tst rs - - PowerPoint PPT Presentation

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✶

❙❛t✐s✜❛❜✐❧✐t② ♦❢ ❇♦♦❧❡❛♥ ❋♦r♠✉❧❛s

❈♦♠❜✐♥❛t♦r✐❝s ❛♥❞ ❆❧❣♦r✐t❤♠s

Pr♦❢✳ ❊♠♦ ❲❡❧③❧ ❆ss✐st❛♥t✿ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐ ✭❈❆❇ ●✸✻✳✶✱ ❝❛♥♥❛♠❛❧❛✐❅✐♥❢✳❡t❤③✳❝❤✮ ❯❘▲✿ ❤tt♣✿✴✴✇✇✇✳t✐✳✐♥❢✳❡t❤③✳❝❤✴❡✇✴❝♦✉rs❡s✴❙❆❚✶✻✴ ❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✷

❚♦❞❛②✬s ❡①❡r❝✐s❡s

• ✶✳✸✿ ❆ss♦❝✐❛t✐✈✐t② ♦❢ ❜♦♦❧❡❛♥ ♦♣❡r❛t♦rs
• ✶✳✶✷✿ ❋♦✉r ❈❧❛✉s❡s ❙✉✣❝❡
• ✶✳✶✺✿ ❉✐s❥✉♥❝t✐✈❡ ◆♦r♠❛❧ ❋♦r♠
• ✶✳✶✽✿ ❈♦✉♥t✐♥❣ ❋♦r♠✉❧❛s
• ✶✳✷✺✿ ◆✉♠❜❡r ♦❢ ❱❡rt❡① ❈♦✈❡rs
• ✶✳✷✽✿ ❍♦r♥ ❋♦r♠✉❧❛s
• ❆❞❞❡♥❞✉♠ t♦ ❡①❡r❝✐s❡ ✶✳✷✺✿ ■♠♣r♦✈❡❞ (≤ 2)✲❈◆❋ ❝♦✉♥t✐♥❣

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✸

✶✳✸✳ ❆ss♦❝✐❛t✐✈✐t② ♦❢ ❜♦♦❧❡❛♥ ♦♣❡r❛t♦rs

❉❡✜♥✐t✐♦♥✳ ❆♥ ♦♣❡r❛t♦r ◦ ✐s ❛ss♦❝✐❛t✐✈❡ ✐❢ ❢♦r ❛❧❧ ♦♣❡r❛♥❞s a, b, c ✇❡ ❤❛✈❡ (a ◦ (b ◦ c)) = ((a ◦ b) ◦ c)✳ ■❢ ❛♥ ♦♣❡r❛t♦r ✐s ❛ss♦❝✐❛t✐✈❡✱ t❤❡♥ ✇❡ ❝❛♥ ✇r✐t❡ ❡①♣r❡ss✐♦♥s ✇✐t❤♦✉t ♣❛r❡♥t❤❡s❡s ❜❡❝❛✉s❡ ♦❢ t❤❡ ❧❛❝❦ ♦❢ ❛♠❜✐❣✉✐t②✳ ❲❡ s❛✇ ❛ s❤♦rt ♣r♦♦❢ ♦❢ t❤✐s ✐♥ t❤❡ ❡①❡r❝✐s❡ s❡ss✐♦♥✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ♦♥❧② t❤❡ ♦r❞❡r ♦❢ t❤❡ ♦♣❡r❛♥❞s ♠❛tt❡r ✭❛♥❞ ♥♦t t❤❡ ♦r❞❡r ♦❢ ❡✈❛❧✉❛t✐♦♥✮✳ ❲❡ ❝❛♥ t❤❡♥ ✐♥t❡r♣r❡t ❛♥ ❡①♣r❡ss✐♦♥ ✐♥✈♦❧✈✐♥❣ n ♦♣❡r❛♥❞s✱ x1 ◦ . . . ◦ xn ❛s ❛♥ n✲✈❛r✐❛t❡ ❢✉♥❝t✐♦♥✳ ❈❤❡❝❦ ✇❤✐❝❤ ♦❢ {∧, ∨, →, ↔, ⊕} ❛r❡ ❛ss♦❝✐❛t✐✈❡ ❛♥❞ ❝❤❛r❛❝t❡r✐③❡ t❤❡ n✲✈❛r✐❛t❡ ❢✉♥❝t✐♦♥s t❤❡② ✐♥❞✉❝❡✳

