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❍❛r❞ ❡①❛♠♣❧❡s ❢♦r ❉P▲▲ ❛❧❣♦r✐t❤♠s

❉♠✐tr② ■ts②❦s♦♥

❙t❡❦❧♦✈ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ❛t ❙t✳ P❡t❡rs❜✉r❣

❚❤❡♦r② ❉❛②s ❖❝♦❜❡r ✽✱ ✷✵✶✶

✶ ✴ ✶✷

❖✉t❧✐♥❡

✶ ❉P▲▲ ✷ ▲♦✇❡r ❜♦✉♥❞s ♦♥ ✉♥s❛t✐s✜❛❜❧❡ ✐♥st❛♥❝❡s ✸ ▲♦✇❡r ❜♦✉♥❞s ♦♥ s❛t✐s✜❛❜❧❡ ✐♥st❛♥❝❡s ✹ ●♦❧❞r❡✐❝❤✬s ♦♥❡✲✇❛② ❝❛♥❞✐❞❛t❡ ✺ ❉P▲▲ ❛❧❣♦r✐t❤♠s ✇✐t❤ ❝✉t ❤❡✉r✐st✐❝

✷ ✴ ✶✷

❉P▲▲ ❛❧❣♦r✐t❤♠

(x ∨ y ∨ ¬z) ∧ (¬x ∨ ¬y) ∧ (¬y ∨ z) (x ∨ y ∨ ¬z) ∧ (¬x ∨ ¬y) ∧ (¬y ∨ z)✱ x := 0, y := 1, z := 1

• ❍❡✉r✐st✐❝ A ❝❤♦♦s❡s t❤❡

✈❛r✐❛❜❧❡ x

• ❍❡✉r✐st✐❝ B ❝❤♦♦s❡s t❤❡

❜r❛♥❝❤ t♦ ❜❡ ❡①❛♠✐♥❛t❡❞ ✜rst

• ❙✐♠♣❧✐✜❝❛t✐♦♥ r✉❧❡s

❊①❛♠♣❧❡s ♦❢ ❤❡✉r✐st✐❝s✿

• A ❝❤♦♦s❡s
• t❤❡ ♠♦st ❢r❡q✉❡♥t ✈❛r✐❛❜❧❡
• ❛ ✈❛r✐❛❜❧❡ ❢r♦♠ t❤❡ s❤♦rt❡st

❝❧❛✉s❡

• B ❝❤♦♦s❡s t❤❡ ♠♦st ❢r❡q✉❡♥t s✐❣♥

❙✐♠♣❧✐✜❝❛t✐♦♥ r✉❧❡s✿

• ❯♥✐t ❝❧❛✉s❡

❡❧✐♠✐♥❛t✐♦♥

• P✉r❡ ❧✐t❡r❛❧ r✉❧❡

✸ ✴ ✶✷

❉P▲▲ ♦♥ ✉♥s❛t✐s✜❛❜❧❡ ❢♦r♠✉❧❛s

• ❘❡s♦❧✉t✐♦♥ r✉❧❡✿ (A∨x)

(B∨¬x) A∨B

✳ (x ∨ y ∨ z) ∧ (x ∨ y ∨ ¬z) ∧ (x ∨ t ∨ ¬y) ∧ (¬t ∨ x) ∧ (¬x ∨ y) ∧ (¬y ∨ z) ∧ (¬y ∨ ¬z)

✹ ✴ ✶✷

❍❛r❞ ❡①❛♠♣❧❡s ❢♦r r❡s♦❧✉t✐♦♥s

• ❚s❡✐t✐♥ ❢♦r♠✉❧❛s ❬✶✾✻✽❪✳

✶ e13 ⊕ e14 ⊕ e15 = 1 ✷ e23 ⊕ e25 = 1 ✸ e13 ⊕ e23 ⊕ e34 ⊕ e35 = 0 ✹ e14 ⊕ e34 ⊕ e45 = 1 ✺ e15 ⊕ e25 ⊕ e35 ⊕ e45 = 0

