❊♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s
❡♠ ❣r❛❢♦s ❜✐♣❛rt✐❞♦s
❆❧❡①❛♥❞r❡ ❞❛ ❙✐❧✈❛ ❋r❡✐r❡
❚❡s❡ ❛♣r❡s❡♥t❛❞❛
❛♦
■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛
❞❛
❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦
♣❛r❛
♦❜t❡♥çã♦ ❞♦ tít✉❧♦
❞❡
❉♦✉t♦r ❡♠ ❈✐ê♥❝✐❛s
Pr♦❣r❛♠❛✿ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❊❞✉❛r❞♦ ❋❡rr❡✐r❛
❉✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦ ♦ ❛✉t♦r r❡❝❡❜❡✉ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦ ❞❛ ❈❆P❊❙ ❡ ❈◆Pq
❊♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s
❡♠ ❣r❛❢♦s ❜✐♣❛rt✐❞♦s
❊st❛ ✈❡rsã♦ ❞❛ t❡s❡ ❝♦♥té♠ ❛s ❝♦rr❡çõ❡s ❡ ❛❧t❡r❛ç❝õ❡s s✉❣❡r✐❞❛s ♣❡❧❛ ❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛ ❞✉r❛♥t❡ ❛ ❞❡❢❡s❛ ❞❛ ✈❡rsã♦ ♦r✐❣✐♥❛❧ ❞♦ tr❛❜❛❧❤♦✱ r❡❛❧✐③❛❞❛ ❡♠ ✵✷✴✶✵✴✷✵✶✷✳ ❯♠❛ ❝ó♣✐❛ ❞❛ ✈❡rsã♦ ♦r✐❣✐♥❛❧ ❡stá ❞✐s♣♦♥í✈❡❧ ♥♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✳
❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛✿
• Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❊❞✉❛r❞♦ ❋❡rr❡✐r❛ ✭♦r✐❡♥t❛❞♦r✮ ✲ ■▼❊✲❯❙P • Pr♦❢❛✳ ❉r❛✳ ❨♦s❤✐❦♦ ❲❛❦❛❜❛②❛s❤✐ ✲ ■▼❊✲❯❙P
• Pr♦❢✳ ❉r✳ ◆❡❧s♦♥ ▼❛❝✉❧❛♥ ✲ ❯❋❘❏
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❈❛r❧✐♥❤♦s ✭Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❊❞✉❛r❞♦ ❋❡rr❡✐r❛✮✱ q✉❡ ♠❡ ❛❝♦♠♣❛♥❤♦✉ ❛♦ ❧♦♥❣♦ ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ s❡✐s ❛♥♦s ✭✷ ❛♥♦s ❞❡ ♠❡str❛❞♦ ❡ ✹ ❞❡ ❞♦✉t♦r❛❞♦✮✳ ❉❡♣♦✐s ❞❡ ❝♦♥✈✐✈❡r t❛♥t♦ t❡♠♣♦ ❝♦♠ ✉♠❛ ♣❡ss♦❛✱ t♦r♥❛✲s❡ ♠✉✐t♦ ❞✐❢í❝✐❧ ♥ã♦ ❤❡r❞❛r ❛❧❣✉♠❛s ❞❡ s✉❛s q✉❛❧✐❞❛❞❡s✳ ❆❧❣✉♠❛s ❞❛s q✉❛❧✐❞❛❞❡s q✉❡ ❡✉ ❡♥①❡r❣♦ ♥♦ ❈❛r❧✐♥❤♦s ❡ q✉❡ ❡✉ ❣♦st❛r✐❛ ❞❡ ❤❡r❞❛r sã♦✿ s✐♠♣❧✐❝✐❞❛❞❡✱ ♣❛❝✐ê♥❝✐❛✱ s❡♥s♦ ❞❡ ❤✉♠♦r ❡ r❡s♣❡✐t♦ às ♣♦s✐çõ❡s ❛❧❤❡✐❛s✱ ♠❡s♠♦ q✉❛♥❞♦ ❡❧❛s ✈ã♦ ❜❛st❛♥t❡ ❝♦♥tr❛ ❛s s✉❛s ♣ró♣r✐❛s✳ ❖ ❈❛r❧✐♥❤♦s t❡✈❡ ♠✉✐t❛ ♣❛❝✐ê♥❝✐❛ ❝♦♠✐❣♦ ♥♦ ✐♥í❝✐♦ ❞❡ ♥♦ss♦ tr❛❜❛❧❤♦✱ q✉❛♥❞♦ ❡✉ ♣r❛t✐❝❛♠❡♥t❡ ♥ã♦ s❛❜✐❛ ❞♦ q✉❡ s❡ tr❛t❛✈❛ ❢❛③❡r ✉♠❛ ♣ós✲❣r❛❞✉❛çã♦✳ ▼✉✐t❛s ✈❡③❡s ❡❧❡ r❡s♣❡✐t♦✉ ♠✐♥❤❛s ♣♦s✐çõ❡s✱ ♠❡s♠♦ s❛❜❡♥❞♦ q✉❡ ❡st❛✈❛♠ ❡q✉✐✈♦❝❛❞❛s✱ ❞❡✐①❛♥❞♦ ❛ ❝❛r❣♦ ❞♦ t❡♠♣♦ ❢❛③❡r ♦ tr❛❜❛❧❤♦ ár❞✉♦ ❞❡ ♠♦❞✐✜❝❛r ❛ ♠✐♥❤❛ ❢♦r♠❛ ❞❡ ♣❡♥s❛r✳ ❚♦❞❛s ❛s ✈❡③❡s q✉❡ ❡♥tr❡✐ ♥❛ s❛❧❛ ❞♦ ❈❛r❧✐♥❤♦s ❢✉✐ r❡❝❡❜✐❞♦ ❝♦♠ ✉♠ s♦rr✐s♦✳ ◆✉♥❝❛ ❢✉✐ ❝♦❜r❛❞♦ ♣♦r r❡s✉❧t❛❞♦s ✏♠❡❧❤♦r❡s✑ ❡ ♦ ♣♦✉❝♦ q✉❡ ✜③✱ ❞❛❞❛s ❛s ♠✐♥❤❛s ❧✐♠✐t❛çõ❡s✱ ❢♦✐ s❡♠♣r❡ r❡❝❡❜✐❞♦ ❝♦♠ ❡❧♦❣✐♦s✳ ▼✉✐t♦ ♦❜r✐❣❛❞♦ ♣♦r t✉❞♦✱ ❈❛r❧✐♥❤♦s✦
P♦ss✐✈❡❧♠❡♥t❡✱ ❡♠ ❛❧❣✉♠❛s ♦❝❛s✐õ❡s s❡r❡✐ ❜❡♠ ✈✐st♦ ♣♦r t❡r ❡st✉❞❛❞♦ ♥♦ ■▼❊✲❯❙P✳ ◆♦ ❡♥t❛♥t♦✱ s✐♥t♦ q✉❡ ♣♦✉❝♦ ✐♠♣♦rt❛ ♦♥❞❡ ❡st✉❞❡✐✱ ♣♦✐s ♦ q✉❡ r❡❛❧♠❡♥t❡ ♠❡ ✐♠♣♦rt❛ é ❧❡♠❜r❛r ❝♦♠ q✉❡♠ ❡st✉❞❡✐✳ ❙♦✉ ✐♠❡♥s❛♠❡♥t❡ ❣r❛t♦ às ♣❡ss♦❛s q✉❡ ❛♦ ❧♦♥❣♦ ❞❡ ♠✐♥❤❛ ♣❛ss❛❣❡♠ ♣❡❧♦ ■▼❊✲❯❙P ♠❡ tr❛♥s♠✐t✐r❛♠ ♣❛rt❡ ❞❡ s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s✳ ❆❧é♠ ❞❡ ❣r❛t✐❞ã♦✱ s✐♥t♦ ♣♦r ❡ss❛s ♣❡ss♦❛s ♠✉✐t❛ ❛❞♠✐r❛çã♦✱ r❡s♣❡✐t♦ ❡ ❛❢❡t♦✳ ❙❡♥t✐r❡✐ ♠✉✐t❛s s❛✉❞❛❞❡s ❞♦s ♠❡✉s ❡t❡r♥♦s ♣r♦❢❡ss♦r❡s✿ ❨♦s❤✐❦♦ ❲✳✱ ❏✳ ❈♦❡❧❤♦ ❞❡ P✐♥❛✳✱ P❛✉❧♦ ❋❡♦✜❧♦✛ ✭❡♠❜♦r❛ ♥ã♦ t❡♥❤❛ s✐❞♦ ♠❡✉ ♣r♦❢❡ss♦r✱ ❛♣r❡♥❞✐ ♠✉✐t♦ ❝♦♠ s❡✉s ♠❛♥✉s❝r✐t♦s✮✱ ❈r✐st✐♥❛ ●✳ ❋❡r♥❛♥❞❡s✱ ❈❛r❧✐♥❤♦s ❡ ❏♦sé ❆✉❣✉st♦✳ ❆♦ ❡♥tr❛r ❡♠ ✉♠❛ s❛❧❛ ❞❡ ❛✉❧❛ ❡♠ q✉❡ ✉♠❛ ❞❡ss❛s ♣❡ss♦❛s ❡r❛ ♦ ♣r♦❢❡ss♦r✱ ♠❡ s❡♥t✐❛ ❝♦♠♦ ♦ ❞♦❡♥t❡ q✉❡ ❝❤❡❣❛ ❛♦ ❝♦♥s✉❧tór✐♦ ♠é❞✐❝♦ ❡✱ ❛♦ ♣❡r❝❡❜❡r q✉❡ ❡stá ❞✐❛♥t❡ ❞❡ ✉♠ ❜♦♠ ♠é❞✐❝♦✱ ❥á s❡ s❡♥t❡ ♣r❛t✐❝❛♠❡♥t❡ ❝✉r❛❞♦✳
P♦r s♦rt❡✱ ✜③ ✈ár✐❛s ❛♠✐③❛❞❡s ❞✉r❛♥t❡ ♦ ♣❡rí♦❞♦ ❡♠ q✉❡ ♣❛ss❡✐ ♣❡❧♦ ■▼❊✲❯❙P ❡ t❛♠❜é♠ ❞✉r❛♥t❡ ♦ ❡stá❣✐♦ ❞❡ ✉♠ ❛♥♦ q✉❡ ✜③ ❡♠ ▲②♦♥✱ ❋r❛♥ç❛✳ ❙♦✉ ♠✉✐t♦ ❣r❛t♦ ❛ ❡ss❛s ♣❡ss♦❛s q✉❡r✐❞❛s✱ ♣♦✐s ❝❡rt❛♠❡♥t❡ ❛ ♠✐♥❤❛ ✈✐❞❛ t❡r✐❛ s✐❞♦ ✐♥s✉♣♦rtá✈❡❧ s❡♠ ❛ ♣r❡s❡♥ç❛ ❞❡❧❛s ❞✉r❛♥t❡ ❡ss❡ t❡♠♣♦✳
P♦r ✜♠✱ ❣♦st❛r✐❛ ❞❡ ❞❡✐①❛r r❡❣✐str❛❞♦ ✉♠ ❛❣r❛❞❡❝✐♠❡♥t♦ ❡s♣❡❝✐❛❧ à ♠✐♥❤❛ ▼ã❡✱ q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐♦✉ ✐♥❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡✳
❘❡s✉♠♦
❋❘❊■❘❊✱ ❆✳ ❙✳ ❊♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s ❡♠ ❣r❛❢♦s ❜✐♣❛rt✐❞♦s✳ ✷✵✶✷✱ ✼✹❢✳ ❚❡s❡ ✭❉♦✉t♦r❛❞♦✮ ✕ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✷✳
◆❡st❛ t❡s❡✱ ❡st✉❞❛♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s✳ ❯♠ ❜✐❝❧✐q✉❡ é ✉♠ ❣r❛❢♦ ❜✐♣❛rt✐❞♦ ❝♦♠♣❧❡t♦✳ ◆♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s sã♦ ❞❛❞♦s ✉♠ ✐♥t❡✐r♦ k ❡ ✉♠ ❣r❛❢♦ ❜✐♣❛rt✐❞♦G❡ ❞❡s❡❥❛✲s❡ ❡♥❝♦♥tr❛r ✉♠ ❝♦♥❥✉♥t♦ ❞❡k❜✐❝❧✐q✉❡s✱ s✉❜❣r❛❢♦s ❞❡G✱ ❞♦✐s ❛ ❞♦✐s ❞✐s❥✉♥t♦s ♥♦s ✈ért✐❝❡s✱ t❛❧ q✉❡ ❛ q✉❛♥t✐❞❛❞❡ t♦t❛❧ ❞❡ ❛r❡st❛s ❞♦s ❜✐❝❧✐q✉❡s ❡s❝♦❧❤✐❞♦s s❡❥❛ ♠á①✐♠❛✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ k = 1✱ t❡♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦✳ ❊ss❡s ❞♦✐s ♣r♦❜❧❡♠❛s ♣♦ss✉❡♠ ❛♣❧✐❝❛çõ❡s ♥❛ ár❡❛ ❞❡ ❇✐♦✐♥❢♦r♠át✐❝❛✳ ▼❛♥t❡♠♦s ♥❡st❡ tr❛❜❛❧❤♦ ✉♠ ❡♥❢♦q✉❡ ♣rát✐❝♦✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ♥♦ss♦ ✐♥t❡r❡ss❡ é r❡s♦❧✈❡r ✐♥stâ♥❝✐❛s ❞❡ss❡s ❞♦✐s ♣r♦❜❧❡♠❛s ❝♦♠ t❛♠❛♥❤♦ r❛③♦❛✈❡❧♠❡♥t❡ ❣r❛♥❞❡✳ P❛r❛ ✐ss♦✱ ✉t✐❧✐③❛♠♦s té❝♥✐❝❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ■♥t❡✐r❛✳ P❛r❛ ❛✈❛❧✐❛r ♦s ♠ét♦❞♦s ♣r♦♣♦st♦s ❛q✉✐✱ ♠♦str❛♠♦s r❡s✉❧t❛❞♦s ❞❡ ❡①♣❡r✐♠❡♥t♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❢❡✐t♦s ❝♦♠ ✐♥stâ♥❝✐❛s ✈✐♥❞❛s ❞❡ ❛♣❧✐❝❛çõ❡s ❡ t❛♠❜é♠ ❝♦♠ ✐♥stâ♥❝✐❛s ❣❡r❛❞❛s ❛❧❡❛t♦r✐❛♠❡♥t❡✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❣❡r❛çã♦ ❞❡ ❝♦❧✉♥❛s✱ ❜✐❝❧✐q✉❡ ♠á①✐♠♦✱ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s✱ ♣r♦❣r❛♠❛çã♦ ✐♥t❡✐r❛✳
