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■♥tr♦❞✉❝t✐♦♥ ▲✐♥❡❛r✐③❛t✐♦♥s ◗✉❛❞r❛t✐③❛t✐♦♥s P❡rs♣❡❝t✐✈❡s ▲✐♥❡❛r✐③❛t✐♦♥ ❛♥❞ q✉❛❞r❛t✐③❛t✐♦♥ ❛♣♣r♦❛❝❤❡s ❢♦r ♥♦♥✲❧✐♥❡❛r ✵✲✶ ♦♣t✐♠✐③❛t✐♦♥ ❊❧✐s❛❜❡t❤ ❘♦❞r✐❣✉❡③✲❍❡❝❦ ❛♥❞ ❨✈❡s ❈r❛♠❛ ◗✉❛♥t❖▼✱ ❍❊❈ ▼❛♥❛❣❡♠❡♥t ❙❝❤♦♦❧✱ ❯♥✐✈❡rs✐t② ♦❢ ▲✐è❣❡ P❛rt✐❛❧❧② s✉♣♣♦rt❡❞ ❜② ❇❡❧s♣♦ ✲ ■❆P Pr♦❥❡❝t ❈❖▼❊❳ ❏✉♥❡ ✷✵✶✺✱ ❚❯ ❉♦rt♠✉♥❞

■♥tr♦❞✉❝t✐♦♥ ▲✐♥❡❛r✐③❛t✐♦♥s ◗✉❛❞r❛t✐③❛t✐♦♥s P❡rs♣❡❝t✐✈❡s ❉❡✜♥✐t✐♦♥s ❉❡✜♥✐t✐♦♥✿ Ps❡✉❞♦✲❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❆ ♣s❡✉❞♦✲❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ✐s ❛ ♠❛♣♣✐♥❣ f : { ✵ , ✶ } n → R ✳ ▼✉❧t✐❧✐♥❡❛r r❡♣r❡s❡♥t❛t✐♦♥ ❊✈❡r② ♣s❡✉❞♦✲❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ f ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✉♥✐q✉❡❧② ❜② ❛ ♠✉❧t✐❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ✭❍❛♠♠❡r✱ ❘♦s❡♥❜❡r❣✱ ❘✉❞❡❛♥✉ ❬✹❪✮✳ ❊①❛♠♣❧❡✿ f ( x ✶ , x ✷ , x ✸ ) = ✾ x ✶ x ✷ x ✸ + ✽ x ✶ x ✷ − ✻ x ✷ x ✸ + x ✶ − ✷ x ✷ + x ✸ ✶ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ ▲✐♥❡❛r✐③❛t✐♦♥s ◗✉❛❞r❛t✐③❛t✐♦♥s P❡rs♣❡❝t✐✈❡s ❆♣♣❧✐❝❛t✐♦♥s ❈♦♠♣✉t❡r ✈✐s✐♦♥✿ ✐♠❛❣❡ r❡st♦r❛t✐♦♥ ❙✉♣♣❧② ❈❤❛✐♥ ❉❡s✐❣♥ ✇✐t❤ ❙t♦❝❤❛st✐❝ ■♥✈❡♥t♦r② ▼❛♥❛❣❡♠❡♥t ✭❥♦✐♥t ♠♦❞❡❧ ♦❢ ❋✳ ❨♦✉✱ ■✳ ❊✳ ●r♦ss♠❛♥✮ ❬✻❪ ✷ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ ▲✐♥❡❛r✐③❛t✐♦♥s ◗✉❛❞r❛t✐③❛t✐♦♥s P❡rs♣❡❝t✐✈❡s Ps❡✉❞♦✲❇♦♦❧❡❛♥ ❖♣t✐♠✐③❛t✐♦♥ ▼❛♥② ♣r♦❜❧❡♠s ❢♦r♠✉❧❛t❡❞ ❛s ♦♣t✐♠✐③❛t✐♦♥ ♦❢ ❛ ♣s❡✉❞♦✲❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ Ps❡✉❞♦✲❇♦♦❧❡❛♥ ❖♣t✐♠✐③❛t✐♦♥ x ∈{ ✵ , ✶ } n f ( x ) ♠✐♥ ❖♣t✐♠✐③❛t✐♦♥ ✐s NP ✲❤❛r❞ ✱ ❡✈❡♥ ✐❢ f ✐s q✉❛❞r❛t✐❝ ✭▼❆❳✲✷✲❙❆❚✱ ▼❆❳✲❈❯❚ ♠♦❞❡❧❧❡❞ ❜② q✉❛❞r❛t✐❝ f ✮✳ ❆♣♣r♦❛❝❤❡s✿ ▲✐♥❡❛r✐③❛t✐♦♥ ✿ st❛♥❞❛r❞ ❛♣♣r♦❛❝❤ t♦ s♦❧✈❡ ♥♦♥✲❧✐♥❡❛r ♦♣t✐♠✐③❛t✐♦♥✳ ◗✉❛❞r❛t✐③❛t✐♦♥ ✿ ▼✉❝❤ ♣r♦❣r❡ss ❤❛s ❜❡❡♥ ❞♦♥❡ ❢♦r t❤❡ q✉❛❞r❛t✐❝ ❝❛s❡ ✭❡①❛❝t ❛❧❣♦r✐t❤♠s✱ ❤❡✉r✐st✐❝s✱ ♣♦❧②❤❡❞r❛❧ r❡s✉❧ts✳✳✳✮✳ ✸ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ ▲✐♥❡❛r✐③❛t✐♦♥s ◗✉❛❞r❛t✐③❛t✐♦♥s P❡rs♣❡❝t✐✈❡s Ps❡✉❞♦✲❇♦♦❧❡❛♥ ❖♣t✐♠✐③❛t✐♦♥ ▼❛♥② ♣r♦❜❧❡♠s ❢♦r♠✉❧❛t❡❞ ❛s ♦♣t✐♠✐③❛t✐♦♥ ♦❢ ❛ ♣s❡✉❞♦✲❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ Ps❡✉❞♦✲❇♦♦❧❡❛♥ ❖♣t✐♠✐③❛t✐♦♥ x ∈{ ✵ , ✶ } n f ( x ) ♠✐♥ ❖♣t✐♠✐③❛t✐♦♥ ✐s NP ✲❤❛r❞ ✱ ❡✈❡♥ ✐❢ f ✐s q✉❛❞r❛t✐❝ ✭▼❆❳✲✷✲❙❆❚✱ ▼❆❳✲❈❯❚ ♠♦❞❡❧❧❡❞ ❜② q✉❛❞r❛t✐❝ f ✮✳ ❆♣♣r♦❛❝❤❡s✿ ▲✐♥❡❛r✐③❛t✐♦♥ ✿ st❛♥❞❛r❞ ❛♣♣r♦❛❝❤ t♦ s♦❧✈❡ ♥♦♥✲❧✐♥❡❛r ♦♣t✐♠✐③❛t✐♦♥✳ ◗✉❛❞r❛t✐③❛t✐♦♥ ✿ ▼✉❝❤ ♣r♦❣r❡ss ❤❛s ❜❡❡♥ ❞♦♥❡ ❢♦r t❤❡ q✉❛❞r❛t✐❝ ❝❛s❡ ✭❡①❛❝t ❛❧❣♦r✐t❤♠s✱ ❤❡✉r✐st✐❝s✱ ♣♦❧②❤❡❞r❛❧ r❡s✉❧ts✳✳✳✮✳ ✸ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ ▲✐♥❡❛r✐③❛t✐♦♥s ◗✉❛❞r❛t✐③❛t✐♦♥s P❡rs♣❡❝t✐✈❡s ▲✐♥❡❛r✐③❛t✐♦♥s✿ r❡❞✉❝t✐♦♥s t♦ t❤❡ ❧✐♥❡❛r ❝❛s❡

