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❍❛s❤ ❋✉♥❝t✐♦♥s ❇❛s❡❞ ♦♥ ❚❤r❡❡ P❡r♠✉t❛t✐♦♥s✿ ❆ ●❡♥❡r✐❝ ❙❡❝✉r✐t② ❆♥❛❧②s✐s ❇❛rt ▼❡♥♥✐♥❦ ❛♥❞ ❇❛rt Pr❡♥❡❡❧ ❑❯ ▲❡✉✈❡♥ ❈❘❨P❚❖ ✷✵✶✷ ✖ ❆✉❣✉st ✷✶✱ ✷✵✶✷ ✶ ✴ ✶✽

❘❡✲❦❡②✐♥❣ r❡❧❛t❡❞✲❦❡② s❡❝✉r✐t②✱ ❡✣❝✐❡♥❝② ❧♦ss✱ ✳ ✳ ✳ ■♥st❡❛❞ ✉s❡ ✜①❡❞✲❦❡② ❜❧♦❝❦ ❝✐♣❤❡rs✱ ♦r ♣❡r♠✉t❛t✐♦♥s ▼♦t✐✈❛t✐♦♥ • ❍❛s❤ ❢✉♥❝t✐♦♥s ❜❛s❡❞ ♦♥ ❜❧♦❝❦ ❝✐♣❤❡rs • ❉❛✈✐❡s✲▼❡②❡r ✬✽✹✱ P●❱ ✬✾✸✱ ❚❛♥❞❡♠✲❉▼ ✬✾✷✱ ✳✳✳ • ▼❉✺ ✬✾✷✱ ❙❍❆✲✶ ✬✾✺✱ ❙❍❆✲✷ ✬✵✶✱ ❇❧❛❦❡ ✬✵✽✱ ❙❦❡✐♥ ✬✵✽✱ ✳✳✳ E F ✷ ✴ ✶✽

■♥st❡❛❞ ✉s❡ ✜①❡❞✲❦❡② ❜❧♦❝❦ ❝✐♣❤❡rs✱ ♦r ♣❡r♠✉t❛t✐♦♥s ▼♦t✐✈❛t✐♦♥ • ❍❛s❤ ❢✉♥❝t✐♦♥s ❜❛s❡❞ ♦♥ ❜❧♦❝❦ ❝✐♣❤❡rs • ❉❛✈✐❡s✲▼❡②❡r ✬✽✹✱ P●❱ ✬✾✸✱ ❚❛♥❞❡♠✲❉▼ ✬✾✷✱ ✳✳✳ • ▼❉✺ ✬✾✷✱ ❙❍❆✲✶ ✬✾✺✱ ❙❍❆✲✷ ✬✵✶✱ ❇❧❛❦❡ ✬✵✽✱ ❙❦❡✐♥ ✬✵✽✱ ✳✳✳ • ❘❡✲❦❡②✐♥❣ − → r❡❧❛t❡❞✲❦❡② s❡❝✉r✐t②✱ ❡✣❝✐❡♥❝② ❧♦ss✱ ✳ ✳ ✳ E F ✷ ✴ ✶✽

▼♦t✐✈❛t✐♦♥ • ❍❛s❤ ❢✉♥❝t✐♦♥s ❜❛s❡❞ ♦♥ ❜❧♦❝❦ ❝✐♣❤❡rs • ❉❛✈✐❡s✲▼❡②❡r ✬✽✹✱ P●❱ ✬✾✸✱ ❚❛♥❞❡♠✲❉▼ ✬✾✷✱ ✳✳✳ • ▼❉✺ ✬✾✷✱ ❙❍❆✲✶ ✬✾✺✱ ❙❍❆✲✷ ✬✵✶✱ ❇❧❛❦❡ ✬✵✽✱ ❙❦❡✐♥ ✬✵✽✱ ✳✳✳ • ❘❡✲❦❡②✐♥❣ − → r❡❧❛t❡❞✲❦❡② s❡❝✉r✐t②✱ ❡✣❝✐❡♥❝② ❧♦ss✱ ✳ ✳ ✳ • ■♥st❡❛❞ ✉s❡ ✜①❡❞✲❦❡② ❜❧♦❝❦ ❝✐♣❤❡rs✱ ♦r ♣❡r♠✉t❛t✐♦♥s E π F F ✷ ✴ ✶✽

