[ ] CSP Λ
ε (αΛ-ε, ε)
ε δ (αΛ+ε, δ)
Ω(√n)
Θ(n)
: exp(exp(poly(qst/ε)))

18. • lp(I) = c
1. opt(I) ≦ lp(I) = c
2. opt(I) ≧ αΛlp(I) = αΛc
➡ opt(I) = c, opt(J) < αΛc I, J
BasicLP
• (αΛ-ε, ε)
➡ BasicLP
LP

19. • lp(I) = c
1. opt(I) ≦ lp(I) = c
2. opt(I) ≧ αΛlp(I) = αΛc
➡ opt(I) = c, opt(J) > αΛc I, J
BasicLP
• (αΛ+ε,δ)
➡ BasicLP (SDP )

20. • Dense Model
• [AE02, AdlVKK03]
CSP (1,ε)
• Bounded-degree model
• Max E3LIN2 (1/2+ε,δ) Ω(n)
[BOT02]
• Max Cut (1/2+ε,δ) Ω(√n) [GR08]

21. [Rag08]
[Rag08]
CSP CSP
BasicLP BasicSDP
Unique Games Conjecture
Unconditional

22. •
• I ε-far: I
εtn ( ε )
• CSP Λ : CSP Λ
ε-far 2/3
• CSP Λ
CSP Λ I lp(I) = 1⇒ opt(I) = 1
(integrality gap curve c = 1 )

23. ( )
(αΛ-ε, ε)

24. •
1. BasicLP Packing LP
2. Packing LP LP solver
3. LP
•
• Max Cut

25. BasicLP for Max Cut
• IP e
u v
• xv,i: v i∈{0,1}
µe,β: e β∈{0,1}2
max Σewe(µe,01 + µe,10)
s.t. xv,0 + xv,1 = 1 ∀v
µe,00 + µe,01 = xv,0 ∀ e = (v, u)
µe,10 + µe,11 = xv,1 ∀ e = (v, u)
xv,i ∈ {0,1} ∀ v, i
µe,β ∈ {0,1}2 ∀ e, β

26. BasicLP for Max Cut
• LP e
u v
• xv: v
µe: e
max Σewe(µe,01 + µe,10)
s.t. xv,0 + xv,1 = 1 ∀v
µe,00 + µe,01 = xv,0 ∀ e = (v, u)
µe,10 + µe,11 = xv,1 ∀ e = (v, u)
xv,i ≧ 0 ∀ v, i
µe,β ≧ 0 ∀ e, β

27. Basic LP Packing LP
• Packing LP: LP
max cTx
s.t. Ax ≦ b
A, b, c ≧ 0
•

28. BasicLP
• LP
• xv,i ε xεv,i
• (xεu,0, xεu,0)=(xεv,0, xεv,0) u,v
G G’
(0.41,0.59) (0.39,0.61) ε = 0.1
(0.4,0.6)
(0.22,0.78) (0.81,0.19) (0.2,0.8) (0.8,0.2)

29. BasicLP
• lp(G’) ≈ lp(G)
• G’ (1/ε)2 β
• β G
G’ G
(0.41,0.59) (0.39,0.61)
(0.4,0.6) ε = 0.1
(0.2,0.8) (0.8,0.2) (0.22,0.78) (0.81,0.19)

30. BasicLP
opt(I)
αΛ = inf
I∈Λ lp(I)
val(G,β) = val(G’,β) = opt(G’,β) ≧ αΛlp(G’) ≈ αΛlp(G)

31. • Packing LP ( ) x* = {x*v,i}v∈V,i∈{0,1}
• [KMW06] x* Olp
Olp(v,i) = x*v,i
• G’ β β Olp
val(G, β)

32. ( )
(αΛ+ε, δ) Ω(√n)

33. Yao’s minimax principle
•
• DN: opt(J) ≦ (αΛc+ε) J
• DY: opt(J) ≧ c J
• D: DY DN J
•D J DY,DN (
) 2/3
Ω(√n)

34. • lp(I) = c, opt(I) ≈ αΛc I ( c,I
)
•I
• DIopt: opt(J) ≦ αΛc+ε J 1-o(1)
• DIlp: opt(J) ≧ c J 1
I
opt(I) = 2 / 3, lp(I) = 1
e µe,00=µe,11=0
u v
µe,01=µe,10=1/2

35. DIopt
•I I
•J I ( )
• 1-o(1) opt(J) ≦ αΛc+ε
J
I
e
u v

36. DIlp
•I µ
• I LP J
• opt(J) ≧ c
J
I
e
u
µe,00=µe,11=0
µe,01=µe,10=1/2

37. • DIopt DIlp Ω(√n)
• o(√n)
1-o(1) ( )
•( )

38. • Ω(√n)
• Sherali-Adams
• CSP Λ
• Horn Sat: Θ(1)
• 2-Colorability: Θ*(√n)
• : Θ(n)