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✹

✶✳✸✳ ❆ss♦❝✐❛t✐✈✐t② ♦❢ ❜♦♦❧❡❛♥ ♦♣❡r❛t♦rs ✭✷✮

❇② str❛✐❣❤t❢♦r✇❛r❞ ❝❤❡❝❦✐♥❣✿ ❆ss♦❝✐❛t✐✈❡✿ ∧, ∨, ↔, ⊕ ◆♦t ❛ss♦❝✐❛t✐✈❡✿ →✳ → ✐s ♥♦t ❛ss♦❝✐❛t✐✈❡ ❜❡❝❛✉s❡ x → (y → z) ≡ (x → y) → z ✇❤❡♥ (x → 0, y → 1, z → 0).

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✺

✶✳✸✳ ❆ss♦❝✐❛t✐✈✐t② ♦❢ ❜♦♦❧❡❛♥ ♦♣❡r❛t♦rs ✭✸✮

❈❤❛r❛❝t❡r✐③❛t✐♦♥✿

• n

i=1 ✐s tr✉❡ ✐✛ ❛❧❧ ♦♣❡r❛♥❞s ❛r❡ tr✉❡✳

• n

i=1 ✐s tr✉❡ ✐✛ ❛t ❧❡❛st ♦♥❡ ♦♣❡r❛♥❞ ✐s tr✉❡✳

• n

i=1 ✐s tr✉❡ ✐✛ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ ♦♣❡r❛♥❞s ✐s tr✉❡ ✭t❤✐s ✐s ❝❛❧❧❡❞

t❤❡ ♣❛r✐t② ❢✉♥❝t✐♦♥✮✳

• ↔n

i=1✿ ❲❤❡♥ n ✐s ❡✈❡♥✱ ✐t ✐s tr✉❡ ✐✛ ❛♥ ❡✈❡♥ ♥✉♠❜❡r ♦❢ ♦♣❡r❛♥❞s

❛r❡ tr✉❡✱ ❛♥❞ ✇❤❡♥ n ✐s ♦❞❞✱ ✐t ✐s tr✉❡ ✐✛ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ ♦♣❡r❛♥❞s ❛r❡ tr✉❡✳ ◆♦t❡ t❤❛t t❤✐s ✐s t❤❡ s❛♠❡ ❛s s❛②✐♥❣ t❤❛t t❤❡ ❡①♣r❡ss✐♦♥ ✐s tr✉❡ ✐✛ ❛♥ ❡✈❡♥ ♥✉♠❜❡r ♦❢ ♦♣❡r❛♥❞s ✐s ❢❛❧s❡✳ ❚❤❡ ♣r♦♦❢s ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ❢♦❧❧♦✇ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ n✳

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✻

✶✳✶✷✳ ❋♦✉r ❈❧❛✉s❡s ❙✉✣❝❡

❆s ✐♥ ❙❡❝t✐♦♥ ✶✳✷✱ ✇❡ ❝❛♥ r❡♣r❡s❡♥t ❡✈❡r② ❜♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ❛s ❛ ❈◆❋ ❢♦r♠✉❧❛✳ ❖✈❡r t❤r❡❡ ✈❛r✐❛❜❧❡s✱ t❤✐s ❛✉t♦♠❛t✐❝❛❧❧② ❜❡❝♦♠❡s ❛ 3✲❈◆❋ ❢♦r♠✉❧❛✱ s❛② F✳ F ❝♦✉❧❞ ♣♦t❡♥t✐❛❧❧② ❝♦♥t❛✐♥ 8 ♣♦ss✐❜❧❡ ❝❧❛✉s❡s✳ ▲❡t ✉s ❡♥✉♠❡r❛t❡ t❤❡♠ ❢r♦♠ C1 t❤r♦✉❣❤ C8 ✐♥ ♦r❞❡r ❛♥❞ ♣❛✐r ❝♦♥s❡❝✉t✐✈❡ ❝❧❛✉s❡s✿ {C1, C2}, . . . , {C7, C8}✳ ■❢ F ❝♦♥t❛✐♥s ❜♦t❤ ❝❧❛✉s❡s ❢r♦♠ s♦♠❡ ♣❛✐r t❤❡♥ r❡s♦❧✈❡ t❤❡ t✇♦ ❝❧❛✉s❡s ✉s✐♥❣ t❤❡ ❧❛st ✈❛r✐❛❜❧❡ ✭s✐♥❝❡ ✇❡ ❛r❡ ❣✉❛r❛♥t❡❡❞ t❤❛t t❤✐s ✈❛r✐❛❜❧❡ ❛♣♣❡❛rs ♣♦s✐t✐✈❡❧② ♦♥❝❡ ❛♥❞ ♥❡❣❛t✐✈❡❧② ♦♥❝❡ ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❝❧❛✉s❡s ✐♥

