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❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✸✱ ✷✵✶✻ s❧✐❞❡ ✶ ❚♦❞❛②✬s ❡①❡r❝✐s❡s • ✶✳✷✶✿ ✭❋r♦♠ ❧❛st s❡ss✐♦♥✮ ❘❡♣r❡s❡♥t✐♥❣ ❙❛t✐s❢②✐♥❣ ❆ss✐❣♥♠❡♥ts • ✶✳✷✷✿ ✭❋r♦♠ ❧❛st s❡ss✐♦♥✮ ▼✐♥✐♠❛❧❧② ❙❛t✐s❢②✐♥❣ • ✶✳✷✹✿ ✭❋r♦♠ ❧❛st s❡ss✐♦♥✮ ✷✲❈◆❋ ❛s ❉◆❋ • ✷✳✶✿ ❙❛t✐s❢❛❝t✐♦♥ Pr♦❜❛❜✐❧✐t✐❡s • ✷✳✷✿ ❆❧♠♦st ❙❛t✐s✜❛❜❧❡ • ✷✳✻✿ Pr♦♣❡rt② ❇ • ✷✳✼✿ ✭■♥ ❝❧❛ss✮ k ✲❈◆❋ ✇✐t❤ k ♦❝❝✉rr❡♥❝❡s ♣❡r ✈❛r✐❛❜❧❡ ❝♦✉♥t✲❙❆❚✱ ❡①tr❡♠❛❧ ♣r♦♣❡rt✐❡s ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✸✱ ✷✵✶✻ s❧✐❞❡ ✷ ✶✳✷✶✳ ❘❡♣r❡s❡♥t✐♥❣ ❙❛t✐s❢②✐♥❣ ❆ss✐❣♥♠❡♥ts ❆❧❧ s❛t✐s❢②✐♥❣ ❛ss✐❣♥♠❡♥ts t♦ ❛ ( ≤ 1) ✲❈◆❋ ❢♦r♠✉❧❛ ❝❛♥ ❛❧✇❛②s ❜❡ ❞❡s❝r✐❜❡❞ ❜② ❛t ♠♦st ♦♥❡ ❤②♣❡r ❛ss✐❣♥♠❡♥t✳ ❢✉♥❝t✐♦♥ ❧✐st✶s ( F, V ) ✐❢ ( ✷ ∈ F ♦r ∃ x ∈ V : {{ x } , { x }} ⊆ F ) t❤❡♥ r❡t✉r♥ {} ; ❡❧s❡  0 , { x i } ∈ F   α i := 1 , { x i } ∈ F ❢♦r ❛❧❧ i ∈ V.   ❡❧s❡ ⋆, r❡t✉r♥ { α } ; ❝♦✉♥t✲❙❆❚✱ ❡①tr❡♠❛❧ ♣r♦♣❡rt✐❡s ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✸✱ ✷✵✶✻ s❧✐❞❡ ✸ ✶✳✷✶✳ ❘❡♣r❡s❡♥t✐♥❣ ❙❛t✐s❢②✐♥❣ ❆ss✐❣♥♠❡♥ts ✭✷✮ ❋♦r ( ≤ 2) ✲❈◆❋ ❢♦r♠✉❧❛❡ ✇❡ s❧✐❣❤t❧② ♠♦❞✐❢② ✜❜ ❝✷s✭❋✱ ❱✮✳ ❢✉♥❝t✐♦♥ ✜❜ ❧✐st✷s ( F, V ) ✐❢ F ✐s ✭ ≤ 1 ✮✲❈◆❋ t❤❡♥ r❡t✉r♥ ❧✐st✶s ( F, V ); ❡❧s❡ { u, v } ← some 2 − clause in F ; α ← ( u �→ 1); β ← ( u �→ 0 , v �→ 1); U ← V \ vbl( { u } ); W ← V \ vbl( { u, v } ); r❡t✉r♥ ✜❜ ❧✐st✷s ( F [ α ] , U ) | ( u �→ 1) ∪ ✜❜ ❧✐st✷s ( F [ β ] , W ) | ( u �→ 0 ,v �→ 1) ; ❋♦r ❛ s❡t ♦❢ ❛ss✐❣♥♠❡♥ts A ✱ t❤❡ ♥♦t❛t✐♦♥ A | x i �→ t i r❡❢❡rs t♦ t❤❡ s❡t { α ∪ ( x i �→ t i ) | α ∈ A } ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ # A = # A | x i �→ t i ❝♦✉♥t✲❙❆❚✱ ❡①tr❡♠❛❧ ♣r♦♣❡rt✐❡s ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✸✱ ✷✵✶✻ s❧✐❞❡ ✹ ✶✳✷✶✳ ❘❡♣r❡s❡♥t✐♥❣ ❙❛t✐s❢②✐♥❣ ❆ss✐❣♥♠❡♥ts ✭✸✮ ▲❡t