❆❜str❛❝t
❋❘❊■❘❊✱ ❆✳ ❙✳ ❇✐❝❧✐q✉❡ ♣❛❝❦✐♥❣ ✐♥ ❜✐♣❛rt✐t❡ ❣r❛♣❤s✳ ✷✵✶✷✱ ✼✹❢✳ ❚❡s❡ ✭❉♦✉t♦r❛❞♦✮ ✕ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✷✳
■♥ t❤✐s t❤❡s✐s✱ ✇❡ st✉❞② t❤❡ ❜✐❝❧✐q✉❡ ♣❛❝❦✐♥❣ ♣r♦❜❧❡♠✳ ❆ ❜✐❝❧✐q✉❡ ✐s ❛ ❝♦♠♣❧❡t❡ ❜✐♣❛rt✐t❡ ❣r❛♣❤✳ ■♥ t❤❡ ❜✐❝❧✐q✉❡ ♣❛❝❦✐♥❣ ♣r♦❜❧❡♠ ✇❡ ❛r❡ ❣✐✈❡♥ ❛♥ ✐♥t❡❣❡r k ❛♥❞ ❛ ❜✐♣❛rt✐t❡ ❣r❛♣❤ G ❛♥❞ ✇❡ ✇❛♥t t♦ ✜♥❞ ❛ s❡t ♦❢ k ✈❡rt❡① ❞✐s❥♦✐♥t ❜✐❝❧✐q✉❡s ♦❢ G✱ s✉❝❤ t❤❛t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❜✐❝❧✐q✉❡✬s ❡❞❣❡s ✐s ♠❛①✐♠✉♠✳ ❲❤❡♥ k = 1✱ ✇❡ ❤❛✈❡ t❤❡ ♠❛①✐♠✉♠ ❜✐❝❧✐q✉❡ ♣r♦❜❧❡♠✳ ❚❤❡s❡ t✇♦ ♣r♦❜❧❡♠s ❤❛✈❡ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❇✐♦✐♥❢♦r♠❛t✐❝s✳ ■♥ t❤✐s ✇♦r❦ ✇❡ ❦❡❡♣ ❛ ♣r❛❝t✐❝❛❧ ❢♦❝✉s✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ s♦❧✈✐♥❣ ❧❛r❣❡ s✐③❡ ✐♥st❛♥❝❡s ♦❢ t❤❡s❡ ♣r♦❜❧❡♠s✳ ❚♦ t❛❝❦❧❡ t❤❡s❡ ♣r♦❜❧❡♠s✱ ✇❡ ✉s❡ ■♥t❡❣❡r ▲✐♥❡❛r Pr♦❣r❛♠♠✐♥❣ t❡❝❤♥✐q✉❡s✳ ■♥ ♦r❞❡r t♦ ❡✈❛❧✉❛t❡ t❤❡ ♠❡t❤♦❞s ♣r♦♣♦s❡❞ ❤❡r❡✱ ✇❡ s❤♦✇ r❡s✉❧ts ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts ❝❛rr✐❡❞ ♦✉t ✇✐t❤ ♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥✬s ✐♥st❛♥❝❡s ❛♥❞ ❛❧s♦ ✇✐t❤ r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞ ♦♥❡s✳
❑❡②✇♦r❞s✿ ❝♦❧✉♠♥ ❣❡♥❡r❛t✐♦♥✱ ♠❛①✐♠✉♠ ❜✐❝❧✐q✉❡✱ ❜✐❝❧✐q✉❡ ♣❛❝❦✐♥❣✱ ✐♥t❡❣❡r ♣r♦❣r❛♠♠✐♥❣✳
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶
✶✳✶ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✶✳✶ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✶✳✷ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ■♥t❡✐r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✸ ❆❧❣♦r✐t♠♦s ❞❡ ❆♣r♦①✐♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ◆♦t❛çã♦ ❡ ❉❡✜♥✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ ❆♣❧✐❝❛çõ❡s Prát✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✹ ❘❡s✉❧t❛❞♦s ❊♥❝♦♥tr❛❞♦s ♥❛ ▲✐t❡r❛t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✺ ❘❡s✉❧t❛❞♦s ❖❜t✐❞♦s ❉✉r❛♥t❡ ♦ ❉♦✉t♦r❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✺✳✶ ▼❛♣❡❛♠❡♥t♦ ❞❡ ●r❛❢♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✺✳✷ ❊♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❇✐❝❧✐q✉❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✺✳✸ ❈♦❜❡rt✉r❛ ❞❡ ▼❛tr✐③❡s P♦r ❘❛✐♦s ❊①tr❡♠❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✻ ❖❜❥❡t✐✈♦ ❉❡st❡ ❚r❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✼ ❖r❣❛♥✐③❛çã♦ ❞♦ ❚❡①t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷ ❖ Pr♦❜❧❡♠❛ ❞❡ ❇✐❝❧✐q✉❡ ▼á①✐♠♦ ✶✺
✷✳✶ ❘❡❧❛çã♦ ❈♦♠ ♦ Pr♦❜❧❡♠❛ ❞❡ ❈❧✐q✉❡ ▼á①✐♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✷ ❯♠❛ ❋♦r♠✉❧❛çã♦ ❈♦♠ ▼❡♥♦s ❱❛r✐á✈❡✐s ❡ ❘❡str✐çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✸ ❯♠ ▼ét♦❞♦ ❞❡ ❘❡❧❛①❛çã♦ ❞❛ ❋✉♥çã♦ ❖❜❥❡t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✹ ❯♠ ❆❧❣♦r✐t♠♦ ❞❡ ❇r❛♥❝❤✲❛♥❞✲❇♦✉♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✺ ❯♠❛ ❍❡✉ríst✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✻ ❈♦♥s✐❞❡r❛çõ❡s ❙♦❜r❡ ❛ ■♠♣❧❡♠❡♥t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✼ ❊①♣❡r✐♠❡♥t♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✼✳✶ ■♥stâ♥❝✐❛s ●❡r❛❞❛s ❆❧❡❛t♦r✐❛♠❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✼✳✷ ■♥stâ♥❝✐❛s ❱✐♥❞❛s ❞❡ ❆♣❧✐❝❛çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✽ ❈♦♥s✐❞❡r❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✸ ❊♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❇✐❝❧✐q✉❡s ✹✶
✸✳✶ ❖ ■♥❝♦♥✈❡♥✐❡♥t❡ ❞❡ ❋♦r♠✉❧❛çõ❡s ❈♦♠ ❙✐♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✷ ❯t✐❧✐③❛♥❞♦ ❛ ❚é❝♥✐❝❛ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✷✳✶ ❯♠ Pr♦❝❡❞✐♠❡♥t♦ ❞❡ ❇r❛♥❝❤✲❆♥❞✲Pr✐❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✷✳✷ ❈♦♥s✐❞❡r❛çõ❡s ❙♦❜r❡ ❛ ■♠♣❧❡♠❡♥t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✸✳✸ ❊①♣❡r✐♠❡♥t♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽
✈✐✐✐ ❙❯▼➪❘■❖
✹ ❯♠ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ❇✐❧✐♥❡❛r ■♥t❡✐r❛ ✻✺
✹✳✶ ❚é❝♥✐❝❛ ❞❡ ❉❡❝♦♠♣♦s✐çã♦ ❇✐♥ár✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✹✳✷ ▼ét♦❞♦ ❞❡ ❘❡❧❛①❛çã♦ ❞❛ ❋✉♥çã♦ ❖❜❥❡t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✹✳✸ ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ✹✳✹ ❊①♣❡r✐♠❡♥t♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾
✺ ❈♦♥❝❧✉sã♦ ❡ ❚r❛❜❛❧❤♦s ❋✉t✉r♦s ✼✸
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
