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❙♦❧✈✐♥❣ ❜❛s✐s ♣✉rs✉✐t✿ ✐♥❢❡❛s✐❜❧❡✲♣♦✐♥t s✉❜❣r❛❞✐❡♥t ❛❧❣♦r✐t❤♠✱ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❛r✐s♦♥✱ ❛♥❞ ✐♠♣r♦✈❡♠❡♥ts ❆♥❞r❡❛s ❚✐❧❧♠❛♥♥ ✭ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❉✳ ▲♦r❡♥③ ❛♥❞ ▼✳ P❢❡ts❝❤ ✮ ❚❡❝❤♥✐s❝❤❡ ❯♥✐✈❡rs✐tät ❇r❛✉♥s❝❤✇❡✐❣ ❙■❆▼ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❆♣♣❧✐❡❞ ▲✐♥❡❛r ❆❧❣❡❜r❛ ✷✵✶✷ ❆✳ ❚✐❧❧♠❛♥♥ ✶ ✴ ✷✷ ❚❯ ❇r❛✉♥s❝❤✇❡✐❣

❖✉t❧✐♥❡ ✶ ▼♦t✐✈❛t✐♦♥ ✷ ■♥❢❡❛s✐❜❧❡✲P♦✐♥t ❙✉❜❣r❛❞✐❡♥t ❆❧❣♦r✐t❤♠ ■❙❆▲✶ ✸ ❈♦♠♣❛r✐s♦♥ ♦❢ ℓ ✶ ✲❙♦❧✈❡rs ❚❡sts❡t ❈♦♥str✉❝t✐♦♥ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥❛❧ ❘❡s✉❧ts ■♠♣r♦✈❡♠❡♥ts ✇✐t❤ ❍❡✉r✐st✐❝ ❖♣t✐♠❛❧✐t② ❈❤❡❝❦ ✹ P♦ss✐❜❧❡ ❋✉t✉r❡ ❘❡s❡❛r❝❤ ❆✳ ❚✐❧❧♠❛♥♥ ✷ ✴ ✷✷ ❚❯ ❇r❛✉♥s❝❤✇❡✐❣

▼♦t✐✈❛t✐♦♥ ❖✉t❧✐♥❡ ✶ ▼♦t✐✈❛t✐♦♥ ✷ ■♥❢❡❛s✐❜❧❡✲P♦✐♥t ❙✉❜❣r❛❞✐❡♥t ❆❧❣♦r✐t❤♠ ■❙❆▲✶ ✸ ❈♦♠♣❛r✐s♦♥ ♦❢ ℓ ✶ ✲❙♦❧✈❡rs ❚❡sts❡t ❈♦♥str✉❝t✐♦♥ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥❛❧ ❘❡s✉❧ts ■♠♣r♦✈❡♠❡♥ts ✇✐t❤ ❍❡✉r✐st✐❝ ❖♣t✐♠❛❧✐t② ❈❤❡❝❦ ✹ P♦ss✐❜❧❡ ❋✉t✉r❡ ❘❡s❡❛r❝❤ ❆✳ ❚✐❧❧♠❛♥♥ ✸ ✴ ✷✷ ❚❯ ❇r❛✉♥s❝❤✇❡✐❣

▼♦t✐✈❛t✐♦♥ ❙♣❛rs❡ ❘❡❝♦✈❡r② ✈✐❛ ℓ ✶ ✲▼✐♥✐♠✐③❛t✐♦♥ ❙❡❡❦ s♣❛rs❡st s♦❧✉t✐♦♥ t♦ ✉♥❞❡r❞❡t❡r♠✐♥❡❞ ❧✐♥❡❛r s②st❡♠✿ ( ❆ ∈ R ♠ × ♥ , ♠ < ♥ ) ♠✐♥ � ① � ✵ s✳ t✳ ❆① = ❜ ❋✐♥❞✐♥❣ ♠✐♥✐♠✉♠✲s✉♣♣♦rt s♦❧✉t✐♦♥ ✐s NP ✲❤❛r❞✳ ❈♦♥✈❡① ✏r❡❧❛①❛t✐♦♥✑✿ ℓ ✶ ✲♠✐♥✐♠✐③❛t✐♦♥ ✴ ❇❛s✐s P✉rs✉✐t✿ ♠✐♥ � ① � ✶ s✳ t✳ ❆① = ❜ ✭▲✶✮ ❙❡✈❡r❛❧ ❝♦♥❞✐t✐♦♥s ✭❘■P✱ ◆✉❧❧s♣❛❝❡ Pr♦♣❡rt②✱ ❡t❝✮ ❡♥s✉r❡ ✏ ℓ ✵ ✲ ℓ ✶ ✲❡q✉✐✈❛❧❡♥❝❡✑ ❆✳ ❚✐❧❧♠❛♥♥ ✹ ✴ ✷✷ ❚❯ ❇r❛✉♥s❝❤✇❡✐❣