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✼

t❤❡ ♣❛✐r✱ ❜② ♦✉r ❡♥✉♠❡r❛t✐♦♥✮ t♦ ♦❜t❛✐♥ ❛♥ ❡q✉✐✈❛❧❡♥t ❝❧❛✉s❡ ✇✐t❤ ♦♥❧② t✇♦ ❧✐t❡r❛❧s✳ ❚❤❡r❡❢♦r❡✱ ❢♦r ❡❛❝❤ ♣❛✐r ✇❡ r❡t❛✐♥ ❛t ♠♦st ♦♥❡ ❝❧❛✉s❡ ✇✐t❤ ❡✐t❤❡r 2 ♦r 3 ❧✐t❡r❛❧s✱ ❣✐✈✐♥❣ ✉s ❛ (≤ 3)✲❈◆❋ ❢♦r♠✉❧❛ ✇✐t❤ ❛t ♠♦st 4 ❝❧❛✉s❡s✳ ✷

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✽

✶✳✶✺✳ ❉✐s❥✉♥❝t✐✈❡ ◆♦r♠❛❧ ❋♦r♠

✭✶✮ ▲❡t φ(x1, . . . , xn) ❜❡ t❤❡ ❣✐✈❡♥ ❜♦♦❧❡❛♥ ❢♦r♠✉❧❛ ♦♥ n ✈❛r✐❛❜❧❡s✳ ■♠❛❣✐♥❡ t❤❡ tr✉t❤ t❛❜❧❡ ♦❢ φ✳ ❲❡ s❤♦✇ ♦♥❡ r♦✇ ♦❢ t❤❡ tr✉t❤ t❛❜❧❡ ❤❡r❡ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❉◆❋ ❝❧❛✉s❡✳ x1 x2 . . . xn φ(x1, . . . , xn) ✳ ✳ ✳ 1 . . . 1 = ⇒ (¯ x1 ∧ x2 ∧ · · · ∧ ¯ xn) ✳ ✳ ✳

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✾

✭✷✮ ❋♦r ❡❛❝❤ ❛ss✐❣♥♠❡♥t α := (x1 → t1, . . . , xn → tn) t❤❛t r❡s✉❧ts ✐♥ φ(α) ❡✈❛❧✉❛t✐♥❣ t♦ 1 ✇❡ ♠❛❦❡ ♦♥❡ ❉◆❋✲❝❧❛✉s❡

  

• xi : α(xi)=1

xi ∧

• xi : α(xi)=0

xi

   ,

✇❤✐❝❤ ❤❛s t❤❡ ♣r♦♣❡rt② t❤❛t ✐t ❡✈❛❧✉❛t❡s t♦ 1 ❡①❛❝t❧② ✇❤❡♥ xi = ti ❢♦r ❛❧❧ i ∈ {1, . . . , n}✳ ✭✸✮ ❚❛❦✐♥❣ t❤❡ ❖❘ ♦❢ ❛❧❧ s✉❝❤ ❝❧❛✉s❡s ✇❡ ♦❜t❛✐♥ ❛ ❜♦♦❧❡❛♥ ❢♦r♠✉❧❛ ❡q✉✐✈❛❧❡♥t t♦ φ ✐♥ ❞✐s❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✿

• α∈{0,1}n : φ(α)=1

  

• xi : α(xi)=1

xi ∧

• xi : α(xi)=0

xi

   .

