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❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✶ ❚♦❞❛②✬s ❡①❡r❝✐s❡s • ✷✳✸✿ ❆❧♠♦st ❙❛t✐s✜❛❜❧❡ ✈❡rs✉s ❇❛❧❛♥❝❡❞ • ✷✳✽✿ ▼❛♥② ❱❛r✐❛❜❧❡s • ✷✳✶✷✿ ❍♦✇ ♠✉❝❤ ■♥❞❡♣❡♥❞❡♥❝❡ ✐s ◆❡❝❡ss❛r②❄ • ✷✳✶✹✿ ❙✉♣r❡♠✉♠ ✈s ▼❛①✐♠✉♠ • ✸✳✷✿ ✭■♥ ❝❧❛ss✮ ■♠♣❧❡♠❡♥t✐♥❣ ❘❡s♦❧✉t✐♦♥ t♦ ❉❡❝✐❞❡ ✷✲❙❆❚ • ✭■♥ ❝❧❛ss✮ ❘❡s♦❧✉t✐♦♥ ❊①❡r❝✐s❡ ❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✷ ✷✳✸✳ ❆❧♠♦st ❙❛t✐s✜❛❜❧❡ ❱❡rs✉s ❇❛❧❛♥❝❡❞ ✭✶✮ ❋r♦♠ ✷✳✷ ✇❡ ❦♥♦✇ t❤❛t ❡✈❡r② t♦t❛❧ ❛ss✐❣♥♠❡♥t ❢♦r F s❛t✐s✜❡s ❛❧❧ ❜✉t ♦♥❡ ❝❧❛✉s❡ ✐♥ F ✳ ❚❤❛t ✐s✱ ❢♦r ❛♥② ❛ss✐❣♥♠❡♥t α ✱ ❡①❛❝t❧② ♦♥❡ ❝❧❛✉s❡ ✐s ✏r❡s♣♦♥s✐❜❧❡✑ ❢♦r t❤❡ ✉♥s❛t✐s✜❛❜✐❧✐t② ♦❢ F ✳ ❉❡♥♦t❡ t❤✐s ❝❧❛✉s❡ ❜② C ( α ) ✳ ▲❡t x ∈ vbl( F ) ✳ P❛rt✐t✐♦♥ t❤❡ ❝❧❛✉s❡s ✭✐✳❡✳ F ✮ ❛♥❞ t❤❡ ❛ss✐❣♥♠❡♥ts ✐♥t♦ t❤r❡❡ ♣❛rts✿ ✶✳ F 0 ✿ ❈❧❛✉s❡s ✇✐t❤ x �∈ vbl( C ) ✳ ▲❡t A 0 = { α | C ( α ) ∈ F 0 } ✳ ✷✳ F x ✿ ❈❧❛✉s❡s ✇✐t❤ x ∈ C ✳ ▲❡t A x = { α | C ( α ) ∈ F x } ✳ ✸✳ F ¯ x ✿ ❈❧❛✉s❡s ✇✐t❤ ¯ x ∈ C ✳ ▲❡t A ¯ x = { α | C ( α ) ∈ F ¯ x } ✳ ❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✸ ❋✐rst ✇❡ s❡❡ t❤❛t | A x | = | A ¯ x | ✿ Pr♦♦❢✿ ❋♦r α ∈ A x ✱ α ( x ) = 0 ✱ ❛♥❞ ❢♦r α ∈ A ¯ x ✱ α ( x ) = 1 ✳ A 0 ✐s ✏s②♠♠❡tr✐❝✑ ✇✐t❤ r❡s♣❡❝t t♦ x ✳ ❍❡♥❝❡ ❢♦r { 0 , 1 } V \ A 0 ✱ t❤❡r❡ ❛r❡ ❛s ♠❛♥② ❛ss✐❣♥♠❡♥ts s❡tt✐♥❣ x t♦ 0 ❛s s❡tt✐♥❣ x t♦ 1 ✳ ❖❜s❡r✈❛t✐♦♥✿ ❊❛❝❤ k ✲❝❧❛✉s❡ ✐s ♥♦t s❛t✐s✜❡❞ ❜② ❡①❛❝t❧② 2 | V |− k ❛ss✐❣♥✲ ♠❡♥ts✳ ❚❤❡r❡❢♦r❡ | F x | = | F ¯ x | ✳ ❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✹ ✷✳✸✳ ❆❧♠♦st ❙❛t✐s✜❛❜❧❡ ❱❡rs✉s ❇❛❧❛♥❝❡❞ ✭✷✮ ✭✷✮ ❋♦r ❛♥ ❛r❜✐tr❛r② ✉♥s❛t✐s✜❛❜❧❡ ❈◆❋✱ t❤✐s ✐s ♥♦t tr✉❡✳ ❈♦✉♥t❡r❡①❛♠♣❧❡✿ F = {{ ¯ x } , { x, ¯ y } , { x, y }} ✳ ❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✺ ✷✳✽✿ ▼❛♥② ✈❛r✐❛❜❧❡s ❈♦♥s✐❞❡r t❤❡ ❜✐♣❛rt✐t❡ ❣r❛♣❤ G = ( V, E ) ✇❤❡r❡ t❤❡ ✈❡rt❡① s❡t V ✐s ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ A ∪ B ✱ ❝❧❛✉s❡ ✈❡rt✐❝❡s ❛♥❞ ✈❛r✐❛❜❧❡ ✈❡rt✐❝❡s r❡s♣❡❝t✐✈❡❧②✳ ❋♦r ❡❛❝❤ ❝❧❛✉s❡ C ∈ F ❛♥❞ ❡❛❝❤ ✈❛r✐❛❜❧❡ i ∈ ✈❜❧ ( C ) ✱ E ❝♦♥t❛✐♥s ❛♥ ❡❞❣❡ ❝♦♥♥❡❝t✐♥❣ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❧❛✉s❡ ❛♥❞ ✈❛r✐❛❜❧❡ ♣❛✐rs ✭❛♥❞ ♦♥❧② t❤♦s❡ ❡❞❣❡s✮✳ ❋♦r ❛♥② s✉❜s❡t S ⊆ A ♦❢ ❝❧❛✉s❡ ✈❡rt✐❝❡s✱ ✇❡ ❤❛✈❡ t❤❛t | N ( S ) | ≥ | S | s✐♥❝❡ N ( S ) ❝♦rr❡s♣♦♥❞s ♣r❡❝✐s❡❧② t♦ t❤❡ s❡t ♦❢ ✈❛r✐❛❜❧❡s ❛♣♣❡❛r✐♥❣ ✐♥ t❤❡ ❝❧❛✉s❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ S ✳ ❇② ❍❛❧❧✬s ❚❤❡♦r❡♠✱ t❤❡r❡ ✐s ❛ ♣❡r❢❡❝t ♠❛t❝❤✐♥❣ t❤❛t s❛t✉r❛t❡s A ✱ ❛♥❞ s♦ F ✐s s❛t✐s✜❛❜❧❡ ✭❡❛❝❤ ❝❧❛✉s❡ ❝❛♥ ✉s❡ ❛ ✉♥✐q✉❡ ✈❛r✐❛❜❧❡ t♦ s❛t✐s❢② ✐t ✐♥ ❛♥ ❛ss✐❣♥♠❡♥t✮✳ ❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✻ ✷✳✶✷✿ ❍♦✇ ▼✉❝❤ ■♥❞❡♣❡♥❞❡♥❝❡ ✐s ◆❡❝❡ss❛r②❄ ▼✉t✉❛❧ ✐♥❞❡♣❡♥❞❡♥❝❡✿ ❊❛❝❤ ✈❛r✐❛❜❧❡✬s tr✉t❤ ✈❛❧✉❡ ✐s s❡t t♦ 1 ✇✐t❤ ♣r♦❜❛❜✐❧✐t② p ❛♥❞ t♦ 0 ✇✐t❤ ♣r♦❜❛❜✐❧✐t② (1 − p ) ❝♦♠♣❧❡t❡❧② ✐♥❞❡♣❡♥❞❡♥t❧② ❢r♦♠ ❛❧❧ ♦t❤❡r ✈❛r✐❛❜❧❡s✳ ❋♦r ❡❛❝❤ ♣❛✐r x, y ♦❢ ✈❛r✐❛❜❧❡s✱ ✇❡ ❤❛✈❡ P❛✐r✇✐s❡ ✐♥❞❡♣❡♥❞❡♥❝❡✿ Pr ( y �→ 1) = Pr ( y �→ 1 | x �→ 0) = Pr ( y �→ 1 | x �→ 1) = p ✭t❤❡ ❛♥❛❧♦❣♦✉s st❛t❡♠❡♥t ❢♦r y �→ 0 ❢♦❧❧♦✇s ❜② t❛❦✐♥❣ ❝♦♠♣❧❡♠❡♥ts✮ ❊①❛♠♣❧❡✿ s❛♠♣❧❡ x, y ✉✳❛✳r✳ ✐♥❞❡♣❡♥❞❡♥t❧② ❛♥❞ t❤❡♥ z := x ⊕ y ✳ ◗ ✿ ❈❛♥ ②♦✉ ♠❛❦❡ ❛ ❧❛r❣❡ s♣❛❝❡ ✇✐t❤ ♣❛✐r✇✐s❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s❄ ❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✼ ✷✳✶✷✿ ❍♦✇ ▼✉❝❤ ■♥❞❡♣❡♥❞❡♥❝❡ ✐s ◆❡❝❡ss❛r②❄ ✭✷✮ ■♥ t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳✹✱ ✇❡ ❝❛❧❝✉❧❛t❡ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❡❛❝❤ ❝❧❛✉s❡ t♦ ❜❡ s❛t✐s✜❡❞ ✐♥❞❡♣❡♥❞❡♥t❧②✳✳✳ ✏■❢ C ∈ F ❝♦♥t❛✐♥s ❛ ♣♦s✐t✐✈❡ ❧✐t❡r❛❧ t❤❡♥ C ✐s s❛t✐s✜❡❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❛t ❧❡❛st p ❬✳✳✳❪✑ ✭t❤❡ r❡♠❛✐♥✐♥❣ ❧✐t❡r❛❧s ❝❛♥ ❜❡ ✐❣♥♦r❡❞✦✮✳ ✏❖t❤❡r✇✐s❡ C ❝♦♥t❛✐♥s t✇♦ ♥❡❣❛t✐✈❡ ❧✐t❡r❛❧s ❛♥❞ C ✐s s❛t✐s✜❡❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❛t ❧❡❛st 1 − p 2 ❬✳✳✳❪✑ ✭t❤❡ r❡♠❛✐♥✐♥❣ ❧✐t❡r❛❧s ❛r❡ ✐❣♥♦r❡❞✦✮✳ ❙✐♥❝❡ ✇❡ ❝♦♥s✐❞❡r ❛t ♠♦st t✇♦ ✈❛r✐❛❜❧❡s ❛t t❤❡ s❛♠❡ t✐♠❡✱ ♣❛✐r✇✐s❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ✐s s✉✣❝✐❡♥t✳ ❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✶✺✱ ✷✵✶✻ s❧✐❞❡ ✽ ✷✳✶✹✿ ❙✉♣r❡♠✉♠ ✈s ▼❛①✐♠✉♠ ▲❡t ( F, µ ) ❜❡ ❛ ❣✐✈❡♥ ❈◆❋ ❢♦r♠✉❧❛ ❢♦r ✇❤✐❝❤ ✇❡ ❛✐♠ t♦ ✜♥❞ ❛♥ ❛s✲ s✐❣♥♠❡♥t α s✉❝❤ t❤❛t µ [ α ] ( F ) ≥ r k µ ( F ) ✳ | µ ( S ) − µ ( T ) | ǫ := min S,T ⊆ F,µ ( S ) � = µ ( T ) > 0 ✳ µ ( F ) ❇② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ r k t❤❡r❡ ✐s ❛♥ ❛ss✐❣♥♠❡♥t α s✉❝❤ t❤❛t µ [ α ] ( F ) ≥ ( r k − ǫ ) µ ( F ) ✳ ❊✐t❤❡r α ❛❧r❡❛❞② s❛t✐s✜❡s ❝❧❛✉s❡s ♦❢ ✇❡✐❣❤t ❛t ❧❡❛st r k µ ( F ) ✱ ♦r µ [ α ] ( F ) < r k µ ( F ) ✳ ■♥ t❤❡ ❧❛tt❡r ❝❛s❡✱ ❛❣❛✐♥ ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ r k ✱ t❤❡r❡ ♠✉st ❜❡ ❛♥ ❛ss✐❣♥♠❡♥t β t❤❛t s❛t✐s✜❡s ♠♦r❡ t❤❛♥ t❤❡ ✇❡✐❣❤t ♦❢ ❝❧❛✉s❡s s❛t✐s✜❡❞ ❜② α ✐♥ F ✳ ❇✉t ❛♥② s✉❝❤ ❛ss✐❣♥♠❡♥t s❛t✐s✜❡s ❝❧❛✉s❡s ♦❢ ✇❡✐❣❤t ❛t ❧❡❛st r k µ ( F ) ❜② t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ǫ ✳ ❊①tr❡♠❛❧ ♣r♦♣❡rt✐❡s✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