✷✳ ▲✐♥❡❛r✐③❡ ❝♦♥str❛✐♥ts ♠✐♥ s✳t✳ ✶ ✵ ✶ ✵ ✶ ✶ ■♥tr♦❞✉❝t✐♦♥ ▲✐♥❡❛r✐③❛t✐♦♥s ◗✉❛❞r❛t✐③❛t✐♦♥s P❡rs♣❡❝t✐✈❡s ❙t❛♥❞❛r❞ ❧✐♥❡❛r✐③❛t✐♦♥ ✭❙▲✮ � � ♠✐♥ a S x k , { ✵ , ✶ } n S ∈S k ∈ S S = { S ⊆ { ✶ , . . . , n } | a S � = ✵ } ✭♥♦♥✲❝♦♥st❛♥t ♠♦♥♦♠✐❛❧s✮ ✶✳ ❙✉❜st✐t✉t❡ ♠♦♥♦♠✐❛❧s � ♠✐♥ a S z S S ∈S � s✳t✳ z S = x k , ∀ S ∈ S k ∈ S z S ∈ { ✵ , ✶ } , ∀ S ∈ S x k ∈ { ✵ , ✶ } , ∀ k = ✶ , . . . , n ✹ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ ▲✐♥❡❛r✐③❛t✐♦♥s ◗✉❛❞r❛t✐③❛t✐♦♥s P❡rs♣❡❝t✐✈❡s ❙t❛♥❞❛r❞ ❧✐♥❡❛r✐③❛t✐♦♥ ✭❙▲✮ � � ♠✐♥ a S x k , { ✵ , ✶ } n S ∈S k ∈ S S = { S ⊆ { ✶ , . . . , n } | a S � = ✵ } ✭♥♦♥✲❝♦♥st❛♥t ♠♦♥♦♠✐❛❧s✮ ✷✳ ▲✐♥❡❛r✐③❡ ❝♦♥str❛✐♥ts ✶✳ ❙✉❜st✐t✉t❡ ♠♦♥♦♠✐❛❧s � � ♠✐♥ ♠✐♥ a S z S a S z S S ∈S S ∈S s✳t✳ z S ≤ x k , ∀ k ∈ S , ∀ S ∈ S � s✳t✳ z S = x k , ∀ S ∈ S � k ∈ S z S ≥ x k − ( | S | − ✶ ) , ∀ S ∈ S k ∈ S z S ∈ { ✵ , ✶ } , ∀ S ∈ S z S ∈ { ✵ , ✶ } , ∀ S ∈ S x k ∈ { ✵ , ✶ } , ∀ k = ✶ , . . . , n x k ∈ { ✵ , ✶ } , ∀ k = ✶ , . . . , n ✹ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ ▲✐♥❡❛r✐③❛t✐♦♥s ◗✉❛❞r❛t✐③❛t✐♦♥s P❡rs♣❡❝t✐✈❡s ❙t❛♥❞❛r❞ ❧✐♥❡❛r✐③❛t✐♦♥ ✭❙▲✮ � � ♠✐♥ a S x k , { ✵ , ✶ } n S ∈S k ∈ S S = { S ⊆ { ✶ , . . . , n } | a S � = ✵ } ✭♥♦♥✲❝♦♥st❛♥t ♠♦♥♦♠✐❛❧s✮ ✷✳ ▲✐♥❡❛r✐③❡ ❝♦♥str❛✐♥ts ✶✳ ❙✉❜st✐t✉t❡ ♠♦♥♦♠✐❛❧s � � ♠✐♥ ♠✐♥ a S z S a S z S S ∈S S ∈S s✳t✳ z S ≤ x k , ∀ k ∈ S , ∀ S ∈ S � s✳t✳ z S = x k , ∀ S ∈ S � k ∈ S z S ≥ x k − ( | S | − ✶ ) , ∀ S ∈ S k ∈ S z S ∈ { ✵ , ✶ } , ∀ S ∈ S z S ∈ { ✵ , ✶ } , ∀ S ∈ S x k ∈ { ✵ , ✶ } , ∀ k = ✶ , . . . , n