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✶✵

✶✳✶✽✳ ❈♦✉♥t✐♥❣ ❋♦r♠✉❧❛s

✭✶✮ ❚❤❡ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ✐ts tr✉t❤ t❛❜❧❡ ❛ss♦❝✐❛t✐♥❣ ❛ ✈❛❧✉❡ ❢r♦♠ {0, 1} t♦ ❡❛❝❤ t✉♣❧❡ ✐♥ {0, 1}n✳ ❚❤✐s ②✐❡❧❞s 22n ♣♦ss✐❜✐❧✐t✐❡s✳ ✭✷✮ ❚♦ ❜✉✐❧❞ ❛ k✲❝❧❛✉s❡ ✇❡ ✜rst ♥❡❡❞ t♦ ❝❤♦♦s❡ t❤❡ k ✈❛r✐❛❜❧❡s ✇❤✐❝❤ ❛♣♣❡❛r ✐♥ t❤❡ ❝❧❛✉s❡✱ ❛♥❞ ❢♦r ❡❛❝❤ ❝❤♦s❡♥ ✈❛r✐❛❜❧❡ ✇❡ ♥❡❡❞ t♦ ❞❡❝✐❞❡ ✐❢ ✐t ✇✐❧❧ ❜❡ ♥❡❣❛t❡❞ ♦r ♥♦t✳ ❚❤❡r❡❢♦r❡ t❤❡r❡ ❛r❡

n

k

• · 2k ♠❛♥② ❞✐st✐♥❝t

k✲❝❧❛✉s❡s✳ ❚♦ ❜✉✐❧❞ ❛ k✲❈◆❋ ❢♦r♠✉❧❛✱ ✇❡ ❤❛✈❡ t♦ s❡❧❡❝t ❛ s✉❜s❡t ♦❢ t❤❡♠✳ ❲❡ ❤❡♥❝❡ ❤❛✈❡ 2(n

k)·2k ♣♦ss✐❜✐❧✐t✐❡s✳

✭✸✮ ❙✐♥❝❡ t❤❡r❡ ❛r❡ 2210 = 21024 s✉❝❤ ❢✉♥❝t✐♦♥s ❛♥❞ ♦♥❧② 2(10

3 )·23 =

2960 ❢♦r♠✉❧❛s✱ t❤❡r❡ ❛r❡ ♠❛♥② ❜♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ♦✈❡r 10 ✈❛r✐❛❜❧❡s t❤❛t ❝❛♥♥♦t ❜❡ r❡♣r❡s❡♥t❡❞ ❛s 3✲❈◆❋ ❢♦r♠✉❧❛❡✳

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✶✶

◗✉❡st✐♦♥✿ ❈❛♥ ②♦✉ ❡①♣❧❛✐♥ ✇❤② ❛ 3✲❈◆❋ ❝❛♥♥♦t ❡①♣r❡ss ❡✈❡r②t❤✐♥❣❄ ❉♦❡s t❤✐s ❝♦♥tr❛❞✐❝t ✇❤❛t ✇❡ ❞✐❞ ♦♥ t❤❡ ✜rst ❞❛② ✭✏❡✈❡r②t❤✐♥❣ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ 3✲❈◆❋✑✮❄

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✶✷

✶✳✷✺✳ ◆✉♠❜❡r ♦❢ ❱❡rt❡① ❈♦✈❡rs

• ♦❛❧✿ ❉❡s✐❣♥ ❛♥❞ ❛♥❛❧②③❡ ❛♥❞ ❛❧❣♦r✐t❤♠ ❢♦r ❝♦✉♥t✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢

✈❡rt❡① ❝♦✈❡rs✳ ❚r✐✈✐❛❧ ❛❧❣♦r✐t❤♠✿ O(2npoly(n))❀ t❡st ❢♦r ❡❛❝❤ ♦❢ t❤❡ 2n s✉❜s❡ts ♦❢ V ✐❢ ✐t ✐s ❛ ✈❡rt❡① ❝♦✈❡r✳ ■❞❡❛✿ ❊♥❝♦❞❡ s✉❜s❡ts N ♦❢ V ❛s ❛ss✐❣♥♠❡♥ts✱ ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t N ✐s ❛ ✈❡rt❡① ❝♦✈❡r ❛s ❛ 2✲❈◆❋ ❢♦r♠✉❧❛✳