• ❈♦♥st❛♥t ❞❡❣r❡❡
• ∀A ⊆ V ✐❢ |V |

3 ≤ |A| ≤ 2|V | 3 ✱ t❤❡♥ E[A, V \ A] ≥ α|V |✳

• P✐❣❡♦♥❤♦❧❡ Pr✐♥❝✐♣❧❡
• n + 1 ♣✐❣❡♦♥s ❛♥❞ n ❤♦❧❡s
• pij✿ i✲t❤ ♣✐❣❡♦♥ ✐s ✐♥ j✲t❤ ❤♦❧❡
• ∀i ∈ [1 . . . n + 1]✿ (pi1 ∨ pi2 ∨ · · · ∨ pin)
• ∀k ∈ [1 . . . n]∀i, j ∈ [1..n + 1]✿ (¬pik ∨ ¬pjk)

✺ ✴ ✶✷

▲♦✇❡r ❜♦✉♥❞s ♦♥ s❛t✐s✜❛❜❧❡ ❢♦r♠✉❧❛s

• ■❢ P = NP t❤❡♥ ♥♦ s✉♣❡r♣♦❧②♥♦♠✐❛❧ ❧♦✇❡r ❜♦✉♥❞s ❢♦r ❉P▲▲

❛❧❣♦r✐t❤♠s s✐♥❝❡ ❤❡✉r✐st✐❝ B ♠❛② ❝❤♦♦s❡ ❝♦r❡❝t ✈❛❧✉❡✳

• ❙❛t✐s✜❛❜❧❡ ❢♦r♠✉❧❛s ❛r❡ ♠✉❝❤ ❡❛s✐❡r ❢♦r s♦❧✈❡rs
• ❬◆✐❦♦❧❡♥❦♦✱ ✷✵✵✷❪✱ ❬❆❝❤✐❧✐♦♣t❛s✱❇❡❛♠❡✱ ▼♦❧❧♦②✱ ✷✵✵✸✲✷✵✵✹❪

❡①♣♦♥❡♥t✐❛❧ ❧♦✇❡r ❜♦✉♥❞ ❢♦r s♣❡❝✐✜❝ ❉P▲▲ ❛❧❣♦r✐t♠s

• ❬❆❧❡❦❤♥♦✈✐❝❤✱ ❍✐rs❝❤✱ ■ts②❦s♦♥✱ ✷✵✵✺❪ ❊①♣♦♥❡♥t✐❛❧ ❧♦✇❡r ❜♦✉♥❞

❢♦r ♠②♦♣✐❝ ❛♥❞ ❞r✉♥❦❡♥ ❛❧❣♦r✐t❤♠s✳

• ■♥✈❡rt✐♥❣ ♦❢ ❢✉♥❝t✐♦♥s ❝♦rr❡s♣♦♥❞s t♦ s❛t✐s✜❛❜❧❡ ❢♦r♠✉❧❛s

✻ ✴ ✶✷

▼②♦♣✐❝ ❛❧❣♦r✐t❤♠s

• ▼②♦♣✐❝ ❤❡✉r✐st✐❝s A, B✿
• ❘❡❛❞ ❢♦r♠✉❧❛ ✇✐t❤ ❡r❛s❡❞ ♥❡❣❛t✐♦♥s
• ❘❡❛❞ K = n1−ε ❝❧❛✉s❡s
• ◗✉❡r② t❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ♦❝❝✉rr❡♥❝❡s ♦❢

✈❛r✐❛❜❧❡

(x1 ∨ x3 ∨ x5) (x2 ∨ x3) (x2 ∨ x4 ∨ x5) (x1 ∨ x4 ∨ x6) ⇒ (x1 ∨ x3 ∨ x5) (x2 ∨ ¬x3) (x2 ∨ x4 ∨ x5) (x1 ∨ ¬x4 ∨ x6)

• ▲♦✇❡r ❜♦✉♥❞✿
• Ax = b ♦✈❡r F2
• A ✐s ❛ r❛♥❞♦♠❧② ❝♦♥str✉❝t❡❞ 0/1 ♠❛tr✐①
• ❡①❛❝t❧② 3 ♦♥❡s ♣❡r r♦✇❀ ❢✉❧❧ r❛♥❦
• x ⊕ y ⊕ z = 1 ⇐

