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■♥tr♦❞✉❝t✐♦♥ ▼✉❧t✐✇❛② ❈✉t ❈❛♣t✉r✐♥❣ ❆❧❧ ❙♦❧✉t✐♦♥s ❙✉♠♠❛r② ▼❛tr♦✐❞ ❚❤❡♦r② ❛♥❞ ❑❡r♥❡❧✐③❛t✐♦♥ P❛rt ✸✿ ■rr❡❧❡✈❛♥t ✈❡rt❡① r✉❧❡s ❛♥❞ ❝❛♣t✉r✐♥❣ ❛❧❧ s♦❧✉t✐♦♥s ❙❛❦❡t ❙❛✉r❛❜❤ ✶ ❙t❡❢❛♥ ❑r❛ts❝❤ ✷ ▼❛❣♥✉s ❲❛❤❧strö♠ ✸ ✶ ❇❡r❣❡♥ ❯♥✐✈❡rs✐t② ❛♥❞ ■▼❙❝ ❈❤❡♥♥❛✐ ✷ ❚❡❝❤♥✐❝❛❧ ❯♥✐✈❡rs✐t② ❇❡r❧✐♥ ✸ ❘♦②❛❧ ❍♦❧❧♦✇❛②✱ ❯♥✐✈❡rs✐t② ♦❢ ▲♦♥❞♦♥ ❲♦r❑❡r ✷✵✶✸✱ ❲❛rs❛✇ ✶✴✹✽

■♥tr♦❞✉❝t✐♦♥ ▼✉❧t✐✇❛② ❈✉t ❈❛♣t✉r✐♥❣ ❆❧❧ ❙♦❧✉t✐♦♥s ❙✉♠♠❛r② ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ▼❛tr♦✐❞s ❘❡♣r❡s❡♥t❛t✐✈❡ s❡ts ❆♣♣❧✐❝❛t✐♦♥s s❡❡♥ s♦ ❢❛r ▼✉❧t✐✇❛② ❈✉t ✷ ■♥tr♦❞✉❝t✐♦♥ ■rr❡❧❡✈❛♥t ✈❡rt❡① r✉❧❡s ❋✉rt❤❡r ❆♣♣❧✐❝❛t✐♦♥s ❈❛♣t✉r✐♥❣ ❆❧❧ ❙♦❧✉t✐♦♥s ✸ ❊①❛♠♣❧❡✿ ❞ ✲❍✐tt✐♥❣ ❙❡t ▼✉❧t✐✇❛② ❈✉t✱ r❡✈✐s✐t❡❞ ❈♦✈❡r✐♥❣ ●r❛♣❤ ❈✉ts ❙✉♠♠❛r② ✹ ✷✴✹✽

■♥tr♦❞✉❝t✐♦♥ ▼✉❧t✐✇❛② ❈✉t ❈❛♣t✉r✐♥❣ ❆❧❧ ❙♦❧✉t✐♦♥s ❙✉♠♠❛r② ▼❛tr♦✐❞s ❆ ♠❛tr♦✐❞ ▼ = ( ❱ , I ) ✱ I ⊆ ✷ ❱ ✱ ✐s ❛♥ ✐♥❞❡♣❡♥❞❡♥❝❡ s②st❡♠ ✇✐t❤ ✐♥❞❡♣❡♥❞❡♥t s❡ts I s❛t✐s❢②✐♥❣✿ ✶ ❚❤❡ ❡♠♣t② s❡t ✐s ✐♥❞❡♣❡♥❞❡♥t ✷ ❆ s✉❜s❡t ♦❢ ❛♥ ✐♥❞❡♣❡♥❞❡♥t s❡t ✐s ✐♥❞❡♣❡♥❞❡♥t ✸ ❆✉❣♠❡♥t❛t✐♦♥ ♣r♦♣❡rt②✿ ■❢ ❆ , ❇ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ | ❇ | > | ❆ | ✱ t❤❡♥ t❤❡r❡ ✐s s♦♠❡ ❜ ∈ ❇ − ❆ s✉❝❤ t❤❛t ❆ + ❜ ✐s ✐♥❞❡♣❡♥❞❡♥t ❘❛♥❦ r ( ❳ ) ✿ ❙✐③❡ ♦❢ ❧❛r❣❡st ✐♥❞❡♣❡♥❞❡♥t s✉❜s❡t ♦❢ ❳ ✸✴✹✽

