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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❡ αi :=

    

0, {xi} ∈ F 1, {xi} ∈ F ⋆, ❡❧s❡ ❢♦r ❛❧❧ i ∈ V. 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|xi→ti r❡❢❡rs t♦ t❤❡ s❡t {α ∪ (xi → ti) | α ∈ A}✳ ■♥ ♣❛rt✐❝✉❧❛r✱ #A = #A|xi→ti

❝♦✉♥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(1.619n)

❝♦✉♥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 (x1 → 0, x2 → 1, x3 → ⋆) ❧❡❛❞s t♦ t❤❡ ❉◆❋✲❝❧❛✉s❡ (x1 ∧ x2)✳ ❋✐♥❛❧❧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(1.619n)✳

❝♦✉♥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✱ αi ✐s ✐♥❞❡♣❡♥❞❡♥t❧② 0 ♦r 1 ✇✐t❤ ♣r♦❜❛❜✐❧✐t② 1

2✳

✭✶✮ Pr(α s❛t✐s✜❡s {x1, x3, x7} ) = 1 − 1

2 · 1 2 · 1 2 = 7 8✳

✭✷✮ Pr(α s❛t✐s✜❡s {{x1, x5, x11}, {x2, x4}, {x6, x12, x13}} ) = 7

8 · 3 4 · 7 8✳

✭✸✮ Pr(α s❛t✐s✜❡s {{x1, x2, x3}, {x3, x4, x5}} ) = Pr(α s❛t✐s✜❡s {{x1, x2, x3}, {x3, x4, x5}} |x3 → 0 ) · Pr(x3 → 0) + Pr(α s❛t✐s✜❡s {{x1, x2, x3}, {x3, x4, x5}} |x3 → 1 ) · Pr(x3 → 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 ✭✷✮

✭✹✮ Pr(α s❛t✐s✜❡s {{x1, x2, x3}, {x3, x4, x5}} ) = 1

2 · 3

4 · 3 4

✐❢ x3→0

+1

2 ·

1

• ✐❢ x3→1

✳ ✭✺✮ Pr(α s❛t✐s✜❡s {{x1, x2, x3}, {x1, x3, x4, x5, x7}} ) = 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❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✸✱ ✷✵✶✻ s❧✐❞❡ ✾

✷✳✻✳ Pr♦♣❡rt② ❇

❙❡❧❡❝t ❛ ❝♦❧♦✉r✐♥❣ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠ ❛♥❞ ❝❛❧❝✉❧❛t❡ t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ ♠♦♥♦❝❤r♦♠❛t✐❝ ❤②♣❡r❡❞❣❡s✳ ❆ ✜①❡❞ ❤②♣❡r❡❞❣❡ ❜❡❝♦♠❡s ♠♦♥♦❝❤r♦♠❛t✐❝ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② 2 · 2−k = 2−(k−1)✳ ■❢ t❤❡r❡ ❛r❡ ✐♥ t♦t❛❧ ❢❡✇❡r t❤❛♥ 2k−1 ❤②♣❡r❡❞❣❡s✱ t❤❡♥ t❤❡ ❡①♣❡❝t❡❞ ♥✉♠❜❡r ♦❢ ♠♦♥♦❝❤r♦♠❛t✐❝ ♦♥❡s ✐s < 1✱ ✐♠♣❧②✐♥❣ ❡①✐st❡♥❝❡ ♦❢ ❛ ❝♦❧♦✉r✲ ✐♥❣ t❤❛t ❛✈♦✐❞s ❛❧❧ ♦❢ t❤❡♠✳

❝♦✉♥t✲❙❆❚✱ ❡①tr❡♠❛❧ ♣r♦♣❡rt✐❡s ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✸✱ ✷✵✶✻ s❧✐❞❡ ✶✵

