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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✿ ❙✐♠♣❧✐✜❝❛t✐♦♥ r✉❧❡s✿ • A ❝❤♦♦s❡s • t❤❡ ♠♦st ❢r❡q✉❡♥t ✈❛r✐❛❜❧❡ • ❯♥✐t ❝❧❛✉s❡ • ❛ ✈❛r✐❛❜❧❡ ❢r♦♠ t❤❡ s❤♦rt❡st ❡❧✐♠✐♥❛t✐♦♥ ❝❧❛✉s❡ • P✉r❡ ❧✐t❡r❛❧ r✉❧❡ • B ❝❤♦♦s❡s t❤❡ ♠♦st ❢r❡q✉❡♥t s✐❣♥ ✸ ✴ ✶✷

❉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 ❬✶✾✻✽❪✳ ✶ e 13 ⊕ e 14 ⊕ e 15 = 1 ✷ e 23 ⊕ e 25 = 1 ✸ e 13 ⊕ e 23 ⊕ e 34 ⊕ e 35 = 0 ✹ e 14 ⊕ e 34 ⊕ e 45 = 1 ✺ e 15 ⊕ e 25 ⊕ e 35 ⊕ e 45 = 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 • p ij ✿ i ✲t❤ ♣✐❣❡♦♥ ✐s ✐♥ j ✲t❤ ❤♦❧❡ • ∀ i ∈ [1 . . . n + 1] ✿ ( p i 1 ∨ p i 2 ∨ · · · ∨ p in ) • ∀ k ∈ [1 . . . n ] ∀ i , j ∈ [1 .. n + 1] ✿ ( ¬ p ik ∨ ¬ p jk ) ✺ ✴ ✶✷

▲♦✇❡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 = n 1 − ε ❝❧❛✉s❡s • ◗✉❡r② t❤❡ ♥✉♠❜❡r ♦❢ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ♦❝❝✉rr❡♥❝❡s ♦❢ ✈❛r✐❛❜❧❡ ( x 1 ∨ x 3 ∨ x 5 ) ( x 1 ∨ x 3 ∨ x 5 ) ( x 2 ∨ x 3 ) ( x 2 ∨ ¬ x 3 ) ⇒ ( x 2 ∨ x 4 ∨ x 5 ) ( x 2 ∨ x 4 ∨ x 5 ) ( x 1 ∨ x 4 ∨ x 6 ) ( x 1 ∨ ¬ x 4 ∨ x 6 ) • ▲♦✇❡r ❜♦✉♥❞✿ • Ax = b ♦✈❡r F 2 • 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 x 1 x 2 P ( x i 1 , x i 2 , . . . , x id ) • G ( X , Y , E ) ✐s ❛ ❜✐♣❛rt✐t❡ x 3 ✳ ✳ ❣r❛♣❤❀ ✳ ✳ • ∀ y ∈ Y deg ( y ) = d ✳ ✳ x n • d ✐s ❛ ❝♦♥st❛♥t✳ X Y ●♦❧❞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 ( x 1 , . . . x d ) = x 1 ⊕ · · · ⊕ x d − k ⊕ Q ( x d − k +1 , . . . , x d ) , 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✐❝ C = 1 φ x i := c x i := 1 − c φ ′ φ ′′ C = 1 C = 0 x j := c 1 x k := c 2 x j := 1 − c 1 x k := 1 − c 2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ • ❍❡✉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 Φ n s✉❝❤ t❤❛t ∀ ❞❡t❡r♠✐♥✐st✐❝ ♠②♦♣✐❝ A , C t❤❡r❡ ❡①✐sts ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ s❛♠♣❧❛❜❧❡ ❡♥s❡♠❜❧❡ ♦❢ ❞✐str✐❜✉t✐♦♥s D n ✇✐t❤ supp D n ⊂ SAT s✉❝❤ t❤❛t ∀ B ❡✐t❤❡r • Pr ϕ ← D n [ DPLL A , B , C ( ϕ ) = 1] < 1 / 100 ♦r • ❘✉♥♥✐♥❣ t✐♠❡ ♦❢ DPLL A , B , C (Φ n ) ✐s 2 Ω( n ) ✳ ❚❤❡♦r❡♠✳❚❤❡r❡ ❡①✐sts ❢❛♠✐❧② ♦❢ ✉♥s❛t✐s✜❛❜❧❡ ❢♦r♠✉❧❛s Φ n ❛♥❞ ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ s❛♠♣❧❛❜❧❡ ❡♥s❡♠❜❧❡ ♦❢ ❞✐str✐❜✉t✐♦♥s R n ✇✐t❤ supp R n ⊂ SAT s✉❝❤ t❤❛t ∀ ❞❡t❡r♠✐♥✐st✐❝ ♠②♦♣✐❝ A , C ❛♥❞ ∀ B ✐❢ Pr ϕ ← D n [ DPLL A , B , C ( ϕ ) = 1] = 1 − o (1) ✱ t❤❡♥ r✉♥♥✐♥❣ t✐♠❡ ♦❢ DPLL A , B , C (Φ n ) ✐s 2 Ω( n ) ✳ ✶✷ ✴ ✶✷