■♥tr♦❞✉❝t✐♦♥ ▼✉❧t✐✇❛② ❈✉t ❈❛♣t✉r✐♥❣ ❆❧❧ ❙♦❧✉t✐♦♥s ❙✉♠♠❛r② ❊①❛♠♣❧❡s ❈❛♥♦♥✐❝❛❧ ❡①❛♠♣❧❡s✿ ✶ ●r❛♣❤✐❝ ♠❛tr♦✐❞s✿ ▲❡t ● = ( ❱ , ❊ ) ❜❡ ❛ ❣r❛♣❤✳ ▼ = ( ❊ , I ) ✱ I ❝♦♥t❛✐♥s ❝②❝❧❡✲❢r❡❡ ❡❞❣❡ s❡ts ❘❛♥❦✿ ♥✉♠❜❡r ♦❢ ✈❡rt✐❝❡s ♠✐♥✉s ♥✉♠❜❡r ♦❢ ❝♦♠♣♦♥❡♥ts ✷ ▲✐♥❡❛r ♠❛tr♦✐❞s ▼ = ( ❱ , I ) ✿ ❱ ✐s ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✈❡❝t♦rs ✐♥ F ❞ ❢♦r s♦♠❡ ✜❡❧❞ F ■♥❞❡♣❡♥❞❡♥❝❡ ❝♦♥❝❡♣t ✐s ❧✐♥❡❛r ✐♥❞❡♣❡♥❞❡♥❝❡ ❘❛♥❦✿ ❞✐♠❡♥s✐♦♥ ❘❡♣r❡s❡♥t❛❜❧❡ ♠❛tr♦✐❞✿ ❈❛♥ ❜❡ ❣✐✈❡♥ ❛s s❡t ♦❢ ✈❡❝t♦rs✳ ✹✴✹✽

■♥tr♦❞✉❝t✐♦♥ ▼✉❧t✐✇❛② ❈✉t ❈❛♣t✉r✐♥❣ ❆❧❧ ❙♦❧✉t✐♦♥s ❙✉♠♠❛r② ●r❛♣❤ ❝✉t ♠❛tr♦✐❞s✿ ●❛♠♠♦✐❞s ▲❡t ● = ( ❱ , ❊ ) ❜❡ ❛ ❣r❛♣❤ ✭♣♦ss✐❜❧② ❞✐r❡❝t❡❞✮✳ ❙❛② t❤❛t ❚ ⊆ ❱ ✐s ❧✐♥❦❡❞ t♦ ❙ ✐❢ t❤❡r❡ ❛r❡ | ❚ | ✈❡rt❡①✲❞✐s❥♦✐♥t ♣❛t❤s ❢r♦♠ ❙ t♦ ❚ ✭✇✐t❤ ❞✐st✐♥❝t ❡♥❞♣♦✐♥ts✮✳ ❚❤❡ ❣❛♠♠♦✐❞ ( ● , ❙ ) ❞❡✜♥❡❞ ❜② ● ❛♥❞ ❙ ✐s ▼ = ( ❱ , I ) ✇❤❡r❡✿ I ❝♦♥t❛✐♥s ❛❧❧ s❡ts ❧✐♥❦❡❞ t♦ ❙ ❚❤❡ r❛♥❦ r ( ❳ ) ❡q✉❛❧s t❤❡ s✐③❡ ♦❢ ❛♥ ( ❙ , ❳ ) ✲❝✉t ❙ ❳ ●❛♠♠♦✐❞s ❛r❡ r❡♣r❡s❡♥t❛❜❧❡✳ ✺✴✹✽