●❡♥❡r❛❧✐③❡❞ ❜② ❘♦❣❛✇❛②✲❙t❡✐♥❜❡r❣❡r ✬✵✽✱ ❙t❛♠ ✬✵✽✱ ❙t❡✐♥❜❡r❣❡r ✬✶✵ ✲t♦✲ ✲❜✐t ❢✉♥❝t✐♦♥ ✉s✐♥❣ ✲❜✐t ♣❡r♠✉t❛t✐♦♥s✿ ❝♦❧❧✐s✐♦♥s ✐♥ q✉❡r✐❡s ✭❛❧♠♦st ❛❧✇❛②s✮ ✷ ✸ ✹ ✺ ▼♦t✐✈❛t✐♦♥ • ❇❧❛❝❦✲❈♦❝❤r❛♥✲❙❤r✐♠♣t♦♥ ✬✵✺✿ ♥♦ s❡❝✉r❡ 2 n ✲t♦✲ n ✲❜✐t ❢✉♥❝t✐♦♥ π ✉s✐♥❣ ✶ n ✲❜✐t ♣❡r♠✉t❛t✐♦♥ ❝❛❧❧ F ✸ ✴ ✶✽

▼♦t✐✈❛t✐♦♥ • ❇❧❛❝❦✲❈♦❝❤r❛♥✲❙❤r✐♠♣t♦♥ ✬✵✺✿ ♥♦ s❡❝✉r❡ 2 n ✲t♦✲ n ✲❜✐t ❢✉♥❝t✐♦♥ π ✉s✐♥❣ ✶ n ✲❜✐t ♣❡r♠✉t❛t✐♦♥ ❝❛❧❧ F • ●❡♥❡r❛❧✐③❡❞ ❜② ❘♦❣❛✇❛②✲❙t❡✐♥❜❡r❣❡r ✬✵✽✱ ❙t❛♠ ✬✵✽✱ ❙t❡✐♥❜❡r❣❡r ✬✶✵ • mn ✲t♦✲ rn ✲❜✐t ❢✉♥❝t✐♦♥ ✉s✐♥❣ k n ✲❜✐t ♣❡r♠✉t❛t✐♦♥s✿ ❝♦❧❧✐s✐♦♥s ✐♥ (2 n ) 1 − ( m − r +1) / ( k +1) q✉❡r✐❡s ✭❛❧♠♦st ❛❧✇❛②s✮ F ✷ π ✸ π ✹ π ✺ π 2 n/ 3 2 n/ 2 2 n → n 5 2 n/ 6 2 3 n/ 8 2 n/ 2 2 n → n 2 n/ 4 2 2 n/ 5 2 n/ 2 4 n → 2 n 1 ✸ ✴ ✶✽

▼♦t✐✈❛t✐♦♥ • ❇❧❛❝❦✲❈♦❝❤r❛♥✲❙❤r✐♠♣t♦♥ ✬✵✺✿ ♥♦ s❡❝✉r❡ 2 n ✲t♦✲ n ✲❜✐t ❢✉♥❝t✐♦♥ π ✉s✐♥❣ ✶ n ✲❜✐t ♣❡r♠✉t❛t✐♦♥ ❝❛❧❧ F • ●❡♥❡r❛❧✐③❡❞ ❜② ❘♦❣❛✇❛②✲❙t❡✐♥❜❡r❣❡r ✬✵✽✱ ❙t❛♠ ✬✵✽✱ ❙t❡✐♥❜❡r❣❡r ✬✶✵ • mn ✲t♦✲ rn ✲❜✐t ❢✉♥❝t✐♦♥ ✉s✐♥❣ k n ✲❜✐t ♣❡r♠✉t❛t✐♦♥s✿ ❝♦❧❧✐s✐♦♥s ✐♥ (2 n ) 1 − ( m − r +1) / ( k +1) q✉❡r✐❡s ✭❛❧♠♦st ❛❧✇❛②s✮ F ✷ π ✸ π ✹ π ✺ π 2 n/ 3 2 n/ 2 2 n → n 5 2 n/ 6 2 3 n/ 8 2 n/ 2 2 n → n 2 n/ 4 2 2 n/ 5 2 n/ 2 4 n → 2 n 1 ✸ ✴ ✶✽