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✶✸

❚❤❡ ❡♥❝♦❞✐♥❣ ✐s str❛✐❣❤t✲❢♦r✇❛r❞✿ ❊❛❝❤ ✈❡rt❡① v ❜❡❝♦♠❡s ❛ ✈❛r✐❛❜❧❡✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❛ss✐❣♥♠❡♥t αN ✐s ❞❡✜♥❡❞ ❛s αN(v) :=

  

0, v ∈ N 1, v ∈ N . ❋♦r ❡❛❝❤ ❡❞❣❡ e = {v1, v2}✱ ✇❡ ✇❛♥t t❤❛t ❛t ❧❡❛st ♦♥❡ ✈❡rt❡① ✐s ♣r❡s❡♥t✳ ❚❤✐s ✐s ❡♥❝♦❞❡❞ ❜② t❤❡ ❝❧❛✉s❡ {v1, v2}✳ ❚❤❡r❡❢♦r❡ F := E ❛♥❞ ❛ s✉❜s❡t N ✐s ❛ ✈❡rt❡① ❝♦✈❡r ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❛ss✐❣♥♠❡♥t αN s❛t✐s✜❡s F✳ ❆♥ ❛ss✐❣♥♠❡♥t ♦♥ V ❡♥❝♦❞❡s ❡①❛❝t❧② ♦♥❡ s✉❜s❡t N ♦❢ V ❀ t❤❡r❡❢♦r❡ t❤❡ ♥✉♠❜❡r ♦❢ s❛t✐s❢②✐♥❣ ❛ss✐❣♥♠❡♥ts ✐s t❤❡ ♥✉♠❜❡r ♦❢ ✈❡rt❡① ❝♦✈❡rs✳

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✶✹

✶✳✷✽✳ ❍♦r♥ ❢♦r♠✉❧❛s

❈♦rr❡❝t♥❡ss✳ ❚r✐✈✐❛❧✳ ❲❡ ❛r❡ ✉s✐♥❣ ♦♥❧② ♠♦r❡ r❡str✐❝t✐✈❡ r✉❧❡s✳ ❈♦♠♣❧❡t❡♥❡ss✳ ▲❡t F ❜❡ ❍♦r♥ ❛♥❞ ❧❡t F ′ ❜❡ t❤❡ ❡①t❡♥s✐♦♥ ♦❢ F ❜② ❛❧❧ ♣♦ss✐❜❧❡ ✭✉♥✐t r❡s✳✲✮r❡s♦❧✈❡♥ts✱ s✳t✳ ♥♦ ♣❛✐r C, D ∈ F ′ ❝❛♥ ②✐❡❧❞ ❛ ✭✉♥✐t r❡s✳✲✮r❡s♦❧✈❡♥t ♥♦t ②❡t ❝♦♥t❛✐♥❡❞ ✐♥ F ′✳ ◆♦t❡ t❤❛t F ′ ✐s ❍♦r♥ ❜❡❝❛✉s❡ t✇♦ ❝❧❛✉s❡s ✇✐t❤ ❛t ♠♦st ♦♥❡ ♣♦s✐t✐✈❡ ❧✐t❡r❛❧ ❡❛❝❤ ❝♦♥t❛✐♥ ❛❣❛✐♥ ❛t ♠♦st ♦♥❡ ♣♦s✐t✐✈❡ ❧✐t❡r❛❧✳