◆❡st❛ t❡s❡✱ tr❛t❛♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s✳ ❯♠ ❜✐❝❧✐q✉❡ é ✉♠ ❣r❛❢♦ ❜✐♣❛rt✐❞♦ ❝♦♠♣❧❡t♦✱ ♦✉ s❡❥❛✱ ✉♠ ❣r❛❢♦ ♥♦ q✉❛❧ t♦❞♦s ♦s ✈ért✐❝❡s ❞❡ ✉♠❛ ❝❧❛ss❡ ❞❛ ❜✐♣❛rt✐çã♦ sã♦ ❛❞❥❛❝❡♥t❡s ❛ t♦❞♦s ♦s ✈ért✐❝❡s ❞❛ ♦✉tr❛ ❝❧❛ss❡ ❞❛ ❜✐♣❛rt✐çã♦✳ ◆♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s sã♦ ❞❛❞♦s ✉♠ ✐♥t❡✐r♦ k ❡ ✉♠ ❣r❛❢♦ ❜✐♣❛rt✐❞♦ G✱ ❡ ❞❡s❡❥❛✲s❡ ❡♥❝♦♥tr❛r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ k ❜✐❝❧✐q✉❡s✱ s✉❜❣r❛❢♦s ❞❡ G✱ ❞♦✐s ❛ ❞♦✐s ❞✐s❥✉♥t♦s ♥♦s ✈ért✐❝❡s✱ t❛❧ q✉❡ ❛ q✉❛♥t✐❞❛❞❡ t♦t❛❧ ❞❡ ❛r❡st❛s ❞♦s ❜✐❝❧✐q✉❡s ❡s❝♦❧❤✐❞♦s s❡❥❛ ♠á①✐♠❛✳ ❈♦♥s✐❞❡r❛♠♦s t❛♠❜é♠ ♦ ❝❛s♦ ❡♠ q✉❡ ❛ r❡str✐çã♦ ❞❡ ❞✐s❥✉♥çã♦ ❛♣❧✐❝❛✲s❡ ❛♣❡♥❛s ❛ ✉♠❛ ❞❛s ❝❧❛ss❡s ❞❛ ❜✐♣❛rt✐çã♦✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ k = 1✱ t❡♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦✱ ♣❛r❛ ♦ q✉❛❧ ❡①✐st❡♠ ❞✐✈❡rs♦s r❡s✉❧t❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛✱ ❝♦♥❢♦r♠❡ ❞❡t❛❧❤❛r❡♠♦s ♥❡st❡ ❝❛♣ít✉❧♦✳ ◆ã♦ t❡♠♦s ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ♥❡♥❤✉♠ r❡s✉❧t❛❞♦ ❡①✐st❡♥t❡ ♥❛ ❧✐t❡r❛t✉r❛✱ ❛♥t❡r✐♦r ❛♦ ♥♦ss♦ tr❛❜❛❧❤♦✱ ♣❛r❛ ♦ ❝❛s♦ ❣❡r❛❧ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s✳ ❚♦♠❛♠♦s ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡st❡ ♣r♦❜❧❡♠❛ ❛tr❛✈és ❞❡ ♥♦ss♦s ❝♦❧❛❜♦r❛❞♦r❡s ❝❤✐❧❡♥♦s ❊❞✉❛r❞♦ ▼♦r❡♥♦ ❡ ❱✐❝❡♥t❡ ❆❝✉ñ❛✱ q✉❡ ♥♦s ❛♣r❡s❡♥t❛r❛♠ ✉♠❛ ❛♣❧✐❝❛çã♦ ❡♠ ❜✐♦✐♥❢♦r♠át✐❝❛ q✉❡ ❡❧❡s t❡♥t❛✈❛♠ ♠♦❞❡❧❛r ❝♦♠♦ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♦t✐♠✐③❛çã♦✳ ❋♦r♠❛❧✐③❛♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❡ss❡s ❝♦❧❛❜♦r❛❞♦r❡s✳
◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ s♦❜r❡ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡st❛ t❡s❡✱ ✐♥tr♦❞✉③✐♠♦s ❛ ♥♦t❛çã♦ q✉❡ s❡rá ✉t✐❧✐③❛❞❛ ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦ ♦ t❡①t♦✱ ❡♥✉♥❝✐❛♠♦s ❢♦r♠❛❧♠❡♥t❡ ♦s ♣r♦❜❧❡♠❛s ✐♥✈❡st✐❣❛❞♦s ♥❡st❛ t❡s❡✱ ❞❡s❝r❡✈❡♠♦s ❛s ❛♣❧✐❝❛çõ❡s q✉❡ ♥♦s ♠♦t✐✈❛r❛♠ ❛ ❡st✉❞❛r ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s ❡ ❛♣r❡s❡♥t❛♠♦s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛ s♦❜r❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦✳ ▼❡♥❝✐♦♥❛♠♦s t❛♠❜é♠ ♥♦ss❛s ❝♦♥tr✐❜✉✐çõ❡s ♦❜t✐❞❛s ❞✉r❛♥t❡ ♦ ❞♦✉t♦r❛❞♦✱ ❞❡s❝r❡✈❡♠♦s ♦s ♦❜❥❡t✐✈♦s ❞❡ ♥♦ss♦ tr❛❜❛❧❤♦ ❡✱ ♣♦r ✜♠✱ ❞❡s❝r❡✈❡♠♦s ❝♦♠♦ ❡stá ♦r❣❛♥✐③❛❞♦ ♦ r❡st❛♥t❡ ❞♦ t❡①t♦✳
✶✳✶ ❈♦♥❝❡✐t♦s ❇ás✐❝♦s
P❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦ ❝♦♥t❡ú❞♦ ❡①♣♦st♦ ♥❡st❛ t❡s❡✱ é ♥❡❝❡ssár✐♦ q✉❡ ♦ ❧❡✐t♦r t❡♥❤❛ ✉♠ ❝♦✲ ♥❤❡❝✐♠❡♥t♦ ❜ás✐❝♦ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ■♥t❡✐r❛✳ ◆❡st❛ s❡çã♦✱ ❛♣r❡s❡♥✲ t❛♠♦s ✉♠❛ ♣❡q✉❡♥❛ r❡s❡♥❤❛ s♦❜r❡ ❡ss❡s ❛ss✉♥t♦s✱ ❜❛s❡❛❞❛ ♥♦s ❧✐✈r♦s ❞❡ ❈❤✈át❛❧ ✭✶✾✽✸✮✱ ❙❝❤r✐❥✈❡r
✭✶✾✽✻✮ ❡❲♦❧s❡②✭✶✾✾✽✮✳ ❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❜ás✐❝❛s ❞❛ ár❡❛ ❞❡ ❆❧❣♦r✐t♠♦s ❞❡ ❆♣r♦①✐♠❛çã♦✱ ❜❛s❡❛❞❛s ♥♦ ❧✐✈r♦ ❞❡❈❛r✈❛❧❤♦ ❡t ❛❧✳ ✭✷✵✵✶✮✳
✷ ■◆❚❘❖❉❯➬➹❖ ✶✳✶
✶✳✶✳✶ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r
❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ♣r♦❜❧❡♠❛✳
Pr♦❜❧❡♠❛ ✶✳✶✳✶✳ ❉❛❞♦s ✈❡t♦r❡s c ∈ Rm✱ b ∈ Rn ❡ ✉♠❛ ♠❛tr✐③ A ∈ Rn×m✱ ❡♥❝♦♥tr❛r ✉♠ ✈❡t♦r
x∈Rm q✉❡ ♠❛①✐♠✐③❡ c⊤x✱ t❛❧ q✉❡ Ax≤b ❡ x≥0✳
❉✐③❡♠♦s q✉❡ ❛ tr✐♣❧❛ (c, b, A)é ✉♠ ♣r♦❣r❛♠❛ ❧✐♥❡❛r✱ ♥♦ q✉❛❧c⊤xé ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ❡ ♦ s✐st❡♠❛ ❞❡ ✐♥❡q✉❛çõ❡sAx≤b❡x≥0sã♦ ❛s r❡str✐çõ❡s✳ ❈❛❞❛ ✈❡t♦rxq✉❡ s❛t✐s❢❛③ ❛s r❡str✐çõ❡s ❞♦ ♣r♦❣r❛♠❛ ❧✐♥❡❛r é ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧✳ ❯♠❛ s♦❧✉çã♦ ✈✐á✈❡❧ x∗ é ót✐♠❛ s❡ c⊤x∗ ≥ c⊤x✱ ♣❛r❛ t♦❞❛ s♦❧✉çã♦ ✈✐á✈❡❧ x✳ ❉✐③❡♠♦s q✉❡ c⊤x é ♦ ✈❛❧♦r ❞❛ s♦❧✉çã♦ x✳ ❉✐③❡♠♦s q✉❡ ✉♠ ♣r♦❣r❛♠❛ ❧✐♥❡❛r é ✐♥✈✐á✈❡❧✱ s❡ ❡❧❡ ♥ã♦ ♣♦ss✉✐ s♦❧✉çõ❡s ✈✐á✈❡✐s✱ ❡ ✐❧✐♠✐t❛❞♦✱ s❡ ❡❧❡ ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧✱ ♠❛s ♥ã♦ ♣♦ss✉✐ ✉♠❛ s♦❧✉çã♦ ót✐♠❛ ✭♦✉ s❡❥❛✱ ♣❛r❛ q✉❛❧q✉❡r s♦❧✉çã♦ ✈✐á✈❡❧ x✱ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧ x′✱ t❛❧ q✉❡ c⊤x′ > c⊤x✮✳ P♦r s✐♠♣❧✐❝✐❞❛❞❡✱ tr❛t❛r❡♠♦s ❛♣❡♥❛s ❞❡ ♣r♦❣r❛♠❛s ❧✐♥❡❛r❡s ✈✐á✈❡✐s ❡ ❧✐♠✐t❛❞♦s✳ ❯♠ ♣r♦❣r❛♠❛ ❧✐♥❡❛r ♣♦❞❡ s❡r ❞❡ ♠✐♥✐♠✐③❛çã♦ ♦✉ ♠❛①✐♠✐③❛çã♦✱ s✉❛s r❡str✐çõ❡s sã♦ ❞❡s❝r✐t❛s ❛tr❛✈és ❞❡ ❡q✉❛çõ❡s ❡✴♦✉ ✐♥❡q✉❛çõ❡s ❧✐♥❡❛r❡s✱ ❡ s✉❛s ✈❛r✐á✈❡✐s ♣♦❞❡♠ ❛ss✉♠✐r q✉❛❧q✉❡r ✈❛❧♦r r❡❛❧✳ ◆♦ ❡♥t❛♥t♦✱ ❛❞♦t❛♠♦s ❛ ❢♦r♠❛ ♣❛❞rã♦ ♣❛r❛ ❞❡s❝r❡✈❡r ♣r♦❣r❛♠❛s ❧✐♥❡❛r❡s✱ ♥❛ q✉❛❧ ❡①✐st❡ ✉♠❛ r❡str✐çã♦ ❞❡ ♥ã♦✲ ♥❡❣❛t✐✈✐❞❛❞❡ s♦❜r❡ ❝❛❞❛ ✈❛r✐á✈❡❧ ❡ ❛s ❞❡♠❛✐s r❡str✐çõ❡s sã♦ t♦❞❛s ❞❛ ❢♦r♠❛ Aix ≤ bi✱ ♥♦ ❝❛s♦ ❞❡ ♠❛①✐♠✐③❛çã♦✱ ❡Aix≥bi ♥♦ ❝❛s♦ ❞❡ ♠✐♥✐♠✐③❛çã♦✳ ❉❛❞♦ ✉♠ ♣r♦❣r❛♠❛ ❧✐♥❡❛r q✉❡ ♥ã♦ ❡stá ♥❛ ❢♦r♠❛ ♣❛❞rã♦✱ ♣♦❞❡♠♦s ❢❛❝✐❧♠❡♥t❡ ❡s❝r❡✈❡r ✉♠ ♣r♦❣r❛♠❛ ❧✐♥❡❛r ❡q✉✐✈❛❧❡♥t❡ ♥❛ ❢♦r♠❛ ♣❛❞rã♦✱ ♥♦ q✉❛❧ ❝❛❞❛ ✐❣✉❛❧❞❛❞❡ Aix = bi é s✉❜st✐t✉✐❞❛ ♣❡❧❛s ❞❡s✐❣✉❛❧❞❛❞❡s Aix ≤bi ❡ −Aix ≤ −bi✱ ❡ ❝❛❞❛ ✈❛r✐á✈❡❧ xj ❧✐✈r❡ ❞❡ s✐♥❛❧ é s✉❜st✐t✉í❞❛ ♣♦r ❞✉❛s ✈❛r✐á✈❡✐s x+j ❡ x−j ✱ ❞❡ ❢♦r♠❛ q✉❡ xj =x+j −x
−
j ❡ x+j , x − j ≥0✳ ❈♦♥s✐❞❡r❡ ♦s s❡❣✉✐♥t❡s ♣r♦❣r❛♠❛s ❧✐♥❡❛r❡s✳
♠❛①✐♠✐③❡ c⊤x ♠✐♥✐♠✐③❡ b⊤y
✭P▲✮ s✉❥❡✐t♦ ❛ Ax ≤ b ✭❉▲✮ s✉❥❡✐t♦ ❛ A⊤y ≥ c
x ≥ 0 y ≥ 0
❉✐③❡♠♦s q✉❡ ♦ ♣r♦❣r❛♠❛ ❧✐♥❡❛r ✭P▲✮ é ♣r✐♠❛❧ ❡ ♦ ♣r♦❣r❛♠❛ ❧✐♥❡❛r ✭❉▲✮ é ♦ ❞✉❛❧ ❞❡ ✭P▲✮✳ ❊q✉✐✲ ✈❛❧❡♥t❡♠❡♥t❡✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r q✉❡ ✭❉▲✮ é ♣r✐♠❛❧ ❡ ✭P▲✮ é ♦ s❡✉ ❞✉❛❧✳ ❖✉ s❡❥❛✱ t♦❞♦ ♣r♦❣r❛♠❛ ❧✐♥❡❛r ♣♦ss✉✐ ✉♠ ❞✉❛❧✱ s❡♥❞♦ q✉❡ s❡ ♦ ♣r✐♠❛❧ ❢♦r ❞❡ ♠❛①✐♠✐③❛çã♦✱ ♦ ❞✉❛❧ s❡rá ❞❡ ♠✐♥✐♠✐③❛çã♦✱ ❡ ✈✐❝❡✲✈❡rs❛✳ ❉✐③❡♠♦s q✉❡ (x, y) é ✉♠ ♣❛r ♣r✐♠❛❧✲❞✉❛❧ s❡x é ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧ ❞❡ ✭P▲✮ ❡ y é ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧ ❞❡ ✭❉▲✮✳ P♦❞❡♠♦s r❡❡s❝r❡✈❡r ✭P▲✮ ❡ ✭❉▲✮ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✳