▼♦t✐✈❛t✐♦♥ ❙♦❧✈✐♥❣ t❤❡ ❇❛s✐s P✉rs✉✐t Pr♦❜❧❡♠ ✭▲✶✮ ❝❛♥ ❜❡ r❡❝❛st ❛s ❛ ❧✐♥❡❛r ♣r♦❣r❛♠ ❇r♦❛❞ ✈❛r✐❡t② ♦❢ s♣❡❝✐❛❧✐③❡❞ ❛❧❣♦r✐t❤♠s ❢♦r ✭▲✶✮ � � ❆ ❝❧❛ss✐❝ ❛❧❣♦r✐t❤♠ ❢r♦♠ ♥♦♥s♠♦♦t❤ ♦♣t✐♠✐③❛t✐♦♥✿ ✭♣r♦❥❡❝t❡❞✮ s✉❜❣r❛❞✐❡♥t ♠❡t❤♦❞ ✕ ❝♦♠♣❡t✐t✐✈❡❄ ❲❤✐❝❤ ❛❧❣♦r✐t❤♠ ✐s ✏t❤❡ ❜❡st✑❄ ❆✳ ❚✐❧❧♠❛♥♥ ✺ ✴ ✷✷ ❚❯ ❇r❛✉♥s❝❤✇❡✐❣

■♥❢❡❛s✐❜❧❡✲P♦✐♥t ❙✉❜❣r❛❞✐❡♥t ❆❧❣♦r✐t❤♠ ❖✉t❧✐♥❡ ✶ ▼♦t✐✈❛t✐♦♥ ✷ ■♥❢❡❛s✐❜❧❡✲P♦✐♥t ❙✉❜❣r❛❞✐❡♥t ❆❧❣♦r✐t❤♠ ■❙❆▲✶ ✸ ❈♦♠♣❛r✐s♦♥ ♦❢ ℓ ✶ ✲❙♦❧✈❡rs ❚❡sts❡t ❈♦♥str✉❝t✐♦♥ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥❛❧ ❘❡s✉❧ts ■♠♣r♦✈❡♠❡♥ts ✇✐t❤ ❍❡✉r✐st✐❝ ❖♣t✐♠❛❧✐t② ❈❤❡❝❦ ✹ P♦ss✐❜❧❡ ❋✉t✉r❡ ❘❡s❡❛r❝❤ ❆✳ ❚✐❧❧♠❛♥♥ ✻ ✴ ✷✷ ❚❯ ❇r❛✉♥s❝❤✇❡✐❣

✐❞❡❛✿ r❡♣❧❛❝❡ ❡①❛❝t ♣r♦❥❡❝t✐♦♥ ❜② ❛♣♣r♦①✐♠❛t✐♦♥ ✏✐♥❢❡❛s✐❜❧❡✑ s✉❜❣r❛❞✐❡♥t ✐t❡r❛t✐♦♥ ① ❦ ✶ ① ❦ ❦ ❤ ❦ ❦ ❦ ✷ ❦ ■♥❢❡❛s✐❜❧❡✲P♦✐♥t ❙✉❜❣r❛❞✐❡♥t ❆❧❣♦r✐t❤♠ Pr♦❥❡❝t❡❞ ❙✉❜❣r❛❞✐❡♥t ▼❡t❤♦❞s Pr♦❜❧❡♠✿ ♠✐♥ ❢ ( ① ) s✳t✳ ① ∈ F ✭ ❢ ✱ F ❝♦♥✈❡①✮ st❛♥❞❛r❞ ♣r♦❥❡❝t❡❞ s✉❜❣r❛❞✐❡♥t ✐t❡r❛t✐♦♥ α ❦ > ✵ , ❤ ❦ ∈ ∂ ❢ ( ① ❦ ) ① ❦ + ✶ = P F � ① ❦ − α ❦ ❤ ❦ � , ❛♣♣❧✐❝❛❜✐❧✐t②✿ ♦♥❧② r❡❛s♦♥❛❜❧❡ ✐❢ ♣r♦❥❡❝t✐♦♥ ✐s ✏❡❛s②✑ ❆✳ ❚✐❧❧♠❛♥♥ ✼ ✴ ✷✷ ❚❯ ❇r❛✉♥s❝❤✇❡✐❣