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✶✺

✶✳✷✽✳ ❍♦r♥ ❢♦r♠✉❧❛s ✭✷✮

❍❡♥❝❡ ✐❢ ✷ ∈ F ′✱ t❤❡♥ ❛ s❛t✐s❢②✐♥❣ ❛ss✐❣♥♠❡♥t α ✐s ❣✐✈❡♥ ❜② s❛t✐s❢②✐♥❣ ❛❧❧ 1✲❝❧❛✉s❡s ❛♥❞ s❡tt✐♥❣ t❤❡ r❡♠❛✐♥✐♥❣ ✈❛r✐❛❜❧❡s t♦ 0✳ ❆ ❝♦♥tr❛❞✐❝t♦r② ♣❛✐r {u} ❛♥❞ {¯ u} ❝❛♥♥♦t ❡①✐st s✐♥❝❡ t❤✐s ✇♦✉❧❞ ❤❛✈❡ ❛❧❧♦✇❡❞ t♦ ❞❡r✐✈❡ ✷✳ ❙✉♣♣♦s❡ α ❞♦❡s ♥♦t s❛t✐s❢② F✱ t❤❡♥ ✇❡ ✇✐❧❧ s❤♦✇ t❤✐s ❧❡❛❞s t♦ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ▲❡t C ❜❡ ❛ ✉♥s❛t✐s✜❡❞ ❝❧❛✉s❡ ♦❢ F ✇✐t❤ ❢❡✇❡st ❧✐t❡r❛❧s✳ C ❝♦♥t❛✐♥s ❛t ❧❡❛st 2 ❧✐t❡r❛❧s ✭❛s ✇❡ s❛t✐s✜❡❞ ❛❧❧ 1✲❝❧❛✉s❡s✮✱ ❛♥❞ s♦ ♦♥❡ ❧✐t❡r❛❧ ♠✉st ❜❡ ♥❡❣❛t✐✈❡ ¯ x✳ ❚❤✐s ❧✐t❡r❛❧ ✐s s❛t✐s✜❡❞ ✉♥❧❡ss {x} ∈ F ′✳ ❍♦✇❡✈❡r ✐❢ {x} ∈ F ′ ✇❡ ❝❛♥ ❞♦ ✉♥✐t r❡s♦❧✉t✐♦♥ ✇✐t❤ C ❛♥❞ {x} t♦ ♦❜t❛✐♥ ❛ s♠❛❧❧❡r ✉♥s❛t✐s✜❡❞ ❝❧❛✉s❡✱ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✶✻

❆❞❞❡♥❞✉♠ t♦ ❡①❡r❝✐s❡ ✶✳✷✺✿ ■♠♣r♦✈❡❞ (≤ 2)✲❈◆❋ ❝♦✉♥t✐♥❣

❚❤❡ ❜❡st ❦♥♦✇♥ (≤ 2)✲❈◆❋ ❝♦✉♥t✐♥❣ ❛❧❣♦r✐t❤♠ ✐s ❜② ❲❛❤❧strö♠ ❛♥❞ r✉♥s ✐♥ t✐♠❡ O(1.2377n) ✉s✐♥❣ ❛ ✈❡r② r❡✜♥❡❞ ❛♥❛❧②s✐s✳ ❘❡♠❡♠❜❡r ♦✉r ❛❧❣♦r✐t❤♠ ❢✐❜ ❝✷s✭F✱V ✮✿

• ■❢ F ✐s ❛ (≤ 1)✲❈◆❋✱ t❤❡ ♣r♦❜❧❡♠ ✐s ❡❛s②✳
• ❖t❤❡r✇✐s❡ ❧❡t {u, v} ❜❡ s♦♠❡ 2✲❝❧❛✉s❡✳

❘❡t✉r♥ ❢✐❜ ❝✷s✭F [u→1]✮+❢✐❜ ❝✷s✭F [u→0,v→1]✮✳ ❍♦✇ t♦ ✐♠♣r♦✈❡ t❤✐s❄

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✶✼

■♠♣r♦✈❡❞ (≤ 2)✲❈◆❋ ❝♦✉♥t✐♥❣ ✭✷✮

❲❡ ✐♥tr♦❞✉❝❡ ❛♥ ❛❞❞✐t✐♦♥❛❧ ❝❛s❡ ❞✐st✐♥❝t✐♦♥ t♦ ❜❡tt❡r ❡①♣❧♦✐t t❤❡ str✉❝✲ t✉r❡✳ ❲❡ ✜rst ❛❞❞r❡ss tr✐✈✐❛❧ ❝❛s❡s✿

• ■❢ F ❝♦♥t❛✐♥s t❤❡ 0✲❝❧❛✉s❡ ✷✱ t❤❡♥ |satV (F)| = 0✳
• ■❢ F = {}✱ t❤❡♥ |satV (F)| = 2|V |✳
• ■❢ F ❝♦♥t❛✐♥s ❛ 1✲❝❧❛✉s❡ {u}✱ t❤❡♥

|satV (F)| = |satV \vbl{u}(F [u→1])|. ❍❡♥❝❡ ✇❡ ❝❛♥ ❛ss✉♠❡ t❤❛t F ✐s ❛ ♥♦♥✲❡♠♣t② 2✲❈◆❋ ❢♦r♠✉❧❛✳