♠❛①✐♠✐③❡ c⊤x ♠✐♥✐♠✐③❡ b⊤y
(P▲′) s✉❥❡✐t♦ ❛ Ax+w = b (❉▲′) s✉❥❡✐t♦ ❛ A⊤y−z = c
x, w ≥ 0 y, z ≥ 0
❆s ✈❛r✐á✈❡✐s w❡ z sã♦ ❝❤❛♠❛❞❛s ❞❡ ✈❛r✐á✈❡✐s ❞❡ ❢♦❧❣❛✳ ❆❜❛✐①♦✱ ❡♥✉♥❝✐❛♠♦s ♦s ♣r✐♥❝✐♣❛✐s t❡♦r❡♠❛s ❞❛ ár❡❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✳
❚❡♦r❡♠❛ ✶✳✶✳✶ ✭❉✉❛❧✐❞❛❞❡ ❋r❛❝❛✮✳ ❉❛❞♦ ✉♠ ♣❛r ♣r✐♠❛❧✲❞✉❛❧ (x, y)✱ t❡♠♦s q✉❡ c⊤x≤b⊤y✳ ❚❡♦r❡♠❛ ✶✳✶✳✷ ✭❉✉❛❧✐❞❛❞❡ ❋♦rt❡✮✳ ❉❛❞♦ ✉♠ ♣❛r ♣r✐♠❛❧✲❞✉❛❧ (x∗, y∗)✱ t❡♠♦s q✉❡ s❡ x∗ é ✉♠❛ s♦❧✉çã♦ ót✐♠❛ ❞❡ ✭P▲✮ ❡y∗ é ✉♠❛ s♦❧✉çã♦ ót✐♠❛ ❞❡ ✭❉▲✮✱ ❡♥tã♦c⊤x∗ =b⊤y∗✳
✶✳✶ ❈❖◆❈❊■❚❖❙ ❇➪❙■❈❖❙ ✸
s❡ ❡ s♦♠❡♥t❡ s❡ x∗j ·z∗j = 0✱ ♣❛r❛ j= 1,2, . . . , n✱ ❡ wi∗·yi∗ = 0✱ ♣❛r❛ i= 1,2, . . . , m✳
❉❛❞❛ ✉♠❛ s♦❧✉çã♦ ót✐♠❛ x∗ ❞❡ ✭P▲✮✱ ♣♦❞❡♠♦s ❢❛❝✐❧♠❡♥t❡ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ ót✐♠❛ y∗ ❞❡ ✭❉▲✮✱ ✉t✐❧✐③❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❢♦❧❣❛s ❝♦♠♣❧❡♠❡♥t❛r❡s ❡✱ ♣♦r ❡ss❛ r❛③ã♦✱ ❞✐③❡♠♦s q✉❡ t❛❧ s♦❧✉çã♦y∗ é ❞✉❛❧ ❝♦♠♣❧❡♠❡♥t❛r ❞❡x∗✳ P❛r❛ ❝❛❞❛ ✈❛r✐á✈❡❧ ❞❡ ❢♦❧❣❛w∗i ❝♦♠ ✈❛❧♦r ♣♦s✐t✐✈♦✱ t❡♠♦s q✉❡ ♦ ✈❛❧♦r ❞❛ ✈❛r✐á✈❡❧ ❞✉❛❧yi∗ ❞❡✈❡ s❡r ✐❣✉❛❧ ❛ ③❡r♦✳ P❛r❛ q✉❡ t❛❧ ❝♦♥❞✐çã♦ s❡❥❛ s❛t✐s❢❡✐t❛✱ ❛❞✐❝✐♦♥❛♠♦s ❛ r❡str✐çã♦ yi = 0 ❡♠ ✭❉▲✮✱ ♣❛r❛ t♦❞♦ i✱ t❛❧ q✉❡ wi∗ > 0✱ ♦❜t❡♥❞♦ ♦ ♣r♦❣r❛♠❛ ❧✐♥❡❛r (❉▲∗)✳ P❛r❛ ♦❜t❡r ♦s ✈❛❧♦r❡s ❞❛s ❞❡♠❛✐s ✈❛r✐á✈❡✐s ❞✉❛✐sy∗✱ ❜❛st❛ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧ ❞❡(❉▲∗)✳
❊♠ ✶✾✹✼✱ ●❡♦r❣❡ ❇✳ ❉❛♥t③✐❣ ♣✉❜❧✐❝♦✉ ♦ ♣r✐♠❡✐r♦ ❛❧❣♦r✐t♠♦✱ ❞❡♥♦♠✐♥❛❞♦ s✐♠♣❧❡①✱ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❣r❛♠❛s ❧✐♥❡❛r❡s✳ ◆❛ é♣♦❝❛✱ ♥ã♦ ❤❛✈✐❛ ✉♠ ❡st✉❞♦ ❞❡t❛❧❤❛❞♦ s♦❜r❡ ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦ s✐♠♣❧❡①✳ ❊♠ ✶✾✼✷✱ ❱✐❝t♦r ❑❧❡❡ ❡ ●❡♦r❣❡ ❏✳ ▼✐♥t② ❛♣r❡s❡♥t❛r❛♠ ✉♠ ❡①❡♠♣❧♦ ♥♦ q✉❛❧ ♦ s✐♠♣❧❡① ❝♦♥s✉♠✐❛ t❡♠♣♦ ❡①♣♦♥❡♥❝✐❛❧ ♥♦ t❛♠❛♥❤♦ ❞❛ ❡♥tr❛❞❛✳ ❚r✐♥t❛ ❡ ❞♦✐s ❛♥♦s ❛♣ós ❛ ♣✉❜❧✐❝❛çã♦ ❞♦ s✐♠♣❧❡①✱ ❡♠ ✶✾✼✾✱ ▲❡♦♥✐❞ ❑❤❛❝❤✐②❛♥ ♠♦str♦✉ q✉❡ ❡r❛ ♣♦ssí✈❡❧ r❡s♦❧✈❡r q✉❛❧q✉❡r ♣r♦❣r❛♠❛ ❧✐♥❡❛r ❡♠ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧✱ ❛tr❛✈és ❞♦ ♠ét♦❞♦ ❞♦s ❡❧✐♣s♦✐❞❡s✳ ◆♦ ❡♥t❛♥t♦✱ ❡ss❛ ❞❡s❝♦❜❡rt❛ ❢♦✐ ❛♣❡♥❛s ❞❡ ✐♥t❡r❡ss❡ t❡ór✐❝♦✱ ♣♦✐s ♥❛ ♣rát✐❝❛ ♦ ♠ét♦❞♦ ❞♦s ❡❧✐♣s♦✐❞❡s ❡r❛ ♣♦✉❝♦ ❡✜❝✐❡♥t❡✳ ❊♠ ✶✾✽✹✱ ◆❛r❡♥❞r❛ ❑❛r♠❛r❦❛r ✐♥tr♦❞✉③✐✉ ♦ ♠ét♦❞♦ ❞❡ ♣♦♥t♦s ✐♥t❡r✐♦r❡s✱ q✉❡ t❛♠❜é♠ t❡♠ ❝♦♥s✉♠♦ ❞❡ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧ ♥♦ t❛♠❛♥❤♦ ❞❛ ❡♥tr❛❞❛✱ ♠❛s ❛❧é♠ ❞✐ss♦✱ ♦❜té♠ ✉♠ ❜♦♠ ❞❡s❡♠♣❡♥❤♦ ♥❛ ♣rát✐❝❛✱ s✉♣❡r❛♥❞♦ ♦ s✐♠♣❧❡① ❡♠ ❛❧❣✉♠❛s s✐t✉❛çõ❡s✳ ❆♣❡s❛r ❞❡ ♦ ❝♦♥s✉♠♦ ❞❡ t❡♠♣♦ ♥♦ ♣✐♦r ❝❛s♦ ❞♦ ❛❧❣♦r✐t♠♦ s✐♠♣❧❡① s❡r ❡①♣♦♥❡♥❝✐❛❧ ♥♦ t❛♠❛♥❤♦ ❞❛ ❡♥tr❛❞❛✱ ❛té ♦s ❞✐❛s ❞❡ ❤♦❥❡ ❡❧❡ é ♦ ❛❧❣♦r✐t♠♦ ♠❛✐s ✉t✐❧✐③❛❞♦ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❣r❛♠❛s ❧✐♥❡❛r❡s✳ ■ss♦ ❞❡✈❡✲s❡ ❛♦ ❢❛t♦ ❞❡ q✉❡ ♦s ❝❛s♦s ❡♠ q✉❡ s❡✉ ❝♦♥s✉♠♦ ❞❡ t❡♠♣♦ é ❡①♣♦♥❡♥❝✐❛❧ ❛♣❛r❡❝❡♠ ♠✉✐t♦ r❛r❛♠❡♥t❡ ♥❛ ♣rát✐❝❛✳
✶✳✶✳✷ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ■♥t❡✐r❛
◆❛ ár❡❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ■♥t❡✐r❛✱ ❝♦♥s✐❞❡r❛♠♦s ♣r♦❣r❛♠❛s ❧✐♥❡❛r❡s ♥♦s q✉❛✐s ❛s ✈❛r✐á✈❡✐s ❞❡✈❡♠ ❛ss✉♠✐r ❛♣❡♥❛s ✈❛❧♦r❡s ✐♥t❡✐r♦s✳ ❆❜❛✐①♦✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ Pr♦❣r❛♠❛ ▲✐♥❡❛r ■♥t❡✐r♦ ✭P▲■✮✳
♠❛①✐♠✐③❡ c⊤x (P■) s✉❥❡✐t♦ ❛ Ax ≤ b
x ∈ Nn
✹ ■◆❚❘❖❉❯➬➹❖ ✶✳✷
✶✳✶✳✸ ❆❧❣♦r✐t♠♦s ❞❡ ❆♣r♦①✐♠❛çã♦
❯♠ ♣r♦❜❧❡♠❛ ❞❡ ♦t✐♠✐③❛çã♦ ✭♠✐♥✐♠✐③❛çã♦ ♦✉ ♠❛①✐♠✐③❛çã♦✮ é ❝♦♠♣♦st♦ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✐♥stâ♥❝✐❛s✱ ✉♠ ❝♦♥❥✉♥t♦Sol(I)❞❡ s♦❧✉çõ❡s ✈✐á✈❡✐s ♣❛r❛ ❝❛❞❛ ✐♥stâ♥❝✐❛ I✱ ❡ ✉♠❛ ❢✉♥çã♦ q✉❡ ❛tr✐❜✉✐ ✉♠ ♥ú♠❡r♦ ♥ã♦✲♥❡❣❛t✐✈♦ val(S) ❛ ❝❛❞❛ s♦❧✉çã♦ ✈✐á✈❡❧ S✳ ❆ss✉♠❛ q✉❡ ❡st❛♠♦s tr❛t❛♥❞♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♠✐♥✐♠✐③❛çã♦✳ ❯♠❛ s♦❧✉çã♦ S é ót✐♠❛ s❡ val(S) ≤val(S′) ✭♥♦ ❝❛s♦ ❞❡ ♠❛①✐♠✐③❛çã♦✱ val(S) ≥ val(S′)✮✱ ♣❛r❛ t♦❞❛ s♦❧✉çã♦ ✈✐á✈❡❧ S′✳ ❖ ✈❛❧♦r ❞❡ q✉❛❧q✉❡r ❞❛s s♦❧✉çõ❡s ót✐♠❛s ♣❛r❛ ✉♠❛ ✐♥stâ♥❝✐❛ I s❡rá ❞❡♥♦t❛❞♦ ♣♦r opt(I)✳ ❙❡❥❛ A ✉♠ ❛❧❣♦r✐t♠♦ q✉❡✱ ♣❛r❛ t♦❞❛ ✐♥stá♥❝✐❛ I ❞♦ ♣r♦❜❧❡♠❛✱ ❞❡✈♦❧✈❡ ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧ A(I)❞❡ Sol(I)✳ ❖❜s❡r✈❡ q✉❡ val(A(I))≥opt(I) ✭♥♦ ❝❛s♦ ❞❡ ♠❛①✐♠✐③❛çã♦✱ val(A(I))≤opt(I)✮✱ ♣❛r❛ t♦❞❛ ✐♥stâ♥❝✐❛ I✳ ❉✐③❡♠♦s q✉❡A é ✉♠❛α✲❛♣r♦①✐♠❛çã♦ s❡ val(A(I))≤α·opt(I) ✭♥♦ ❝❛s♦ ❞❡ ♠❛①✐♠✐③❛çã♦✱val(A(I))≥α·opt(I)✮✱ ♣❛r❛ t♦❞❛ ✐♥stâ♥❝✐❛ I✳ ❖ ✈❛❧♦r ❞❡α é ♦ ❢❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦ ❞♦ ❛❧❣♦r✐t♠♦ A✳ ❖❜s❡r✈❡ q✉❡ α≥1 ✭♥♦ ❝❛s♦ ❞❡ ♠❛①✐♠✐③❛çã♦✱ α≤1✮✳
❖ ✐♥t❡r❡ss❡ ♣♦r ❛❧❣♦r✐t♠♦s ❞❡ ❛♣r♦①✐♠❛çã♦ s✉r❣❡ ❡♠ ✉♠ ❝♦♥t❡①t♦ ♥♦ q✉❛❧ ♦ ♣r♦❜❧❡♠❛ ❛ s❡r r❡s♦❧✈✐❞♦ é ◆P✲❞✐❢í❝✐❧ ❡ ❞❡s❡❥❛✲s❡ ❡♥❝♦♥tr❛r ✉♠ ❛❧❣♦r✐t♠♦ ❝♦♠ ❝♦♥s✉♠♦ ❞❡ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧ ♥♦ t❛♠❛♥❤♦ ❞❛ ❡♥tr❛❞❛ q✉❡ ❣❛r❛♥t❛ ✉♠ ❜♦♠ ❢❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦✳ ❯♠ ♣r♦❜❧❡♠❛ ❞❡ ♦t✐♠✐③❛çã♦ é ❞✐t♦ ✐♥❛♣r♦①✐♠á✈❡❧ s❡ ❡❧❡ ♥ã♦ ❛❞♠✐t❡ ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧ ❝♦♠ ❢❛t♦r ❞❡ ❛♣r♦①✐♠❛çã♦ O(nǫ)✭♥♦ ❝❛s♦ ❞❡ ♠❛①✐♠✐③❛çã♦ O(1nǫ)✮✱ ♦♥❞❡né ♦ t❛♠❛♥❤♦ ❞❛ ❡♥tr❛❞❛✱ ♣❛r❛ ❛❧❣✉♠ǫ >0✱ ❛ ♠❡♥♦s
q✉❡ P=◆P✳
✶✳✷ ◆♦t❛çã♦ ❡ ❉❡✜♥✐çõ❡s