■♥❢❡❛s✐❜❧❡✲P♦✐♥t ❙✉❜❣r❛❞✐❡♥t ❆❧❣♦r✐t❤♠ Pr♦❥❡❝t❡❞ ❙✉❜❣r❛❞✐❡♥t ▼❡t❤♦❞s Pr♦❜❧❡♠✿ ♠✐♥ ❢ ( ① ) s✳t✳ ① ∈ F ✭ ❢ ✱ F ❝♦♥✈❡①✮ st❛♥❞❛r❞ ♣r♦❥❡❝t❡❞ s✉❜❣r❛❞✐❡♥t ✐t❡r❛t✐♦♥ α ❦ > ✵ , ❤ ❦ ∈ ∂ ❢ ( ① ❦ ) ① ❦ + ✶ = P F � ① ❦ − α ❦ ❤ ❦ � , ❛♣♣❧✐❝❛❜✐❧✐t②✿ ♦♥❧② r❡❛s♦♥❛❜❧❡ ✐❢ ♣r♦❥❡❝t✐♦♥ ✐s ✏❡❛s②✑ ✐❞❡❛✿ r❡♣❧❛❝❡ ❡①❛❝t ♣r♦❥❡❝t✐♦♥ ❜② ❛♣♣r♦①✐♠❛t✐♦♥ � ✏✐♥❢❡❛s✐❜❧❡✑ s✉❜❣r❛❞✐❡♥t ✐t❡r❛t✐♦♥ ① ❦ + ✶ = P ε ❦ ① ❦ − α ❦ ❤ ❦ � � �P ε ❦ , F − P F � ✷ ≤ ε ❦ F ❆✳ ❚✐❧❧♠❛♥♥ ✼ ✴ ✷✷ ❚❯ ❇r❛✉♥s❝❤✇❡✐❣

■♥❢❡❛s✐❜❧❡✲P♦✐♥t ❙✉❜❣r❛❞✐❡♥t ❆❧❣♦r✐t❤♠ ■❙❆▲✶ ■❙❆ ❂ ■♥❢❡❛s✐❜❧❡✲P♦✐♥t ❙✉❜❣r❛❞✐❡♥t ❆❧❣♦r✐t❤♠ ✳✳✳ ✇♦r❦s ❢♦r ❛r❜✐tr❛r② ❝♦♥✈❡① ♦❜❥❡❝t✐✈❡s ❛♥❞ ❝♦♥str❛✐♥t s❡ts ✳✳✳ ✐♥❝♦r♣♦r❛t❡s ❛❞❛♣t✐✈❡ ❛♣♣r♦①✐♠❛t❡ ♣r♦❥❡❝t✐♦♥s P ε F s✉❝❤ t❤❛t �P ε F ( ② ) − P F ( ② ) � ✷ ≤ ε ❢♦r ❡✈❡r② ε ≥ ✵ ✳✳✳ ❝♦♥✈❡r❣❡s t♦ ♦♣t✐♠❛❧✐t② ✭✉♥❞❡r r❡❛s♦♥❛❜❧❡ ❛ss✉♠♣t✐♦♥s✮ ✇❤❡♥❡✈❡r ♣r♦❥❡❝t✐♦♥ ❛❝❝✉r❛❝✐❡s ( ε ❦ ) s✉✣❝✐❡♥t❧② s♠❛❧❧✱ ❢♦r st❡♣s✐③❡s α ❦ > ✵ ✇✐t❤ � ∞ ❦ = ✵ α ❦ = ∞ ✱ � ∞ ❦ = ✵ α ✷ ❦ < ∞ / � ❤ ❦ � ✷ � ❢ ( ① ❦ ) − ϕ � ✷ ✇✐t❤ ϕ ≤ ϕ ∗ ❢♦r ❞②♥❛♠✐❝ st❡♣s✐③❡s α ❦ = λ ❦ ✳✳✳ ❝♦♥✈❡r❣❡s t♦ ϕ ✇✐t❤ ❞②♥❛♠✐❝ st❡♣s✐③❡s ✉s✐♥❣ ϕ ≥ ϕ ∗ ❆✳ ❚✐❧❧♠❛♥♥ ✽ ✴ ✷✷ ❚❯ ❇r❛✉♥s❝❤✇❡✐❣