⇒ (¬x ∨ ¬y ∨ ¬z) ∧ (¬x ∨ y ∨ z) ∧ (x ∨ ¬y ∨ z) ∧ (x ∨ y ∨ ¬z)

• x ⊕ y ⊕ z = 0 ⇐

⇒ (x ∨ y ∨ z) ∧ (¬x ∨ ¬y ∨ z) ∧ (¬x ∨ y ∨ ¬z) ∧ (x ∨ ¬y ∨ ¬z)

✼ ✴ ✶✷

❉r✉♥❦❡♥ ❛❧❣♦r✐t❤♠s

❉r✉♥❦❡♥ ❤❡✉r✐st✐❝s✿

• A✿ ❛♥②✦
• B✿ r❛♥❞♦♠ ✺✵✿✺✵

▲♦✇❡r ❜♦✉♥❞✿

• F ✐s ❛ ❤❛r❞ ✉♥s❛t✐s✜❛❜❧❡ ❢♦r♠✉❧❛
• F ′ = F + ♦♥❡ s❛t✐s❢②✐♥❣ ❛ss✐❣♥♠❡t
• ❲r♦♥❣ s✉❜st✐t✉t✐♦♥ ❞✉r✐♥❣ ✜rst s❡✈❡r❛❧ st❡♣ ✇✳❤✳♣✳
• ❋❛❧❧ t♦ ❤❛r❞ ✉♥s❛t✐s✜❛❜❧❡ ❢♦r♠✉❧❛

❈♦♠❜✐♥❛t✐♦♥✿

• A✿ ❛♥②✦
• B✿ ♠②♦♣✐❝

❈❤❡❛t✐♥❣✿

• A ❝❤♦♦s❡s ✈❛r✐❛❜❧❡ t❤❛t s❛t✐s❢② ✜rst ❝❧❛✉s❡✳

✽ ✴ ✶✷

• ♦❧❞r❡✐❝❤✬s ♦♥❡✲✇❛② ❝❛♥❞✐❞❛t❡

f : {0, 1}n → {0, 1}n xn ✳ ✳ ✳ ✳ ✳ ✳ x3 x2 x1 X Y P(xi1, xi2, . . . , xid)

• G(X, Y , E) ✐s ❛ ❜✐♣❛rt✐t❡

❣r❛♣❤❀

• ∀y ∈ Y

deg(y) = d

• d ✐s ❛ ❝♦♥st❛♥t✳
• ♦❧❞r❡✐❝❤✬s ❝♦♥❥❡❝t✉r❡✿
• P ✐s ❛ r❛♥❞♦♠ ♣r❡❞✐❝❛t❡❀
• G ✐s ❛♥ ❡①♣❛♥❞❡r❀

t❤❡♥ ❢✉♥❝t✐♦♥ f ✐s ❛ ♦♥❡✲✇❛②✳

• f ✐s ❝♦♠♣✉t❡❞ ❜② ❝♦♥st❛♥t ❞❡♣t❤ ❝✐r❝✉✐t❀
• ❬❆♣♣❧❡❜❛✉♠✱ ■s❤❛✐✱ ❑✉s❤✐❧❡✈✐t③ ✷✵✵✻❪ ■❢ ♦♥❡✲✇❛② ❢✉♥❝t✐♦♥s ❡①✐st

t❤❡♥ t❤❡r❡ ✐s ❛ ♦♥❡✲✇❛② ❢✉♥❝t✐♦♥ t❤❛t ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❜② ❝♦♥st❛♥t ❞❡♣t❤ ❝✐r❝✉✐t✳

✾ ✴ ✶✷

❊①♣♦♥❡♥t✐❛❧ ❧♦✇❡r ❜♦✉♥❞s ♦♥ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ✐♥✈❡rt✐♦♥ ♦❢ ●♦❧❞r❡✐❝❤✬s ❢✉♥❝t✐♦♥