❙❡❥❛ G= (V, E) ✉♠ ❣r❛❢♦ ❜✐♣❛rt✐❞♦✳ ❆s ❞✉❛s ❝❧❛ss❡s ❞❛ ❜✐♣❛rt✐çã♦ ❞❡V sã♦ ❞❡♥♦t❛❞❛s ♣♦rL ❡ R ✭♦✉ s❡❥❛✱ V =L∪R ❡ L∩R =∅✮✳ P♦r s✐♠♣❧✐❝✐❞❛❞❡✱ ❞❛❞❛ ✉♠❛ ❛r❡st❛uv ❞❡ E✱ ❛ss✉♠✐♠♦s ♣♦r ❝♦♥✈❡♥çã♦ q✉❡u∈L❡v∈R✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♦ rót✉❧♦ ❞♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❝♦rr❡s♣♦♥❞❡ ❛♦ ✈ért✐❝❡
q✉❡ ♣❡rt❡♥❝❡ à ❝❧❛ss❡ L ❡ ♦ rót✉❧♦ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❝♦rr❡s♣♦♥❞❡ ❛♦ ✈ért✐❝❡ q✉❡ ♣❡rt❡♥❝❡ à ❝❧❛ss❡ R✳ ❉❛❞♦ ✉♠ ✈ért✐❝❡ u ∈ V✱ ❞❡♥♦t❛♠♦s ♣♦r N(u) ♦ ❝♦♥❥✉♥t♦ ❞♦s ✈ért✐❝❡s ❛❞❥❛❝❡♥t❡s ❛ u✳ ❉❛❞♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s V′ ⊂V✱ ❞❡♥♦t❛♠♦s ♣♦r N∩(V′) =∩u∈V′N(u)♦ ❝♦♥❥✉♥t♦ ❞♦s ✈ért✐❝❡s q✉❡ sã♦ ❛❞❥❛❝❡♥t❡s ❛ t♦❞♦s ♦s ✈ért✐❝❡s ❞❡V′✱ ❡ ❞❡♥♦t❛♠♦s ♣♦r G[V′]♦ s✉❜❣r❛❢♦ ❞❡ G✐♥❞✉③✐❞♦ ♣♦r V′✳ ❉❛❞♦ ✉♠ s✉❜❣r❛❢♦ B ❞❡ G✱ ❞❡♥♦t❛♠♦s s❡✉ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s ♣♦r EB ❡ s❡✉ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s ♣♦rVB✳ ❆s ❞✉❛s ❝❧❛ss❡s ❞❛ ❜✐♣❛rt✐çã♦ ❞❡VB sã♦ ❞❡♥♦t❛❞❛s ♣♦rLB =VB∩L❡ RB =VB∩R✳
❉✐③❡♠♦s q✉❡ ✉♠ ❣r❛❢♦ ❜✐♣❛rt✐❞♦ B é ✉♠ ❜✐❝❧✐q✉❡ s❡ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ✈ért✐❝❡s u ❡ v✱ t❛❧ q✉❡ u∈LB ❡v∈RB✱ t❡♠♦s q✉❡uv ∈EB✳ ❉❛❞♦s s✉❜❝♦♥❥✉♥t♦sA⊆L❡ C⊆R✱ t❛✐s q✉❡uv ∈E✱ ♣❛r❛ t♦❞♦ u ∈ A ❡ v ∈ C✱ ❞❡♥♦t❛♠♦s ♣♦r (A, C) ♦ ❜✐❝❧✐q✉❡ B✱ ♦♥❞❡ LB = A ❡ RB =C✱ s❡♥❞♦ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛sEB={uv∈E |u∈A ❡v∈C}✜❝❛ ✐♠♣❧í❝✐t♦✳ P♦r s✐♠♣❧✐❝✐❞❛❞❡✱ ✉t✐❧✐③❛r❡♠♦s ❛ ❢r❛s❡ ✏✉♠ ❜✐❝❧✐q✉❡B ❡♠G✑✱ ❝♦♠ ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ✏✉♠ ❜✐❝❧✐q✉❡B s✉❜❣r❛❢♦ ❞❡G✑✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❜✐❝❧✐q✉❡B ❡♠Gé ♠❛①✐♠❛❧✱ s❡ ♥ã♦ ❡①✐st❡ ❜✐❝❧✐q✉❡B′ ❡♠G✱ t❛❧ q✉❡E
B ⊂EB′✱ ♦✉ ♠á①✐♠♦✱ s❡ ♥ã♦ ❡①✐st❡ ❜✐❝❧✐q✉❡ B′ ❡♠G✱ t❛❧ q✉❡|EB′|>|EB|✳
❯♠ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s ❡♠Gé ✉♠ ❝♦♥❥✉♥t♦P ❞❡ ❜✐❝❧✐q✉❡s ❡♠G✱ ❞♦✐s ❛ ❞♦✐s ❞✐s❥✉♥t♦s ♥♦s ✈ért✐❝❡s q✉❡ ❡stã♦ ❝♦♥t✐❞♦s ❡♠ S✱ ♦♥❞❡ S = V ♦✉ S ∈ {L, R}✳ ❙❡ S = V✱ ❝❛❞❛ ✈ért✐❝❡ ♣♦❞❡ ❡st❛r ❝♦♥t✐❞♦ ❡♠ ♥♦ ♠á①✐♠♦ ✉♠ ❜✐❝❧✐q✉❡ ❞❡P✱ ❡♥q✉❛♥t♦ q✉❡ s❡S ∈ {L, R}✱ ❡st❛ r❡str✐çã♦ ❛♣❧✐❝❛✲s❡
✶✳✸ ❆P▲■❈❆➬Õ❊❙ P❘➪❚■❈❆❙ ✺
Pr♦❜❧❡♠❛ ✶✳✷✳✶✳ ❉❛❞♦ ✉♠ ❣r❛❢♦ ❜✐♣❛rt✐❞♦ G ❡ ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ k✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♠♣❛❝♦✲ t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s ❝♦♥s✐st❡ ❡♠ ❡♥❝♦♥tr❛r ✉♠ ❡♠♣❛❝♦t❛♠❡♥t♦ P ❞❡ ❜✐❝❧✐q✉❡s ❡♠ G✱ t❛❧ q✉❡
|P| ≤k ❡ P
B∈P|EB|s❡❥❛ ♠á①✐♠♦✳
◆♦ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s ❡♠ q✉❡ k = 1✱ t❡♠♦s ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦✱ q✉❡ ♣♦❞❡ s❡r ❡♥✉♥❝✐❛❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✳
Pr♦❜❧❡♠❛ ✶✳✷✳✷✳ ❉❛❞♦ ✉♠ ❣r❛❢♦ ❜✐♣❛rt✐❞♦ G✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦ ❝♦♥s✐st❡ ❡♠ ❡♥✲ ❝♦♥tr❛r ✉♠ ❜✐❝❧✐q✉❡ ♠á①✐♠♦ ❡♠ G✳
❯♠ ❝❛s♦ ♠❛✐s ❣❡r❛❧ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦ é ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ❞❡ ♣❡s♦ ♠á①✐♠♦✱ ❡♥✉♥❝✐❛❞♦ ❛ s❡❣✉✐r✳
Pr♦❜❧❡♠❛ ✶✳✷✳✸✳ ❉❛❞♦ ✉♠ ❣r❛❢♦ ❜✐♣❛rt✐❞♦ G ❡ ✉♠❛ ❢✉♥çã♦ w :E →R✱ t❛❧ q✉❡ w(uv)≥0✱ ♣❛r❛ t♦❞❛ ❛r❡st❛uv ❞❡ E✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ❞❡ ♣❡s♦ ♠á①✐♠♦ ❝♦♥s✐st❡ ❡♠ ❡♥❝♦♥tr❛r ✉♠ ❜✐❝❧✐q✉❡ B ❡♠ G✱ t❛❧ q✉❡ P
uv∈EBw(uv) s❡❥❛ ♠á①✐♠♦✳
❆♣❡s❛r ❞❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ❞❡ ♣❡s♦ ♠á①✐♠♦ ❡st❛r ❢♦r❛ ❞♦ ❡s❝♦♣♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦✱ ❥✉❧❣❛♠♦s ✐♥t❡r❡ss❛♥t❡ ❢❛③❡r ❛❧❣✉♠❛s ❝♦♥s✐❞❡r❛çõ❡s s♦❜r❡ ❡st❡ ♣r♦❜❧❡♠❛ ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✱ ♣♦✐s ♠✉✐t❛s ❞❛s ❛❜♦r❞❛❣❡♥s q✉❡ ❛♣r❡s❡♥t❛♠♦s ♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦ sã♦ ❢❛❝✐❧♠❡♥t❡ ❛❞❛♣tá✈❡✐s ♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ❞❡ ♣❡s♦ ♠á①✐♠♦✳
❉❛❞♦ ✉♠ ❣r❛❢♦ ❛r❜✐trár✐♦ H✱ ❞❡♥♦t❛♠♦s s❡✉ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s ♣♦r V(H) ❡ s❡✉ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s ♣♦r A(H)✳ ❆ s❡❣✉✐r✱ ❡♥✉♥❝✐❛♠♦s ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ❝❧áss✐❝♦s ❞❡ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛ q✉❡✱ ❝♦♥❢♦r♠❡ ✈❡r❡♠♦s ♥❛s ♣ró①✐♠❛s s❡çõ❡s✱ ♣♦ss✉❡♠ ❛❧❣✉♠❛ r❡❧❛çã♦ ❝♦♠ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦✳
Pr♦❜❧❡♠❛ ✶✳✷✳✹✳ ❉❛❞♦ ✉♠ ❣r❛❢♦H✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ ♠á①✐♠♦ ❝♦♥s✐st❡ ❡♠ ❡♥❝♦♥tr❛r ✉♠ ❝♦♥❥✉♥t♦ Q❞❡ ✈ért✐❝❡s ❞❡V(H)❞❡ ❝❛r❞✐♥❛❧✐❞❛❞❡ ♠á①✐♠❛✱ t❛❧ q✉❡{u, v}∈/ A(H)✱ ♣❛r❛ t♦❞♦ ♣❛r u, v ❡♠Q✳
Pr♦❜❧❡♠❛ ✶✳✷✳✺✳ ❉❛❞♦ ✉♠ ❣r❛❢♦ H✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❝❧✐q✉❡ ♠á①✐♠♦ ❝♦♥s✐st❡ ❡♠ ❡♥❝♦♥tr❛r ✉♠ ❝♦♥❥✉♥t♦Q ❞❡ ✈ért✐❝❡s ❞❡ V(H) ❞❡ ❝❛r❞✐♥❛❧✐❞❛❞❡ ♠á①✐♠❛✱ t❛❧ q✉❡ {u, v} ∈A(H)✱ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ✈ért✐❝❡s ❞✐st✐♥t♦su, v ❡♠ Q✳
Pr♦❜❧❡♠❛ ✶✳✷✳✻✳ ❉❛❞♦ ✉♠ ❣r❛❢♦H ❡ ✉♠❛ ❢✉♥çã♦w:V(H)→R✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❝❧✐q✉❡ ❞❡ ♣❡s♦
♠á①✐♠♦ ❝♦♥s✐st❡ ❡♠ ❡♥❝♦♥tr❛r ✉♠ ❝♦♥❥✉♥t♦ Q ❞❡ ✈ért✐❝❡s ❞❡ V(H)✱ t❛❧ q✉❡ {u, v} ∈A(H)✱ ♣❛r❛ t♦❞♦ ♣❛r ❞❡ ✈ért✐❝❡s ❞✐st✐♥t♦s u, v ❡♠Q✱ ❡ P