❆❆ ❚ ✐s s✳♣✳❞✳ ♠❛② ❡♠♣❧♦② ❈● t♦ s♦❧✈❡ ❡q✉❛t✐♦♥ s②st❡♠ ❆♣♣r♦①✐♠❛t✐♦♥✿ ❙t♦♣ ❛❢t❡r ❛ ❢❡✇ ❈● ✐t❡r❛t✐♦♥s ❦ ✜ts ■❙❆ ❢r❛♠❡✇♦r❦✮ ✭❈● r❡s✐❞✉❛❧ ♥♦r♠ ♠✐♥ ❆ ❦ ■♥❢❡❛s✐❜❧❡✲P♦✐♥t ❙✉❜❣r❛❞✐❡♥t ❆❧❣♦r✐t❤♠ ■❙❆▲✶ ■❙❆▲✶ ❂ ❙♣❡③✐❛❧✐③❛t✐♦♥ ♦❢ ■❙❆ t♦ ℓ ✶ ✲▼✐♥✐♠✐③❛t✐♦♥ ❢ ( ① ) = � ① � ✶ ✱ F = { ① | ❆① = ❜ } ✱ s✐❣♥ ( ① ) ∈ ∂ � ① � ✶ ❡①❛❝t ♣r♦❥❡❝t❡❞ s✉❜❣r❛❞✐❡♥t st❡♣ ❢♦r ✭▲✶✮✿ ① ❦ − α ❦ ❤ ❦ � ① ❦ + ✶ = P F � = ( ① ❦ − α ❦ ❤ ❦ ) − ❆ ❚ ( ❆❆ ❚ ) − ✶ � ❆ ( ① ❦ − α ❦ ❤ ❦ ) − ❜ � ❆✳ ❚✐❧❧♠❛♥♥ ✾ ✴ ✷✷ ❚❯ ❇r❛✉♥s❝❤✇❡✐❣

❆♣♣r♦①✐♠❛t✐♦♥✿ ❙t♦♣ ❛❢t❡r ❛ ❢❡✇ ❈● ✐t❡r❛t✐♦♥s ❦ ✜ts ■❙❆ ❢r❛♠❡✇♦r❦✮ ✭❈● r❡s✐❞✉❛❧ ♥♦r♠ ♠✐♥ ❆ ❦ ■♥❢❡❛s✐❜❧❡✲P♦✐♥t ❙✉❜❣r❛❞✐❡♥t ❆❧❣♦r✐t❤♠ ■❙❆▲✶ ■❙❆▲✶ ❂ ❙♣❡③✐❛❧✐③❛t✐♦♥ ♦❢ ■❙❆ t♦ ℓ ✶ ✲▼✐♥✐♠✐③❛t✐♦♥ ❢ ( ① ) = � ① � ✶ ✱ F = { ① | ❆① = ❜ } ✱ s✐❣♥ ( ① ) ∈ ∂ � ① � ✶ ❡①❛❝t ♣r♦❥❡❝t❡❞ s✉❜❣r❛❞✐❡♥t st❡♣ ❢♦r ✭▲✶✮✿ ② ❦ ← ① ❦ − α ❦ ❤ ❦ ③ ❦ ← ❙♦❧✉t✐♦♥ ♦❢ ❆❆ ❚ ③ = ❆② ❦ − ❜ ① ❦ + ✶ ← ② ❦ − ❆ ❚ ③ ❦ = P F ( ② ❦ ) ❆❆ ❚ ✐s s✳♣✳❞✳ ⇒ ♠❛② ❡♠♣❧♦② ❈● t♦ s♦❧✈❡ ❡q✉❛t✐♦♥ s②st❡♠ ❆✳ ❚✐❧❧♠❛♥♥ ✾ ✴ ✷✷ ❚❯ ❇r❛✉♥s❝❤✇❡✐❣