s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② ❙❡❝✉r✐t② ▼♦❞❡❧ π i , π − 1 i ✻ q q✉❡r✐❡s ❄ ❛❞✈❡rs❛r② A • ■❞❡❛❧ ♣❡r♠✉t❛t✐♦♥ ♠♦❞❡❧✿ π i ✬s r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞ • ❆❞✈❡rs❛r② q✉❡r② ❛❝❝❡ss t♦ π i ✬s ✹ ✴ ✶✽

s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② ❙❡❝✉r✐t② ▼♦❞❡❧ π i , π − 1 i ✻ q q✉❡r✐❡s ❄ ❞✐st✐♥❝t ( x 1 , x 2 ) , ( x ′ 2 ) s✳t✳ 1 , x ′ ✲ ❛❞✈❡rs❛r② A F ( x 1 , x 2 ) = F ( x ′ 2 ) 1 , x ′ • ■❞❡❛❧ ♣❡r♠✉t❛t✐♦♥ ♠♦❞❡❧✿ π i ✬s r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞ • ❆❞✈❡rs❛r② q✉❡r② ❛❝❝❡ss t♦ π i ✬s Adv col F ( q ) = max s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② A A ✹ ✴ ✶✽

❙❡❝✉r✐t② ▼♦❞❡❧ π i , π − 1 i ✻ q q✉❡r✐❡s ❄ ✲ ✲ z ∈ { 0 , 1 } n ❛❞✈❡rs❛r② A ( x 1 , x 2 ) s✳t✳ F ( x 1 , x 2 ) = z • ■❞❡❛❧ ♣❡r♠✉t❛t✐♦♥ ♠♦❞❡❧✿ π i ✬s r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞ • ❆❞✈❡rs❛r② q✉❡r② ❛❝❝❡ss t♦ π i ✬s Adv col F ( q ) = max s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② A A Adv epre ( q ) = max z ∈{ 0 , 1 } n s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② A max F A ✹ ✴ ✶✽

Pr✐♦r ❈♦♥str✉❝t✐♦♥s ✖ ❙❤r✐♠♣t♦♥✲❙t❛♠ ✬✵✽ f 1 x 1 n f 2 f 3 x 2 z n n • 2 n ✲t♦✲ n ✲❜✐t ❢✉♥❝t✐♦♥ ✉s✐♥❣ 3 ♦♥❡✲✇❛② ❢✉♥❝t✐♦♥s • ❖♣t✐♠❛❧ ❝♦❧❧✐s✐♦♥ s❡❝✉r✐t② • ❈♦❧❧✐s✐♦♥ s❡❝✉r✐t② ✐❢ f i ( x ) = π i ( x ) ⊕ x ✭s❤♦✇❡❞ ❜② ❛✉t♦♠❛t❡❞ ❛♥❛❧②s✐s✮ ✺ ✴ ✶✽

❈♦❧❧✐s✐♦♥✴♣r❡✐♠❛❣❡ s❡❝✉r✐t② ✐❢ s❛t✐s❢② ✏✐♥❞❡♣❡♥❞❡♥❝❡ ❝r✐t❡r✐♦♥✑ ❊①❝❧✉❞❡s ❜✐♥❛r② Pr✐♦r ❈♦♥str✉❝t✐♦♥s ✖ ❘♦❣❛✇❛②✲❙t❡✐♥❜❡r❣❡r ✬✵✽ x 1 n a 11 a 21 a 31 a 41 x 2 n a 12 a 22 a 32 a 42 π 1 a 23 a 33 a 43 π 2 a 34 a 44 π 3 a 45 z n • 2 n ✲t♦✲ n ✲❜✐t ❢✉♥❝t✐♦♥ ✭♦✈❡r F 2 n ✮ ✉s✐♥❣ 3 ♣❡r♠✉t❛t✐♦♥s ✻ ✴ ✶✽