v∈Qw(v) é ♠á①✐♠♦✳
Pr♦❜❧❡♠❛ ✶✳✷✳✼✳ ❉❛❞♦ ✉♠ ❣r❛❢♦H✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❝♦❜❡rt✉r❛ ♠í♥✐♠❛ ❞❡ ❛r❡st❛s ♣♦r ✈ér✲ t✐❝❡s ❝♦♥s✐st❡ ❡♠ ❡♥❝♦♥tr❛r ✉♠ ❝♦♥❥✉♥t♦ Q ❞❡ ✈ért✐❝❡s ❞❡V(H) ❞❡ ❝❛r❞✐♥❛❧✐❞❛❞❡ ♠í♥✐♠❛✱ t❛❧ q✉❡ ♣❛r❛ t♦❞❛ ❛r❡st❛ {u, v} ❡♠ A(H)✱ t❡♠♦s q✉❡ u∈Q ♦✉ v∈Q✳
✶✳✸ ❆♣❧✐❝❛çõ❡s Prát✐❝❛s
✻ ■◆❚❘❖❉❯➬➹❖ ✶✳✸
♥❛ ♠❡s♠❛ ♣r♦♣♦rçã♦✳ ❯♠ ♠♦❞♦ é ❞✐t♦ ❡❧❡♠❡♥t❛r q✉❛♥❞♦ ♥ã♦ ❡①✐st❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ ♣ró♣r✐♦ ❞❡ s✉❛s r❡❛çõ❡s q✉❡ s❡❥❛ t❛♠❜é♠ ✉♠ ♠♦❞♦✳ ◆♦ ❛rt✐❣♦ ❞❡❙❝❤✉st❡r ❡ ❍✐❧❣❡t❛❣✭✶✾✾✹✮✱ é ❞❛❞❛ ✉♠❛ ❡①♣❧❛♥❛çã♦ ♠❛✐s ❞❡t❛❧❤❛❞❛ ❞❡ss❡s ❝♦♥❝❡✐t♦s✱ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❡ ❜✐♦❧♦❣✐❛✳ ◆ã♦ r❛r♦✱ ♦ ♥ú♠❡r♦ ❞❡ ♠♦❞♦s ❡❧❡✲ ♠❡♥t❛r❡s ♣r❡s❡♥t❡s ❡♠ ✉♠❛ r❡❞❡ ♠❡t❛❜ó❧✐❝❛ é ❡♥♦r♠❡ ❡✱ ♣♦rt❛♥t♦✱ t♦r♥❛✲s❡ ✐♠♣r❛t✐❝á✈❡❧ ❛♥❛❧✐sá✲❧♦s ❞✐r❡t❛♠❡♥t❡✱ s❡♠ ❛❣r✉♣á✲❧♦s ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛✳ P♦r ❡①❡♠♣❧♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠❚❡r③❡r ❡ ❙t❡❧❧✐♥❣✭✷✵✵✽✮✱ ♦ ♠❡t❛❜♦❧✐s♠♦ ❝❡♥tr❛❧ ❞❛ ❜❛❝tér✐❛ ❊s❝❤❡r✐❝❤✐❛ ❈♦❧✐ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ r❡❞❡ ♠❡t❛❜ó❧✐❝❛ ❝♦♠ ❛♣r♦①✐✲ ♠❛❞❛♠❡♥t❡ ❝❡♠ r❡❛çõ❡s q✉í♠✐❝❛s ❡ ✈✐♥t❡ ❡ s❡✐s ♠✐❧❤õ❡s ❞❡ ♠♦❞♦s ❡❧❡♠❡♥t❛r❡s✳ ◆❡st❡ ❝♦♥t❡①t♦✱ ✉♠❛ ❢♦r♠❛ ❞❡ ❢❛❝✐❧✐t❛r ❛ ✐♥t❡r♣r❡t❛çã♦ ❞♦s ❞❛❞♦s é s❡♣❛r❛r k ❝♦♥❥✉♥t♦s ❞❡ r❡❛çõ❡s q✉í♠✐❝❛s ❞❡ ❢♦r♠❛ q✉❡ ❝❛❞❛ ❝♦♥❥✉♥t♦ ❝♦♥t❡♥❤❛ r❡❛çõ❡s q✉❡ ♣❛rt✐❝✐♣❛♠ ❥✉♥t❛s ❞❡ ♠✉✐t♦s ♠♦❞♦s ❡❧❡♠❡♥t❛r❡s✱ ❝♦♥❢♦r♠❡ ❞❡s❝r✐t♦ ♥♦ tr❛❜❛❧❤♦ ❞❡◆♦t❡❜❛❛rt ❡t ❛❧✳✭✷✵✵✽✮✳ ■st♦ ❝♦rr❡s♣♦♥❞❡ ❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♠♣❛❝♦✲ t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s ♥♦ ❣r❛❢♦ ❡♠ q✉❡Lr❡♣r❡s❡♥t❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ r❡❛çõ❡s q✉í♠✐❝❛s ❡Rr❡♣r❡s❡♥t❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ ♠♦❞♦s ❡❧❡♠❡♥t❛r❡s✳ ❊♠ t❛❧ ❣r❛❢♦✱ ❡①✐st❡ ✉♠❛ ❛r❡st❛uv∈E s❡ ❡ só s❡ ❛ r❡❛çã♦u❡stá ♣r❡s❡♥t❡ ♥♦ ♠♦❞♦v✱ ❡ ♦s ❜✐❝❧✐q✉❡s ❡♠♣❛❝♦t❛❞♦s sã♦ ❞♦✐s ❛ ❞♦✐s ❞✐s❥✉♥t♦s ♥♦s s❡✉s s✉❜❝♦♥❥✉♥t♦s ❞❡ r❡❛çõ❡s q✉í♠✐❝❛s ✭♦✉ s❡❥❛✱S=L✮✳ ◆♦t❡ q✉❡ ♥❡st❛ ❛♣❧✐❝❛çã♦ ♦ ❣r❛❢♦Gé ❜❛st❛♥t❡ ❞❡s❜❛❧❛♥❝❡❛❞♦ ✭♥♦ ❡①❡♠♣❧♦ ❝✐t❛❞♦✱|L| ≃100❡ |R| ≃26.000.000✮✳
❖✉tr❛ ❛♣❧✐❝❛çã♦ q✉❡ ❝♦♥s✐❞❡r❛♠♦s é ❛ ❞❡ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ♣r♦❞✉t♦s✱ ♥❛ q✉❛❧ sã♦ ❞❛❞♦s ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝❧✐❡♥t❡s ❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣r♦❞✉t♦s✱ ❜❡♠ ❝♦♠♦ ❛ ❞❡♠❛♥❞❛ ❞❡ ❝❛❞❛ ❝❧✐❡♥t❡ ♣♦r ❝❛❞❛ ♣r♦❞✉t♦✱ ❡ ♦ ♦❜❥❡t✐✈♦ é ❞❡✜♥✐r ♥♦ ♠á①✐♠♦ k ♣❛❝♦t❡s ❞❡ ♣r♦❞✉t♦s ❡ ❡st✐♣✉❧❛r ❛s q✉❛♥t✐❞❛❞❡s ❞❡ ❝❛❞❛ ♣r♦❞✉t♦ ❡♠ ❝❛❞❛ ♣❛❝♦t❡✱ ❞❡ ❢♦r♠❛ ❛ ♠❛①✐♠✐③❛r ❛ ✈❡♥❞❛ t♦t❛❧ ❞❡ ♣r♦❞✉t♦s✳ ❆ss✉♠❡✲s❡ q✉❡ ❝❛❞❛ ❝❧✐❡♥t❡ ❝♦♠♣r❛ ❛ q✉❛♥t✐❞❛❞❡ ♠á①✐♠❛ ❞❡ ♣❛❝♦t❡s t❛❧ q✉❡ ♥ã♦ ✉❧tr❛♣❛ss❡ s✉❛ ❞❡♠❛♥❞❛ ♣♦r ♥❡♥❤✉♠ ♣r♦❞✉t♦✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ♣r♦❞✉t♦s é ❝♦♥s✐❞❡r❛❞♦ ♥♦s tr❛❜❛❧❤♦s ❞❡
▼❝❈❛r❞❧❡ ❡t ❛❧✳ ✭✷✵✵✼✮❀ ❙tr❡♠❡rs❝❤ ❡ ❚❡❧❧✐s ✭✷✵✵✷✮✳ ◆❡st❛ ❛♣❧✐❝❛çã♦✱ ♦s ✈ért✐❝❡s ❞❡ L r❡♣r❡s❡♥t❛♠ ♦s ❝❧✐❡♥t❡s✱ ♦s ✈ért✐❝❡s ❞❡ R r❡♣r❡s❡♥t❛♠ ♦s ♣r♦❞✉t♦s ❡ ❝❛❞❛ ❛r❡st❛uv ❡♠ E ✐♥❞✐❝❛ q✉❡ ♦ ❝❧✐❡♥t❡ u ♣♦ss✉✐ ❞❡♠❛♥❞❛ ♣♦s✐t✐✈❛ ♣❡❧♦ ♣r♦❞✉t♦ v✱ s❡♥❞♦ q✉❡ ♦ ✈❛❧♦r ❞❛ ❞❡♠❛♥❞❛ é ❞❛❞♦ ❛tr❛✈és ❞❡ ✉♠❛ ❢✉♥çã♦ ❞❡ ♣❡s♦s w : E → N ❛ss♦❝✐❛❞❛ às ❛r❡st❛s ❞❡ E✳ ❯♠ ❜✐❝❧✐q✉❡ ♥❡st❡ ❣r❛❢♦ ❝♦rr❡s♣♦♥❞❡ ❛
✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝❧✐❡♥t❡s q✉❡ ❞❡s❡❥❛♠ ❝♦♠♣r❛r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣r♦❞✉t♦s✳ ❖ ♦❜❥❡t✐✈♦ é ❡♥❝♦♥tr❛r k ♣❛❝♦t❡s ❞❡ ♣r♦❞✉t♦s q✉❡ ♠❛①✐♠✐③❛♠ ❛ ✈❡♥❞❛ t♦t❛❧ ❞❡ ♣r♦❞✉t♦s✱ s✉❥❡✐t♦ ❛♦ ❧✐♠✐t❡ ❞❛ ❞❡♠❛♥❞❛ ❞❡ ❝❛❞❛ ❝❧✐❡♥t❡ ♣♦r ❝❛❞❛ ♣r♦❞✉t♦✳ ◆♦ ❝❛s♦ ❡♠ q✉❡S =R✱ ❝❛❞❛ ❝❧✐❡♥t❡ ♣♦❞❡ ❝♦♠♣r❛r ♥♦ ♠á①✐♠♦ ✉♠ ♣❛❝♦t❡ ❞❡ ♣r♦❞✉t♦s✱ ❡ q✉❛♥❞♦S=L✱ ❝❛❞❛ ♣r♦❞✉t♦ ♣♦❞❡ ❛♣❛r❡❝❡r ❡♠ ♥♦ ♠á①✐♠♦ ✉♠ ♣❛❝♦t❡✳ ◆❡st❛ ❛♣❧✐❝❛çã♦ t❛♠❜é♠ ♦❝♦rr❡ q✉❡ ♦ ❣r❛❢♦ G é ❜❛st❛♥t❡ ❞❡s❜❛❧❛♥❝❡❛❞♦ ✭♦ ♥ú♠❡r♦ ❞❡ ❝❧✐❡♥t❡s é ♠✉✐t♦ ♠❛✐♦r ❞♦ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ♣r♦❞✉t♦s✮✳ ◆♦ ❈❛♣ít✉❧♦ ✹✱ ❝♦♥s✐❞❡r❛♠♦s ♦ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❝r✐❛çã♦ ❞❡ ♣❛❝♦t❡s ❡♠ q✉❡ k= 1✳ ❈♦♥s✐❞❡r❡ ✉♠❛ ✈❛r✐❛♥t❡ s✐♠♣❧✐✜❝❛❞❛ ❞❡st❛ ❛♣❧✐❝❛çã♦ ❡♠ q✉❡ ❛s ❞❡♠❛♥❞❛s sã♦ ❜✐♥ár✐❛s✳ ◆❡st❡ ❝❛s♦✱ t❡♠♦s ✉♠❛ ✐♥stâ♥❝✐❛ ❞♦ ♣r♦❜❧❡♠ ❞❡ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s✳
✶✳✹ ❘❊❙❯▲❚❆❉❖❙ ❊◆❈❖◆❚❘❆❉❖❙ ◆❆ ▲■❚❊❘❆❚❯❘❆ ✼
P♦rt❛♥t♦✱ q✉❛♥t♦ ♠❛✐♦r ❢♦r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♠♣♦♥❡♥t❡s ❝♦♥t✐❞♦s ❡♠ ✉♠ ✈❛♥✐❧❧❛ ❜♦①✱ ♠❡♥♦r ♦ ❝♦♥s✉♠♦ ❞❡ r❡❝✉rs♦s ♥❛ ♠♦♥t❛❣❡♠ ❞♦s ♣r♦❞✉t♦s q✉❡ ♦ ✉t✐❧✐③❛♠✳ ❯♠ ✈❛♥✐❧❧❛ ❜♦① só ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♥❛ ♠♦♥t❛❣❡♠ ❞❡ ✉♠ ♣r♦❞✉t♦ q✉❡ ❝♦♥té♠ t♦❞♦s ♦s s❡✉s ❝♦♠♣♦♥❡♥t❡s✳ P♦rt❛♥t♦✱ q✉❛♥t♦ ♠❛✐♦r ❢♦r ♦ ♥ú♠❡r♦ ❞❡ ❝♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ ✈❛♥✐❧❧❛ ❜♦①✱ ♠❡♥♦r t❡♥❞❡ ❛ s❡r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♣r♦❞✉t♦s q✉❡ ♣♦❞❡♠ ✉t✐❧✐③á✲❧♦ ❡♠ s✉❛ ♠♦♥t❛❣❡♠✳ ❯♠ ❡♠♣❛❝♦t❛♠❡♥t♦ ❞❡ ❜✐❝❧✐q✉❡s✱ ❝♦♥s✐❞❡r❛♥❞♦S =R✱ r❡♣r❡s❡♥t❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡k ✈❛♥✐❧❧❛ ❜♦①❡s q✉❡ r❡s✉❧t❛♠ ♥❛ ♠❛✐♦r ❡❝♦♥♦♠✐❛ ❞❡ r❡❝✉rs♦s✱ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ♣r♦❞✉çã♦ t♦t❛❧ ❞♦s ♣r♦❞✉t♦s✳ ◆♦ ❛rt✐❣♦ ❞❡❙✇❛♠✐♥❛t❤❛♥ ❡ ❚❛②✉r✭✶✾✾✽✮✱ sã♦ ❛♣r❡s❡♥t❛❞❛s ❛♣❡♥❛s ❤❡✉ríst✐❝❛s ♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❡ sã♦ ❝♦♥s✐❞❡r❛❞♦s ♦✉tr♦s ❢❛t♦r❡s✱ t❛✐s ❝♦♠♦ ❛ ❞❡♠❛♥❞❛ ❡s♣❡r❛❞❛ ❞❡ ❝❛❞❛ ♣r♦❞✉t♦✱ ❝✉st♦ ♣❛r❛ ♠❛♥t❡r ✉♠ ✈❛♥✐❧❧❛ ❜♦① ❡♠ ❡st♦q✉❡✱ ❡t❝✳ ❊ss❛ ❛♣❧✐❝❛çã♦ s✉r❣✐✉ ❝♦♠♦ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ❞✐♠✐♥✉✐r ♦ t❡♠♣♦ ❞❡ ❡s♣❡r❛ ♣❡❧❛ ♠♦♥t❛❣❡♠ ❞❡ ❝♦♠♣✉t❛❞♦r❡s ♥❛ ❢á❜r✐❝❛ ❞❛ ■❇▼✱ ❛t❡♥❞❡♥❞♦ ❛ ♣❡❞✐❞♦s ❝✉st♦♠✐③❛❞♦s✱ ♥♦s q✉❛✐s ♦s ❝❧✐❡♥t❡s ♣♦❞✐❛♠ ❡s❝♦❧❤❡r ❝♦♠ ❣r❛♥❞❡ ✢❡①✐❜✐❧✐❞❛❞❡ ❛s ❝♦♥✜❣✉r❛çõ❡s ❞♦s ❝♦♠♣✉t❛❞♦r❡s q✉❡ ❡st❛✈❛♠ ❛❞q✉✐r✐♥❞♦✳