x k ∈ { ✵ , ✶ } , ∀ k = ✶ , . . . , n ✹ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ ▲✐♥❡❛r✐③❛t✐♦♥s ◗✉❛❞r❛t✐③❛t✐♦♥s P❡rs♣❡❝t✐✈❡s ❙t❛♥❞❛r❞ ❧✐♥❡❛r✐③❛t✐♦♥ ✭❙▲✮ � � ♠✐♥ a S x k , { ✵ , ✶ } n S ∈S k ∈ S S = { S ⊆ { ✶ , . . . , n } | a S � = ✵ } ✭♥♦♥✲❝♦♥st❛♥t ♠♦♥♦♠✐❛❧s✮ ✶✳ ❙✉❜st✐t✉t❡ ♠♦♥♦♠✐❛❧s ✸✳ ▲✐♥❡❛r r❡❧❛①❛t✐♦♥ � � ♠✐♥ ♠✐♥ a S z S a S z S S ∈S S ∈S s✳t✳ z S ≤ x k , ∀ k ∈ S , ∀ S ∈ S � s✳t✳ z S = ∀ S ∈ S x k , k ∈ S � z S ≥ x k − ( | S | − ✶ ) , ∀ S ∈ S k ∈ S z S ∈ { ✵ , ✶ } , ∀ S ∈ S ✵ ≤ z S ≤ ✶ , ∀ S ∈ S x k ∈ { ✵ , ✶ } , ∀ k = ✶ , . . . , n ✵ ≤ x k ≤ ✶ , ∀ k = ✶ , . . . , n ✹ ✴ ✸✷

■❙ ❧✐♥❡❛r✐③❛t✐♦♥ ✶ ■♥tr♦❞✉❝t✐♦♥ ▲✐♥❡❛r✐③❛t✐♦♥s ◗✉❛❞r❛t✐③❛t✐♦♥s P❡rs♣❡❝t✐✈❡s ■♥t❡r♠❡❞✐❛t❡ s✉❜st✐t✉t✐♦♥s ✭■❙✮ ✭♦♥❡ ♠♦♥♦♠✐❛❧✮ ❙▲ ❧✐♥❡❛r✐③❛t✐♦♥ ❙▲ s✉❜st✐t✉t✐♦♥ � z S ≤ x k , ∀ k ∈ S z S = x k k ∈ S � z S ≥ x k − ( | S | − ✶ ) k ∈ S ■❙ s✉❜st✐t✉t✐♦♥ � z S = z A x k k ∈ S \ A � z A = x k k ∈ A ✺ ✴ ✸✷

■♥tr♦❞✉❝t✐♦♥ ▲✐♥❡❛r✐③❛t✐♦♥s ◗✉❛❞r❛t✐③❛t✐♦♥s P❡rs♣❡❝t✐✈❡s ■♥t❡r♠❡❞✐❛t❡ s✉❜st✐t✉t✐♦♥s ✭■❙✮ ✭♦♥❡ ♠♦♥♦♠✐❛❧✮ ❙▲ ❧✐♥❡❛r✐③❛t✐♦♥ ❙▲ s✉❜st✐t✉t✐♦♥ � z S ≤ x k , ∀ k ∈ S z S = x k k ∈ S � z S ≥ x k − ( | S | − ✶ ) k ∈ S ■❙ ❧✐♥❡❛r✐③❛t✐♦♥ ■❙ s✉❜st✐t✉t✐♦♥ � z S ≤ x k , ∀ k ∈ S \ A z S = z A x k z S ≤ z A , k ∈ S \ A � z A = � x k z S ≥ z A + x k − | S \ A | , k ∈ A k ∈ S \ A z A ≤ x k , ∀ k ∈ A � z A ≥ x k − ( | A | − ✶ ) . k ∈ A ✺ ✴ ✸✷