Pr✐♦r ❈♦♥str✉❝t✐♦♥s ✖ ❘♦❣❛✇❛②✲❙t❡✐♥❜❡r❣❡r ✬✵✽ x 1 n a 11 a 21 a 31 a 41 x 2 n a 12 a 22 a 32 a 42 π 1 a 23 a 33 a 43 π 2 a 34 a 44 π 3 a 45 z n • 2 n ✲t♦✲ n ✲❜✐t ❢✉♥❝t✐♦♥ ✭♦✈❡r F 2 n ✮ ✉s✐♥❣ 3 ♣❡r♠✉t❛t✐♦♥s • ❈♦❧❧✐s✐♦♥✴♣r❡✐♠❛❣❡ s❡❝✉r✐t② ✐❢ a ij s❛t✐s❢② ✏✐♥❞❡♣❡♥❞❡♥❝❡ ❝r✐t❡r✐♦♥✑ − → ❊①❝❧✉❞❡s ❜✐♥❛r② a ij ✻ ✴ ✶✽

▼✉❧t✐✲♣❡r♠✉t❛t✐♦♥ s❡tt✐♥❣✿ ✬s ❛❧❧ ❞✐✛❡r❡♥t ❙✐♥❣❧❡✲♣❡r♠✉t❛t✐♦♥ s❡tt✐♥❣✿ ❖✉r ❈♦♠♣r❡ss✐♦♥ ❋✉♥❝t✐♦♥ ❉❡s✐❣♥ • 2 n ✲t♦✲ n ❝♦♠♣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ✉s✐♥❣ ♣❡r♠✉t❛t✐♦♥s ❛♥❞ � ✲♦♣❡r❛t♦rs x 1 n a 21 a 11 a 31 a 41 x 2 n a 12 a 22 a 32 a 42 π 1 a 23 a 33 a 43 π 2 a 34 a 44 π 3 a 45 a ij ∈ { 0 , 1 } z n ✼ ✴ ✶✽

❖✉r ❈♦♠♣r❡ss✐♦♥ ❋✉♥❝t✐♦♥ ❉❡s✐❣♥ • 2 n ✲t♦✲ n ❝♦♠♣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ✉s✐♥❣ ♣❡r♠✉t❛t✐♦♥s ❛♥❞ � ✲♦♣❡r❛t♦rs x 1 n a 21 a 11 a 31 a 41 x 2 n a 12 a 22 a 32 a 42 π 1 a 23 a 33 a 43 π 2 a 34 a 44 π 3 a 45 a ij ∈ { 0 , 1 } z n • ▼✉❧t✐✲♣❡r♠✉t❛t✐♦♥ s❡tt✐♥❣✿ π i ✬s ❛❧❧ ❞✐✛❡r❡♥t • ❙✐♥❣❧❡✲♣❡r♠✉t❛t✐♦♥ s❡tt✐♥❣✿ π 1 = π 2 = π 3 ✼ ✴ ✶✽

x 1 n a 11 a 21 a 31 a 41 x 2 n a 12 a 22 a 32 a 42 π 1 a 23 a 33 a 43 π 2 a 34 a 44 π 3 a 45 z n ✽ ✴ ✶✽

x 1 n a 11 a 21 a 31 a 41 x 2 n a 12 a 22 a 32 a 42 π 1 a 23 a 33 a 43 π 2 a 34 a 44 π 3 a 45 x 1 n z a 12 a 22 a 32 a 42 n x 2 n a 11 a 21 a 31 a 41 π 1 a 23 a 33 a 43 π 2 a 34 a 44 π 3 a 45 z n ✽ ✴ ✶✽

x 1 n a 11 a 21 a 31 a 41 x 2 n a 12 a 22 a 32 a 42 π 1 a 23 a 33 a 43 π 2 a 34 a 44 π 3 a 45 x 2 n z a 12 a 22 a 32 a 42 n x 1 n a 11 a 21 a 31 a 41 π 1 a 23 a 33 a 43 π 2 a 34 a 44 π 3 a 45 z n ✽ ✴ ✶✽