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
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
•