■♥tr♦❞✉❝t✐♦♥ ▼✉❧t✐✇❛② ❈✉t ❈❛♣t✉r✐♥❣ ❆❧❧ ❙♦❧✉t✐♦♥s ❙✉♠♠❛r② ❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❣❛♠♠♦✐❞s ●❛♠♠♦✐❞✱ ✜①❡❞✲s♦✉r❝❡ ❢♦r♠ ▲❡t ● = ( ❱ , ❊ ) ❛♥❞ ❙ , ❚ ⊆ ❱ ✳ ❚❤❡ ❣❛♠♠♦✐❞ ✇✐t❤ ❙ ❛s s♦✉r❝❡s ❛♥❞ ❚ ❛s ♣♦t❡♥t✐❛❧ s✐♥❦s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ s♣❛❝❡ | ❙ | · | ❚ | · ❖ ( ♠✐♥ ( | ❚ | , | ❙ | ❧♦❣ | ❚ | ) + ❧♦❣ | ❱ | + ❧♦❣ ( ✶ /ε )) ✐♥ r❛♥❞♦♠✐s❡❞ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✇✐t❤ s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② ✶ − ε ✳ ●❛♠♠♦✐❞✱ ✢❡①✐❜❧❡ ❢♦r♠ ▲❡t ● = ( ❱ , ❊ ) ❛♥❞ ❳ ⊆ ❱ ✳ ❚❤❡r❡ ✐s ❛ ❣❛♠♠♦✐❞ ✇❤✐❝❤ ❢♦r ❡✈❡r② ♣❛rt✐t✐♦♥ ❳ = ❆ ∪ ❇ ∪ ❈ ∪ ❉ ❡♥❝♦❞❡s t❤❡ s✐③❡ ♦❢ ❛♥ ❆ ✲ ❇ ♠✐♥✲❝✉t ✐♥ ● \ ❉ ✳ ■t ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ s♣❛❝❡ ❖ ( | ❳ | ✷ · ( | ❳ | + ❧♦❣ | ❱ | + ❧♦❣ ( ✶ /ε ))) ✐♥ r❛♥❞♦♠✐s❡❞ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✇✐t❤ s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② ✶ − ε ✳ ✻✴✹✽

■♥tr♦❞✉❝t✐♦♥ ▼✉❧t✐✇❛② ❈✉t ❈❛♣t✉r✐♥❣ ❆❧❧ ❙♦❧✉t✐♦♥s ❙✉♠♠❛r② ❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❣❛♠♠♦✐❞s ●❛♠♠♦✐❞✱ ✜①❡❞✲s♦✉r❝❡ ❢♦r♠ ▲❡t ● = ( ❱ , ❊ ) ❛♥❞ ❙ , ❚ ⊆ ❱ ✳ ❚❤❡ ❣❛♠♠♦✐❞ ✇✐t❤ ❙ ❛s s♦✉r❝❡s ❛♥❞ ❚ ❛s ♣♦t❡♥t✐❛❧ s✐♥❦s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ s♣❛❝❡ | ❙ | · | ❚ | · ❖ ( ♠✐♥ ( | ❚ | , | ❙ | ❧♦❣ | ❚ | ) + ❧♦❣ | ❱ | + ❧♦❣ ( ✶ /ε )) ✐♥ r❛♥❞♦♠✐s❡❞ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✇✐t❤ s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② ✶ − ε ✳ ●❛♠♠♦✐❞✱ ✢❡①✐❜❧❡ ❢♦r♠ ▲❡t ● = ( ❱ , ❊ ) ❛♥❞ ❳ ⊆ ❱ ✳ ❚❤❡r❡ ✐s ❛ ❣❛♠♠♦✐❞ ✇❤✐❝❤ ❢♦r ❡✈❡r② ♣❛rt✐t✐♦♥ ❳ = ❆ ∪ ❇ ∪ ❈ ∪ ❉ ❡♥❝♦❞❡s t❤❡ s✐③❡ ♦❢ ❛♥ ❆ ✲ ❇ ♠✐♥✲❝✉t ✐♥ ● \ ❉ ✳ ■t ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ s♣❛❝❡ ❖ ( | ❳ | ✷ · ( | ❳ | + ❧♦❣ | ❱ | + ❧♦❣ ( ✶ /ε ))) ✐♥ r❛♥❞♦♠✐s❡❞ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✇✐t❤ s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② ✶ − ε ✳ ✻✴✹✽