◆♦ ❛rt✐❣♦ ❞❡❉✐♥❣ ❡t ❛❧✳✭✷✵✵✻✮✱ é ❛♣r❡s❡♥t❛❞❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦ ♥❛ ❛♥á❧✐s❡ ❞❡ r❡❞❡s ❞❡ ✐♥t❡r❛çã♦ ❞❡ ❣❡♥❡s✳ ◆❡st❛ ❛♣❧✐❝❛çã♦✱ ♦s ✈ért✐❝❡s ❞❡Lr❡♣r❡s❡♥t❛♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡♥❡s✱ ♦s ✈ért✐❝❡s ❞❡ R r❡♣r❡s❡♥t❛♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝♦♥❞✐çõ❡s ♦❜s❡r✈❛❞❛s ❡♠ ❡①♣❡r✐♠❡♥t♦s ❡ ✉♠❛ ❛r❡st❛uv ❡stá ♣r❡s❡♥t❡ ❡♠E s❡ ❡ só s❡ ♦ ❣❡♥❡ué ❡①♣r❡ss♦ ♥❛ ❝♦♥❞✐çã♦v✳ ❉❡s❡❥❛✲s❡ ❡♥❝♦♥tr❛r ✉♠ ❜✐❝❧✐q✉❡ ♠á①✐♠♦ ❡♠ G✱ q✉❡ r❡♣r❡s❡♥t❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡♥❡s ❡ ❝♦♥❞✐çõ❡s ❞❡ ♠❛✐♦r ✐♥t❡r❛çã♦✳
▼❛❞❡✐r❛ ❡ ❖❧✐✈❡✐r❛ ✭✷✵✵✹✮ ❛♣r❡s❡♥t❛♠ ✉♠ r❡s✉♠♦ ♥♦ q✉❛❧ ❧✐st❛♠ ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦ ❡♠ ❛♥á❧✐s❡ ❞❡ ❞❛❞♦s ❜✐♦❧ó❣✐❝♦s ❡ t❛♠❜é♠ ❡♠ ♦✉tr❛s ár❡❛s✳
✶✳✹ ❘❡s✉❧t❛❞♦s ❊♥❝♦♥tr❛❞♦s ♥❛ ▲✐t❡r❛t✉r❛
❙❡❥❛ G = (V, E) ✉♠ ❣r❛❢♦ ❜✐♣❛rt✐❞♦✳ ◆♦ ❧✐✈r♦ ❞❡ ●❛r❡② ❡ ❏♦❤♥s♦♥ ✭✶✾✼✾✮✱ sã♦ ❞✐s❝✉t✐❞❛s ❛s ❝♦♠♣❧❡①✐❞❛❞❡s ❞❡ ❞✐❢❡r❡♥t❡s ♣r♦❜❧❡♠❛s r❡❧❛❝✐♦♥❛❞♦s ❛ ❡♥❝♦♥tr❛r ✉♠ ❜✐❝❧✐q✉❡ ❡♠G✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❡♥❝♦♥tr❛r ✉♠ ❜✐❝❧✐q✉❡ ❜❛❧❛♥❝❡❛❞♦ B ❡♠ G✱ t❛❧ q✉❡ |LB| = |RB| = k✱ ♣❛r❛ ✉♠ ❞❛❞♦ ✐♥t❡✐r♦ k✱ é ◆P✲❝♦♠♣❧❡t♦✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ ❞❡s❡❥❛✲s❡ ❡♥❝♦♥tr❛r ✉♠ ❜✐❝❧✐q✉❡ ❡♠G✱ t❛❧ q✉❡|LB|+|RB| ≥k✱ ❡①✐st❡ ❛❧❣♦r✐t♠♦ ❝♦♠ ❝♦♥s✉♠♦ ❞❡ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧ ♥♦ t❛♠❛♥❤♦ ❞❛ ❡♥tr❛❞❛✳ ❊ss❛ ❛✜r♠❛çã♦ ❜❛s❡✐❛✲s❡ ♥♦ ❢❛t♦ ❞❡ q✉❡ ❡♥❝♦♥tr❛r ✉♠ ❜✐❝❧✐q✉❡ ❝♦♠ k ✈ért✐❝❡s ❡♠ G é ❡q✉✐✈❛❧❡♥t❡ ❛ ❡♥❝♦♥tr❛r ✉♠ ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ ♠á①✐♠♦ ♥♦ ❣r❛❢♦ ❜✐♣❛rt✐❞♦ G′ = (V′, E′)✱ ♦♥❞❡ V′ =V ❡ E′ ={uv |u∈L, v ∈R ❡ uv /∈E}✳ ❈♦♠♦G′ é ❜✐♣❛rt✐❞♦✱ ❝♦♥❢♦r♠❡ ♦ t❡♦r❡♠❛ ❞❡ ❑ö♥✐❣✱ ♥❡st❡ ❝❛s♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ ♠á①✐♠♦ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♠♣❛r❡❧❤❛♠❡♥t♦ ♠á①✐♠♦✱ q✉❡ ♣♦❞❡ s❡r r❡s♦❧✈✐❞♦ ❡♠ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧✳ ▼❡s♠♦ ♥♦ ❝❛s♦ ❡♠ q✉❡ é ❞❛❞♦ ✉♠ ❣r❛❢♦ ❛r❜✐trár✐♦ H ✭♦✉ s❡❥❛✱H ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ é ❜✐♣❛rt✐❞♦✮✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♥❝♦♥tr❛r ✉♠ ❜✐❝❧✐q✉❡ ❝♦♠ ♥ú♠❡r♦ ♠á①✐♠♦ ❞❡ ✈ért✐❝❡s ♣♦❞❡ s❡r r❡s♦❧✈✐❞♦ ❡♠ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧✳ ❊ss❛ ❛✜r♠❛çã♦ ❜❛s❡✐❛✲s❡ ♥♦ ❢❛t♦ ❞❡ q✉❡ ❡①✐st❡ ✉♠❛ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡ ♦s ❜✐❝❧✐q✉❡s ❞❡H ❡ ♦s ❜✐❝❧✐q✉❡s ❞♦ ❣r❛❢♦ ❜✐♣❛rt✐❞♦G✱ ♦♥❞❡ ❝❛❞❛ ❝❧❛ss❡ L ❡ R ❞♦s ✈ért✐❝❡s ❞❡ G ❝♦♥s✐st❡ ❡♠ ✉♠❛ ❝ó♣✐❛ ❞❡ V(H) ❡ ♣❛r❛ ❝❛❞❛ ❛r❡st❛ {u, v} ❡♠ A(H)✱ t❡♠♦s ❛s ❛r❡st❛s uv ❡ vu ❡♠ E✳ ◆♦ ❡♥t❛♥t♦✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♥❝♦♥tr❛r ✉♠ ❜✐❝❧✐q✉❡ ✐♥❞✉③✐❞♦ ❝♦♠ ♥ú♠❡r♦ ♠á①✐♠♦ ❞❡ ✈ért✐❝❡s ❡♠ ✉♠ ❣r❛❢♦ ❛r❜✐trár✐♦ é ◆P✲❞✐❢í❝✐❧✱ ❝♦♥❢♦r♠❡ ♠♦str❛❞♦ ♥♦ tr❛❜❛❧❤♦ ❞❡
✽ ■◆❚❘❖❉❯➬➹❖ ✶✳✹
❜❧❡♠❛ ◆P✲❞✐❢í❝✐❧✱ ♣♦ré♠ s❡♠ ❢♦r♥❡❝❡r ♥❡♥❤✉♠❛ ♣r♦✈❛ ❞❡ q✉❡ t❛❧ ❛✜r♠❛çã♦ ❢♦ss❡ ✈❡r❞❛❞❡✐r❛✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♥♦ ❛rt✐❣♦ ❞❡ ❉❛✇❛♥❞❡ ❡t ❛❧✳ ✭✶✾✾✼✮ é ❛♣r❡s❡♥t❛❞♦ ✉♠ ❛❧❣♦r✐t♠♦ ❝♦♠ r❛③ã♦ ❞❡ ❛♣r♦✲ ①✐♠❛çã♦ ❡s♣❡r❛❞❛ ✐❣✉❛❧ ❛ ✷✱ ♣❛r❛ ❣r❛❢♦s ❛❧❡❛tór✐♦s s✉✜❝✐❡♥t❡♠❡♥t❡ ❞❡♥s♦s✳ ◆❡ss❡ ♠❡s♠♦ ❛rt✐❣♦✱ é ♠♦str❛❞♦ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ❞❡ ♣❡s♦ ♠á①✐♠♦ é ◆P✲❞✐❢í❝✐❧✳ ❆❧❡①❡ ❡t ❛❧✳ ✭✷✵✵✹✮ ❡
◆✉ss❜❛✉♠ ❡t ❛❧✳✭✷✵✶✵✮ ✐♥tr♦❞✉③✐r❛♠✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ ❛❧❣♦r✐t♠♦s ❝♦♠ ❝♦♥s✉♠♦ ❞❡ t❡♠♣♦ ♣♦❧✐✲ ♥♦♠✐❛❧ ♥♦ t❛♠❛♥❤♦ ❞❛ ❡♥tr❛❞❛ ♣❛r❛ ♦ ❝❛s♦ ❡s♣❡❝✐❛❧ ❡♠ q✉❡ G é ❝♦♥✈❡①♦✱ s❡♥❞♦ q✉❡ G é ❞✐t♦ ❝♦♥✈❡①♦ s❡ ❡①✐st❡ ✉♠❛ ♣❡r♠✉t❛çã♦ (v1, v2, . . . , vm) ❞♦s ✈ért✐❝❡s ❞❡ R✱ t❛❧ q✉❡ N(u) = (vi, vi+1, . . . ,
vi+|N(u)|−1) é ✉♠ s❡❣♠❡♥t♦ ❞❡ (v1, v2, . . . , vm)✱ ♣❛r❛ t♦❞♦ u ❡♠L✳ ◆♦ ❛rt✐❣♦ ❞❡ ❉✐♥❣ ❡t ❛❧✳✭✷✵✵✻✮✱ é ❛♣r❡s❡♥t❛❞♦ ✉♠ ♠ét♦❞♦ ❞❡ ❖t✐♠✐③❛çã♦ ❈♦♥tí♥✉❛ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦✳ ❚❛❧ ❛❧❣♦r✐t♠♦ é ❜❛s❡❛❞♦ ❡♠ ✉♠ t❡♦r❡♠❛ ❞❡ ▼♦t③❦✐♥✲❙tr❛✉s q✉❡ r❡❧❛❝✐♦♥❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❝❧✐q✉❡ ♠á✲ ①✐♠♦ ❛♦ ♣r♦❜❧❡♠❛ ❞❡ ♦t✐♠✐③❛r ✉♠❛ ❝❡rt❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✳ ❖s ❛✉t♦r❡s ❢❛③❡♠ ✉♠❛ ❛❞❛♣t❛çã♦ ❞❡ss❡ t❡♦r❡♠❛ ❡ ♠♦str❛♠ q✉❡ ♦ ❛❧❣♦r✐t♠♦ ❛♣r❡s❡♥t❛❞♦ ❝♦♥✈❡r❣❡ ❡♠ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧✱ ❝❛s♦ ❡①✐st❛ ❛♣❡♥❛s ✉♠ ❜✐❝❧✐q✉❡ ♠á①✐♠♦ ❡♠G✳ ◆❛ ♣rát✐❝❛✱ ❡ss❡ ❛❧❣♦r✐t♠♦ ♦❜té♠ ✉♠ ❞❡s❡♠♣❡♥❤♦ r❛③♦á✈❡❧ ❛♣❡♥❛s ♣❛r❛ ✐♥stâ♥❝✐❛s ❞❡ ♣❡q✉❡♥♦ t❛♠❛♥❤♦✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦ ♣♦ss✉✐ ✉♠❛ ❢♦rt❡ r❡❧❛çã♦ ❝♦♠ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ ♠á①✐♠♦ ♦✉✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ❝♦♠ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❝❧✐q✉❡ ♠á①✐♠♦✱ ♣❛r❛ ♦s q✉❛✐s ❡♥❝♦♥tr❛✲s❡ ✉♠❛ ✈❛st❛ ❧✐t❡r❛t✉r❛✳ ❆ s❡❣✉✐r✱ ♠♦str❛♠♦s ❡♠ ♠❛✐s ❞❡t❛❧❤❡s ❛ r❡❧❛çã♦ ❡♥tr❡ ❡ss❡s ♣r♦❜❧❡♠❛s✳