P(x1, . . . xd) = x1 ⊕ · · · ⊕ xd−k ⊕ Q(xd−k+1, . . . , xd), k < d/4✳ P❛♣❡r

• r❛♣❤

❉P▲▲ ❈♦♦❦✱ ❊t❡s❛♠✐✱ ❘❛♥❞♦♠ ▼②♦♣✐❝ ▼✐❧❧❡r✱ ❚r❡✈✐s❛♥✱ ✷✵✵✾ ■ts②❦s♦♥✱ ✷✵✶✵ ❘❛♥❞♦♠ ❉r✉♥❦❡♥ ■ts②❦s♦♥✱ ❙♦❦♦❧♦✈✱ ✷✵✶✶ ❊①♣❧✐❝✐t ❉r✉♥❦❡♥ ✭❜❛s❡❞ ♦♥ ▼②♦♣✐❝ ❡①♣❛♥❞❡r✮

✶✵ ✴ ✶✷

❉P▲▲ ✇✐t❤ ❝✉t ❤❡✉r✐st✐❝

φ φ′ ✳ ✳ ✳ xj := c1 ✳ ✳ ✳ xj := 1 − c1 C = 1 xi := c φ′′ ✳ ✳ ✳ xk := c2 ✳ ✳ ✳ xk := 1 − c2 C = 0 xi := 1 − c C = 1

• ❍❡✉r✐st✐❝ A ❝❤♦♦s❡s ❛ ✈❛r✐❛❜❧❡ ❢♦r s♣❧✐tt✐♥❣✳
• ❍❡✉r✐st✐❝ B ❝❤♦♦s❡s ✜rst ✈❛❧✉❡✳
• ❍❡✉r✐st✐❝ C ❝✉ts ✉♥♣r♦♠✐s✐♥❣ ❜r❛♥❝❤❡s✳
• ❆❧❣♦r✐t❤♠ ✐s ❝♦rr❡❝t ♦♥ ✉♥s❛t✐s✜❛❜❧❡ ❢♦r♠✉❧❛s✳
• P♦ss✐❜❧❡ ❡rr♦rs ♦♥ s❛t✐s✜❛❜❧❡ ❢♦r♠✉❧❛s
• ❈♦rr❡❝t♥❡s ✈s✳ ❡✛❡❝t✐✈❡♥❡ss tr❛❞❡♦✛

✶✶ ✴ ✶✷

❈♦rr❡❝t♥❡s ✈s✳ ❡✛❡❝t✐✈❡♥❡ss

❬■ts②❦s♦♥✱ ❙♦❦♦❧♦✈ ✷✵✶✶❪ ❚❤❡♦r❡♠✳❚❤❡r❡ ❡①✐sts ❢❛♠✐❧② ♦❢ ✉♥s❛t✐s✜❛❜❧❡ ❢♦r♠✉❧❛s Φn s✉❝❤ t❤❛t ∀ ❞❡t❡r♠✐♥✐st✐❝ ♠②♦♣✐❝ A, C t❤❡r❡ ❡①✐sts ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ s❛♠♣❧❛❜❧❡ ❡♥s❡♠❜❧❡ ♦❢ ❞✐str✐❜✉t✐♦♥s Dn ✇✐t❤ supp Dn ⊂ SAT s✉❝❤ t❤❛t ∀B ❡✐t❤❡r

• Prϕ←Dn[DPLLA,B,C(ϕ) = 1] < 1/100 ♦r
• ❘✉♥♥✐♥❣ t✐♠❡ ♦❢ DPLLA,B,C(Φn) ✐s 2Ω(n)✳

❚❤❡♦r❡♠✳❚❤❡r❡ ❡①✐sts ❢❛♠✐❧② ♦❢ ✉♥s❛t✐s✜❛❜❧❡ ❢♦r♠✉❧❛s Φn ❛♥❞ ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ s❛♠♣❧❛❜❧❡ ❡♥s❡♠❜❧❡ ♦❢ ❞✐str✐❜✉t✐♦♥s Rn ✇✐t❤ supp Rn ⊂ SAT s✉❝❤ t❤❛t ∀ ❞❡t❡r♠✐♥✐st✐❝ ♠②♦♣✐❝ A, C ❛♥❞ ∀B ✐❢ Prϕ←Dn[DPLLA,B,C(ϕ) = 1] = 1 − o(1)✱ t❤❡♥ r✉♥♥✐♥❣ t✐♠❡ ♦❢ DPLLA,B,C(Φn) ✐s 2Ω(n)✳

✶✷ ✴ ✶✷