✷✳✼✿ k✲❈◆❋ ✇✐t❤ k ♦❝❝✉rr❡♥❝❡s ♣❡r ✈❛r✐❛❜❧❡

❚❤❡♦r❡♠ ✶ ✭❍❛❧❧✬s ❝♦♥❞✐t✐♦♥✮ ❆ ❜✐♣❛rt✐t❡ ❣r❛♣❤ G = (A, B, E) ❤❛s ❛ ♠❛t❝❤✐♥❣ t❤❛t s❛t✉r❛t❡s A ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❢♦r ❡❛❝❤ s✉❜s❡t S ⊆ A✱ t❤❡ ♥✉♠❜❡r ♦❢ ♥❡✐❣❤❜♦rs ♦❢ S ✐♥ G ✐s ❛t ❧❡❛st ❛s ❧❛r❣❡ ❛s t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ S✳

• ✐✈❡♥ ❛ k✲❈◆❋ ❢♦r♠✉❧❛ F ✇❤❡r❡ ❡❛❝❤ ✈❛r✐❛❜❧❡ ❛♣♣❡❛rs ❛t ♠♦st k

t✐♠❡s✱ ❝♦♥str✉❝t ❛ ❜✐♣❛rt✐t❡ ❣r❛♣❤ G ✇✐t❤ t✇♦ s❡ts ♦❢ ✈❡rt✐❝❡s✿ ♦♥❡ ❢♦r ❝❧❛✉s❡s ❛♥❞ ♦♥❡ ❢♦r ✈❡rt✐❝❡s✳ ▼❛❦❡ ❛♥ ❡❞❣❡ ❢♦r ❡❛❝❤ ❝❧❛✉s❡✲✈❛r✐❛❜❧❡ ♣❛✐r s✉❝❤ t❤❛t t❤❡ ✈❛r✐❛❜❧❡ ❛♣♣❡❛rs ✐♥ t❤❡ ❝❧❛✉s❡ ✐♥ F✳ ◆♦✇ ✇❡ ❝❛♥ ✈❡r✐❢② t❤❛t G s❛t✐s✜❡s ❍❛❧❧✬s ❝♦♥❞✐t✐♦♥✳ ❚❛❦❡ ❛♥② s✉❜s❡t ♦❢ ❝❧❛✉s❡ ✈❡rt✐❝❡s S✳ ❆s F ✐s ❛ k✲❈◆❋ ❢♦r♠✉❧❛✱ t❤❡ ♥✉♠❜❡r ♦❢ ❡❞❣❡s

❝♦✉♥t✲❙❆❚✱ ❡①tr❡♠❛❧ ♣r♦♣❡rt✐❡s ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✸✱ ✷✵✶✻ s❧✐❞❡ ✶✶

✐♥❝✐❞❡♥t ♦♥ S ✐s ❡①❛❝t❧② k|S|✳ ■❢ t❤❡ ♥✉♠❜❡r ♦❢ ♥❡✐❣❤❜♦rs ♦❢ S ✐♥ G ✐s s♠❛❧❧❡r t❤❛♥ |S| t❤✐s ✐♠♣❧✐❡s t❤❛t s♦♠❡ ♥❡✐❣❤❜♦r ♦❢ S ❤❛s ❞❡❣r❡❡ ♠♦r❡ t❤❛♥ k ❛♥❞ t❤✉s ❛ ✈❛r✐❛❜❧❡ ❛♣♣❡❛rs ♠♦r❡ t❤❛♥ k t✐♠❡s ✐♥ F✳ ❚❤✉s✱ t❤❡r❡ ✐s ❛ ♠❛t❝❤✐♥❣ M t❤❛t s❛t✉r❛t❡s ❛❧❧ t❤❡ ❝❧❛✉s❡ ✈❡rt✐❝❡s✳ ❚❤✐s ♠❡❛♥s t❤❛t ✇❡ ❝❛♥ s❡t ❡❛❝❤ ✈❛r✐❛❜❧❡ ✐♥ t❤❡ ♠❛t❝❤✐♥❣ t♦ s❛t✐s❢② t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❧❛✉s❡ ✐t ✐s ♠❛t❝❤❡❞ ✇✐t❤ ✐♥ M✱ t❤❡r❡❜② s❛t✐s❢②✐♥❣ ❛❧❧ t❤❡ ❝❧❛✉s❡s✳ ✷

❝♦✉♥t✲❙❆❚✱ ❡①tr❡♠❛❧ ♣r♦♣❡rt✐❡s ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