❉✐③❡♠♦s q✉❡ ❞✉❛s ❛r❡st❛suv ❡pq ❞❡ E sã♦ ✐♥❝♦♠♣❛tí✈❡✐s s❡u6=p✱v6=q ❡ uq /∈E ♦✉pv /∈E✳
❙❡❥❛ I ={{uv, pq} ∈E×E |uv ❡ pqsã♦ ✐♥❝♦♠♣❛tí✈❡✐s⑥ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ♣❛r❡s ❞❡ ❛r❡st❛s ✐♥❝♦♠♣❛tí✈❡✐s ❞❡ E✳ ❙❡❥❛ H ♦ ❣r❛❢♦ ❡♠ q✉❡ ♣❛r❛ ❝❛❞❛ ❛r❡st❛uv ❞❡ E ❡①✐st❡ ✉♠ ✈ért✐❝❡ ❞❡ rót✉❧♦ l(uv) ❡♠V(H)✱ ❡{l(uv), l(pq)} ∈A(H)s❡ ❡ só s❡{uv, pq} ∈ I✳ ❖❜s❡r✈❡ q✉❡ ❡♥❝♦♥tr❛r ✉♠ ❜✐❝❧✐q✉❡
♠á①✐♠♦ ❡♠G❡q✉✐✈❛❧❡ ❛ ❡♥❝♦♥tr❛r ✉♠ ❝♦♥❥✉♥t♦ ✐♥❞❡♣❡♥❞❡♥t❡ ♠á①✐♠♦ ❡♠H♦✉✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ❡♥❝♦♥tr❛r ✉♠ ❝❧✐q✉❡ ♠á①✐♠♦ ♥♦ ❣r❛❢♦ H✱ ♦♥❞❡H é ♦ ❝♦♠♣❧❡♠❡♥t♦ ❞❡ H ✭♦✉ s❡❥❛✱V(H) =V(H) ❡ e ∈ A(H) s❡ ❡ só s❡ e /∈ A(H)✮✳ ◆❛ ❙❡çã♦ ✷✳✶✱ ✈♦❧t❛r❡♠♦s ❛ ❢❛❧❛r s♦❜r❡ ❛ r❡❧❛çã♦ ❡♥tr❡ ❡ss❡s
♣r♦❜❧❡♠❛s✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❞❡❝✐❞✐r s❡ ✉♠ ❣r❛❢♦ ❝♦♥té♠ ✉♠ ❝❧✐q✉❡ ❝♦♠ k ✈ért✐❝❡s ❝♦♥st❛ ♥❛ ❢❛♠♦s❛ ❧✐st❛ ❞❡ ✷✶ ♣r♦❜❧❡♠❛s ◆P✲❝♦♠♣❧❡t♦s ❞❡ ❑❛r♣ ✭✶✾✼✷✮✳ ◆♦ ❛rt✐❣♦ ❞❡ ❇❡r♠❛♥ ❡ ❙❝❤♥✐t❣❡r ✭✶✾✾✷✮✱ é ♣r♦✈❛❞♦ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❝❧✐q✉❡ ♠á①✐♠♦ é ✐♥❛♣r♦①✐♠á✈❡❧✳ ❈♦♥s✐❞❡r❡ ❛ ✈❡rsã♦ ❞❡ ❞❡❝✐sã♦ ❞♦s ♣r♦❜❧❡♠❛s ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦ ❡ ❝❧✐q✉❡ ♠á①✐♠♦✱ ❝♦♥❢♦r♠❡ ❡♥✉♥❝✐❛♠♦s ❛❜❛✐①♦✳
Pr♦❜❧❡♠❛ ✶✳✹✳✶✳ ❉❛❞♦s ✉♠ ❣r❛❢♦H❡ ✉♠ ✐♥t❡✐r♦q✱ ❞❡❝✐❞✐r s❡H ❝♦♥té♠ ✉♠ ❝❧✐q✉❡ ❝♦♠q ✈ért✐❝❡s✳ Pr♦❜❧❡♠❛ ✶✳✹✳✷✳ ❉❛❞♦s ✉♠ ❣r❛❢♦ ❜✐♣❛rt✐❞♦ G ❡ ✉♠ ✐♥t❡✐r♦ k✱ ❞❡❝✐❞✐r s❡ G ❝♦♥té♠ ✉♠ ❜✐❝❧✐q✉❡ ❝♦♠ ♣❡❧♦ ♠❡♥♦s k ❛r❡st❛s✳
❆ ❝♦♠♣❧❡①✐❞❛❞❡ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦ ✜❝♦✉ ❡♠ ❛❜❡rt♦ ❛té ♦ ❛♥♦ ❞❡ ✷✵✵✸✳ ❆ s❡❣✉✐r✱ ❛♣r❡s❡♥t❛♠♦s ❛ ♣r♦✈❛ ❝♦♥t✐❞❛ ♥♦ ❛rt✐❣♦ ❞❡P❡❡t❡rs ✭✷✵✵✸✮✱ ♣❛r❛ ♠♦str❛r q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦ é ◆P✲❞✐❢í❝✐❧✱ ❛tr❛✈és ❞❛ r❡❞✉çã♦ ❞♦ Pr♦❜❧❡♠❛✶✳✹✳✶❛♦ Pr♦❜❧❡♠❛✶✳✹✳✷✳ ❊ss❛ ♣r♦✈❛ ❢♦✐ ✐♥s♣✐r❛❞❛ ♥❛ r❡❞✉çã♦ ❞❡ ●❛r❡② ❡ ❏♦❤♥s♦♥ ✭✶✾✼✾✮ ❞♦ Pr♦❜❧❡♠❛ ✶✳✹✳✶ ❛♦ ♣r♦❜❧❡♠❛ ❞❡ ❡♥❝♦♥tr❛r ✉♠ ❜✐❝❧✐q✉❡B ❜❛❧❛♥❝❡❛❞♦ ❝♦♠|LB|=|RB|=k ❡♠ ✉♠ ❣r❛❢♦ ❜✐♣❛rt✐❞♦✳
▲❡♠❛ ✶✳✹✳✶✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❜✐❝❧✐q✉❡ ♠á①✐♠♦ é ◆P✲❞✐❢í❝✐❧✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛(H, q)✉♠❛ ✐♥stâ♥❝✐❛ ❞♦ Pr♦❜❧❡♠❛✶✳✹✳✶✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ❛ss✉♠✐♠♦s q✉❡ |V(H)| é ♣❛r ❡q = |V(2H)|✳ ❈♦♥str✉í♠♦s ✉♠❛ ✐♥stâ♥❝✐❛ (G, k) ❞♦ Pr♦❜❧❡♠❛ ✶✳✹✳✷✱ t❛❧ q✉❡ k= q3−32q2✱L=V(H)✱R=A(H)∪W✱ ♦♥❞❡W ❝♦♥té♠ 12q2−q♥♦✈♦s ❡❧❡♠❡♥t♦s✱ ❡E ={uv |u∈V(H) ❡ v ∈ W} ∪ {ua |u ∈ V(H), a ∈ A(H) ❡ u /∈ a}✳ ❖❜s❡r✈❡ q✉❡ ❡ss❛ ❝♦♥str✉çã♦ é ❢❡✐t❛ ❡♠ t❡♠♣♦
✶✳✹ ❘❊❙❯▲❚❆❉❖❙ ❊◆❈❖◆❚❘❆❉❖❙ ◆❆ ▲■❚❊❘❆❚❯❘❆ ✾
❙✉♣♦♥❤❛ q✉❡ H❝♦♥té♠ ✉♠ ❝❧✐q✉❡C❝♦♠q ✈ért✐❝❡s✳ ❙❡❥❛LB =V(H)\C ❡RB =W∪ {{u, v} ∈ A(H) |u, v ∈ C}✳ ❖❜s❡r✈❡ q✉❡ B = (LB, RB) é ✉♠ ❜✐❝❧✐q✉❡ ❡♠ G✱ t❛❧ q✉❡ |EB|= |LB| · |RB|= (|V(H)| − |C|)·(|W|+|C|22−|C|) = (2q−q)·(12q2−q+q22−q) =q3−3
2q2=k✳ P♦rt❛♥t♦✱ s❡H ❝♦♥té♠
✉♠ ❝❧✐q✉❡ ❝♦♠ q ✈ért✐❝❡s✱ ❡♥tã♦ G❝♦♥té♠ ✉♠ ❜✐❝❧✐q✉❡ ❝♦♠k❛r❡st❛s✳
❆❣♦r❛✱ s✉♣♦♥❤❛ q✉❡ H ♥ã♦ ❝♦♥té♠ ✉♠ ❝❧✐q✉❡ ❝♦♠ q ✈ért✐❝❡s✳ ❙❡❥❛ B = (LB, RB) ✉♠ ❜✐❝❧✐q✉❡ ♠á①✐♠♦ ❡♠G✳ ➱ s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡|EB|< k✳ P❡❧❛ ❡s❝♦❧❤❛ ❞❡B ❡ ♣❡❧❛ ❝♦♥str✉çã♦ ❞❡G✱ t❡♠♦s q✉❡ W ⊆ RB✳ ❙❡❥❛♠ l = |LB| ❡ r = |RB| − |W|✳ ❈❛❞❛ ✈ért✐❝❡ a ❡♠ RB \W ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ ❛r❡st❛a={u, v}❞❡A(H)✱ t❛❧ q✉❡ua, va /∈E ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱u, v /∈LB✳ ❈♦♠♦ ❡①✐st❡♠2q−l ✈ért✐❝❡s ❝♦♥t✐❞♦s ❡♠ L\LB✱ t❡♠♦s q✉❡r ≤ 21(2q−l)(2q−l−1)✱ s❡♥❞♦ q✉❡ ✈❛❧❡ ❛ ✐❣✉❛❧❞❛❞❡ s❡ ❡ só s❡ C=V(H)\LB é ✉♠ ❝❧✐q✉❡ ❡♠ H✱ ♦♥❞❡A(C) =RB\W✳ ❉✐✈✐❞✐♠♦s ♦ r❡st❛♥t❡ ❞❛ ♣r♦✈❛ ♥♦s ❞♦✐s ❝❛s♦s ❛❜❛✐①♦✳
• ❙✉♣♦♥❤❛ q✉❡l > q✳ ◆❡st❡ ❝❛s♦✱ t❡♠♦s q✉❡|L\LB|=q−c✱ ♦♥❞❡c=l−q ❡0≤c≤q✳ ▲♦❣♦✱ r≤ 1
2(q−c)(q−c−1)❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ t❡♠♦s q✉❡ |LB| · |RB| = l·(|W|+r)
≤ (q+c)·1
2q2−q+12(q−c)(q−c−1)
≤ q·1
2q2−q+12(q−c)(q−c−1)
+c·1
2q2−q+12(q−c)(q−c−1)
≤ (q3−32q2) + 12c2q−cq2+12cq+c·1
2q2−q+12(q−c)(q−c−1)
≤ (q3−3
2q2) + 12c
c2−(q−1)c−2q
≤ k+12c
c2−(q−1)c−2q
.
❈♦♠♦0≤c≤q✱ t❡♠♦s q✉❡ 12c[c2−(q−1)c−2q]é ♥❡❣❛t✐✈♦ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱|L
B|·|RB|< k✳
• ❙✉♣♦♥❤❛ q✉❡l≤q✳ ◆❡st❡ ❝❛s♦✱ t❡♠♦s q✉❡|L\LB|=q+c✱ ♦♥❞❡ c=q−l ❡0≤c≤q✳ ❈♦♠♦ H ♥ã♦ ❝♦♥té♠ ♥❡♥❤✉♠ ❝❧✐q✉❡ ❝♦♠q ✈ért✐❝❡s✱ t❡♠♦s q✉❡ ♦ s✉❜❣r❛❢♦ ❞❡H ✐♥❞✉③✐❞♦ ♣♦rL\LB ♣♦ss✉✐ ♠❡♥♦s ❞♦ q✉❡ 1
2(q+c)(q+c−1)−c❛r❡st❛s ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱r < 12(q+c)(q+c−1)−c✳
P♦rt❛♥t♦✱ ❝♦♥❝❧✉✐✲s❡ q✉❡
|LB| · |RB| = l·(|W|+r) < (q−c)·1
2q2−q+12(q+c)(q+c−1)−c
< (q3−3
2q2) + 12c2(−c+ 3−q)
< k+12c2(−c+ 3−q).
❈♦♠♦0≤c≤q ❡ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡q ≥4✱ t❡♠♦s q✉❡ 21c2(−c+ 3−q)≤0❡✱ ❝♦♥s❡q✉❡♥t❡✲
♠❡♥t❡✱ |LB| · |RB|< k✳