■♥tr♦❞✉❝t✐♦♥ ▼✉❧t✐✇❛② ❈✉t ❈❛♣t✉r✐♥❣ ❆❧❧ ❙♦❧✉t✐♦♥s ❙✉♠♠❛r② ❘❡♣r❡s❡♥t❛t✐✈❡ s❡ts ❘❡❝❛❧❧ ♠❛tr♦✐❞ ♥♦t✐♦♥ ♦❢ ❜❛s✐s✳ ▼❛tr♦✐❞ ❜❛s✐s ▲❡t ▼ = ( ❱ , I ) ❜❡ ❛ ♠❛tr♦✐❞✱ ❛♥❞ ❇ ❛ ❜❛s✐s ♦❢ ▼ ✳ ❋♦r ❛♥② ❳ ∈ I ✱ ❳ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ ❜② s♦♠❡ ✈ ∈ ❱ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❳ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ ❜② s♦♠❡ ✈ ∈ ❇ ✳ ∈ ❳ ❛♥❞ ( ❳ + ✈ ) ∈ I ✳ ✈ ❡①t❡♥❞s ❳ ✿ ✈ / ❙♣❡❝✳ ❣❛♠♠♦✐❞ ( ● , ❙ ) ✿ ▲❡t P✉s❤ ( ❳ , ❙ ) ❞❡♥♦t❡ t❤❡ r❡s✉❧t ♦❢ ♣✉s❤✐♥❣ ❳ t♦✇❛r❞s ❙ ✭❝r❡❛t✐♥❣ ❝❧♦s❡st s❡t✮✳ ❚❤❡♥ ✈ ❡①t❡♥❞s ❳ ✐❢ ❛♥❞ ♦♥❧② ✭ ❳ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞✮ ✈ r❡❛❝❤❛❜❧❡ ❢r♦♠ ❙ ✐♥ ● \ P✉s❤ ( ❳ , ❙ ) ✳ ❘❡♣r❡s❡♥t❛t✐✈❡ s❡ts✿ ❊①t❡♥❞ t❤❡ ❛❜♦✈❡ t♦ t✉♣❧❡s t = ( ✈ ✶ , . . . , ✈ s ) ❡①t❡♥❞✐♥❣ ❳ ✳ ✼✴✹✽

■♥tr♦❞✉❝t✐♦♥ ▼✉❧t✐✇❛② ❈✉t ❈❛♣t✉r✐♥❣ ❆❧❧ ❙♦❧✉t✐♦♥s ❙✉♠♠❛r② ❘❡♣r❡s❡♥t❛t✐✈❡ s❡ts✱ ❞❡✜♥✐t✐♦♥s ❉❡✜♥✐t✐♦♥✿ ❊①t❡♥❞s ▲❡t ▼ = ( ❱ , I ) ❜❡ ❛ ♠❛tr♦✐❞✱ ❳ ∈ I ✱ ❛♥❞ t ❛♥❞ s ✲t✉♣❧❡ ❢r♦♠ ❱ � ❱ � ✭✐✳❡✳✱ t ∈ ✮✳ ❲❡ s❛② t❤❛t t ❡①t❡♥❞s ❳ ✐❢ ❳ ∩ t = ∅ ❛♥❞ s ❳ ∪ t ∈ I ✳ ❊①❛♠♣❧❡s✿ ❋r❡❡ ♠❛tr♦✐❞ ▼ = ( ❱ , ✷ ❱ ) ✿ t ❡①t❡♥❞s ❳ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❳ ∩ t = ∅ ✭❝❢✳ ❞ ✲❍✐tt✐♥❣ ❙❡t✮✳ ❉✐s❥♦✐♥t s✉♠ ▼ = ▼ ✶ + . . . + ▼ s ❛♥❞ t ∈ ❱ ( ✶ ) × . . . × ❱ ( s ) ✿ ❚❡sts ✭ t ( ✶ ) ❡①t❡♥❞s ❳ ( ✶ ) ✮ ❆◆❉ ✳ ✳ ✳ ❆◆❉ ✭ t ( s ) ❡①t❡♥❞s ❳ ( s ) ✮ ✽✴✹✽