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✶✽

■♠♣r♦✈❡❞ (≤ 2)✲❈◆❋ ❝♦✉♥t✐♥❣ ✭✸✮

▲❡t C = {u, v} ❜❡ s♦♠❡ 2✲❝❧❛✉s❡✳ ■❢ t❤❡ ✈❛r✐❛❜❧❡s ♦❢ u ❛♥❞ v ❞♦ ♥♦t ♦❝❝✉r ❛♥②✇❤❡r❡ ❡❧s❡ ✐♥ t❤❡ ❢♦r♠✉❧❛✱ t❤❡♥ |satV (F)| = 3 · |satV \vbl{u,v}(F \ {{u, v}})|. ❖t❤❡r✇✐s❡ t❤❡r❡ ✐s D ∈ F ✇✐t❤ vbl(C) ∩ vbl(D) = ∅✳ ❚❤❡r❡ ❛r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✿ ✶✳ vbl(C) = vbl(D) ✭t❤❡ ❝❧❛✉s❡s s❤❛r❡ ❜♦t❤ ✈❛r✐❛❜❧❡s✮ ✷✳ |C ∩ D| = 1 ✭t❤❡ ❝❧❛✉s❡s ❤❛✈❡ ❝♦♠♣❧❡♠❡♥t❛r② ❧✐t❡r❛❧s✮ ✸✳ |C ∩ D| = 1 ✭t❤❡ ❝❧❛✉s❡s s❤❛r❡ ❛ ❧✐t❡r❛❧✮

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ✶ ▼❛r❝❤✱ ✷✵✶✻ s❧✐❞❡ ✶✾

■♠♣r♦✈❡❞ (≤ 2)✲❈◆❋ ❝♦✉♥t✐♥❣ ✭✹✮

✶✳ vbl(C) = vbl(D)✿ ❲❡ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r ✷ ♦✉t ♦❢ ✹ ♣♦ss✐❜✐❧✐t✐❡s ♦❢ ❢♦r♠✉❧❛s ✇✐t❤ ✷ ❢❡✇❡r ✈❛r✐❛❜❧❡s✳ ✷✳ |C ∩ D| = 1 ✭t❤❡ ❝❧❛✉s❡s ❤❛✈❡ ❝♦♠♣❧❡♠❡♥t❛r② ❧✐t❡r❛❧s✮✳ C = {u, v}✱ D = {¯ u, w}✳ ❊✐t❤❡r s❡t u t♦ 1 ❛♥❞ w t♦ 1✱ ♦r s❡t u t♦ 0 ❛♥❞ v t♦ 1✳ ✸✳ |C ∩ D| = 1 ✭t❤❡ ❝❧❛✉s❡s s❤❛r❡ ❛ ❧✐t❡r❛❧✮✳ C = {u, v}✱ C = {u, w}✳ ❊✐t❤❡r s❡t u t♦ 1 ♦r s❡t u t♦ 0 ❛♥❞ v, w t♦ 1✳ ❚❤❡ ✜rst t✇♦ ❝❛s❡s ❣✐✈❡ r✉♥♥✐♥❣ t✐♠❡ O( √ 2npoly(n)) ≈ O(1.415n)✳ ❚❤❡ ❧❛st ❝❛s❡ ✐s t❤❡ ✇♦rst ❝❛s❡ ❛♥❞ ❣✐✈❡s r✉♥♥✐♥❣ t✐♠❡ O(1.466n)✱ ❛ ❧❛r❣❡ ✐♠♣r♦✈❡♠❡♥t ♦✈❡r O(1.619n)✳ ✭s❡❡ ❆♣♣❡♥❞✐① ❆✳✷ ❢♦r ♠❡t❤♦❞s t♦ s♦❧✈❡ r❡❝✉rr❡♥❝❡s✱ ❛❧s♦ ❧♦♦❦ ❛t ❙♣❡❝✐❛❧ ❆ss✐❣♥♠❡♥t ✶✱ ❊①❡r❝✐s❡ ✶ ♦❢ ❙❆❚✶✷✮

❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✱ ❝♦✉♥t✲❙❆❚✱ r❡s♦❧✉t✐♦♥ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