t ( n ) ❞❡♥♦t❡ t❤❡ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ❤②♣❡r❛ss✐❣♥♠❡♥ts ♦✉t♣✉t ❜② ❛ s✐♥❣❧❡ ❝❛❧❧ t♦ ✜❜ ❧✐st✷s✭❋✱❱✮ ✇✐t❤ ❛ ( ≤ 2) ✲❈◆❋ ❢♦r♠✉❧❛ F ❝♦♥t❛✐♥✐♥❣ n ✈❛r✐❛❜❧❡s✳ ❚❤❡♥✱ t ( n ) ≤ t ( n − 1) + t ( n − 2) ✐♥ ❣❡♥❡r❛❧✱ ❛♥❞ t (0) , t (1) ≤ 1 ✳ ❚❤✉s✱ t ( n ) = O (1 . 619 n ) ❝♦✉♥t✲❙❆❚✱ ❡①tr❡♠❛❧ ♣r♦♣❡rt✐❡s ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✸✱ ✷✵✶✻ s❧✐❞❡ ✺ ✶✳✷✷✳ ▼✐♥✐♠❛❧❧② ❙❛t✐s❢②✐♥❣✳ ■s ✐t tr✉❡ t❤❛t ❡❛❝❤ ♦❢ t❤❡ ❤②♣❡r❛ss✐❣♥✲ ♠❡♥ts ✐♥ t❤❡ ♦✉t♣✉t ♦❢ ❛ ❝❛❧❧ t♦ ✜❜ ❧✐st✷s✭❋✱❱✮ ✐s ♠✐♥✐♠❛❧❧② s❛t✐s❢②✐♥❣❄ ✶✳✷✸✳ ✷✲❈◆❋ ❛s ❉◆❋✳ ❚❤✐s r❡s✉❧t ❢♦❧❧♦✇s ❢r♦♠ ❊①❡r❝✐s❡ ✶✳✷✶ ❛♥❞ t❤❡ ❢❛❝t t❤❛t ✇❡ ❝❛♥ ✇r✐t❡ ❛ s✐♥❣❧❡ ❉◆❋✲❝❧❛✉s❡ t❤❛t ❡✈❛❧✉❛t❡s t♦ 1 ✉♥❞❡r ❛♥ ❛ss✐❣♥♠❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❛ss✐❣♥♠❡♥t ♠❛t❝❤❡s ❛ ❤②♣❡r❛s✲ s✐❣♥♠❡♥t✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❤②♣❡r❛ss✐❣♥♠❡♥t ( x 1 �→ 0 , x 2 �→ 1 , x 3 �→ ⋆ ) ❧❡❛❞s t♦ t❤❡ ❉◆❋✲❝❧❛✉s❡ ( x 1 ∧ x 2 ) ✳ ❋✐♥❛❧❧②✱ ❡❛❝❤ s✉❝❤ ❉◆❋✲❝❧❛✉s❡ ❤❛s ❧❡ss t❤❛♥ n ❝♦♥❥✉♥❝t✐♦♥s s♦ t❤❛t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝♦♥❥✉♥❝t✐♦♥s ✐♥ t❤❡ ❉◆❋✲❢♦r♠✉❧❛ ✐s ❛t ♠♦st t❤❡ ♥✉♠❜❡r ♦❢ ❝❧❛✉s❡s t✐♠❡s n ✇❤✐❝❤ ✐s st✐❧❧ O (1 . 619 n ) ✳ ❝♦✉♥t✲❙❆❚✱ ❡①tr❡♠❛❧ ♣r♦♣❡rt✐❡s ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✸✱ ✷✵✶✻ s❧✐❞❡ ✻ ✷✳✶✳ ❙❛t✐s❢❛❝t✐♦♥ Pr♦❜❛❜✐❧✐t✐❡s α ✐s ❞r❛✇♥ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠ ❢r♦♠ { 0 , 1 } 100 ⇐ ⇒ ❢♦r ❡❛❝❤ ❝♦♦r❞✐✲ ♥❛t❡ i ✱ α i ✐s ✐♥❞❡♣❡♥❞❡♥t❧② 0 ♦r 1 ✇✐t❤ ♣r♦❜❛❜✐❧✐t② 1 2 ✳ ✭✶✮ Pr( α s❛t✐s✜❡s { x 1 , x 3 , x 7 } ) = 1 − 1 2 · 1 2 · 1 2 = 7 8 ✳ ✭✷✮ Pr( α s❛t✐s✜❡s {{ x 1 , x 5 , x 11 } , { x 2 , x 4 } , { x 6 , x 12 , x 13 }} ) = 7 8 · 3 4 · 7 8 ✳ ✭✸✮ Pr( α s❛t✐s✜❡s {{ x 1 , x 2 , x 3 } , { x 3 , x 4 , x 5 }} ) = Pr( α s❛t✐s✜❡s {{ x 1 , x 2 , x 3 } , { x 3 , x 4 , x 5 }} | x 3 �→ 0 ) · Pr( x 3 �→ 0) + Pr( α s❛t✐s✜❡s {{ x 1 , x 2 , x 3 } , { x 3 , x 4 , x 5 }} | x 3 �→ 1 ) · Pr( x 3 �→ 1) = 3 4 · 1 2 + 3 4 · 1 2 = 3 4 ✳ ❝♦✉♥t✲❙❆❚✱ ❡①tr❡♠❛❧ ♣r♦♣❡rt✐❡s ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✸✱ ✷✵✶✻ s❧✐❞❡ ✼ ✷✳✶✳ ❙❛t✐s❢❛❝t✐♦♥ Pr♦❜❛❜✐❧✐t✐❡s ✭✷✮ 2 · 3 4 · 3 ✭✹✮ Pr( α s❛t✐s✜❡s {{ x 1 , x 2 , x 3 } , { x 3 , x 4 , x 5 }} ) = 1 + 1 2 · 1 ✳ ���� 4 ✐❢ x 3 �→ 1 � �� � ✐❢ x 3 �→ 0 ✭✺✮ Pr( α s❛t✐s✜❡s {{ x 1 , x 2 , x 3 } , { x 1 , x 3 , x 4 , x 5 , x 7 }} ) = 1 2 · 15 16 + 1 2 · 3 4 = 27 32 ✳ ❝♦✉♥t✲❙❆❚✱ ❡①tr❡♠❛❧ ♣r♦♣❡rt✐❡s ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✸✱ ✷✵✶✻ s❧✐❞❡ ✽ ✷✳✷✳ ❆❧♠♦st ❙❛t✐s✜❛❜❧❡ ✭✶✮ ↔ ✭✷✮✳ ❚❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❞✐ss❛t✐s✜❡❞ ❝❧❛✉s❡s ✐s 1 ✭❝♦♠✲ ♣❛r❡ t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳✷✳✮✳ ❙✐♥❝❡ ♥♦ ❛ss✐❣♥♠❡♥t ❞✐ss❛t✐s✜❡s 0 ❝❧❛✉s❡s✱ ❛❧❧ ♦❢ t❤❡♠ ♠✉st ❞✐ss❛t✐s❢② ❡①❛❝t❧② 1 t♦ ♣r♦❞✉❝❡ t❤✐s ❛✈❡r❛❣❡✳ ❋♦r t❤❡ ❝♦♥✈❡rs❡ ❝♦♠♣✉t❡ t❤❡ ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ ❞✐ss❛t✐s✜❡❞ ❝❧❛✉s❡s✳ ✭✷✮ ↔ ✭✸✮✳ ❆ss✉♠❡ t❤❡r❡ ✇❛s ❛ ♣❛✐r C, D ∈ F ✇✐t❤ ♥♦ s✉❝❤ ❝♦♠♣❧❡✲ ♠❡♥t❛r② ❧✐t❡r❛❧✳ ❚❤❡♥ ✇❡ ❝❛♥ ❞❡✈✐s❡ ❛♥ ❛ss✐❣♥♠❡♥t t❤❛t ✈✐♦❧❛t❡s ❜♦t❤ ❝❧❛✉s❡s✱ ❝♦♥tr❛❞✐❝t✐♥❣ ✭✷✮✳ ❈♦♥✈❡rs❡❧②✱ s✉♣♣♦s❡ ❡✈❡r② ♣❛✐r ♦❢ ❝❧❛✉s❡s ❤❛s ❛ ♣❛✐r ♦❢ ❝♦♠♣❧❡♠❡♥t❛r② ❧✐t❡r❛❧s✳ ❆♥❞ ❧❡t α ❜❡ ❛♥② ❛ss✐❣♥♠❡♥t✳ ❙✐♥❝❡ F ✐s ✉♥s❛t✐s✜❛❜❧❡✱ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ❝❧❛✉s❡ C ✈✐♦❧❛t❡❞ ❜② α ✳ ❙✐♥❝❡ ❛❧❧ ❝❧❛✉s❡s D � = C ❤❛✈❡ ❛ ❧✐t❡r❛❧ ❝♦♠♣❧❡♠❡♥t❛r② t♦ s♦♠❡ ❧✐t❡r❛❧ ✐♥ C ✱ α s❛t✐s✜❡s t❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤❡ ❝❧❛✉s❡s✳ ✷ ❝♦✉♥t✲❙❆❚✱ ❡①tr❡♠❛❧ ♣r♦♣❡rt✐❡s ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