❛
❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙
❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ❈❖▼P❯❚❆➬➹❖
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❈■✃◆❈■❆ ❉❆ ❈❖▼P❯❚❆➬➹❖
P❆❇▲❖ ▲❯■❩ ❇❘❆●❆ ❙❖❆❘❊❙
P❘❖❇▲❊▼❆❙ ◗❯❆❉❘➪❚■❈❖❙ ❇■◆➪❘■❖❙✿ ❆❇❖❘❉❆●❊▼ ❚❊Ó❘■❈❆ ❊ ❈❖▼P❯❚❆❈■❖◆❆▲
P❆❇▲❖ ▲❯■❩ ❇❘❆●❆ ❙❖❆❘❊❙
P❘❖❇▲❊▼❆❙ ◗❯❆❉❘➪❚■❈❖❙ ❇■◆➪❘■❖❙✿ ❆❇❖❘❉❆●❊▼ ❚❊Ó❘■❈❆ ❊ ❈❖▼P❯❚❆❈■❖◆❆▲
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ❣r❛❞✉❛çã♦ ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦ ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈♦♠♣✉t❛çã♦ ❞❛ ❯♥✐✈❡rs✐✲ ❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐✲ s✐t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦✳ ➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ❆❧❣♦r✐t♠♦s ❡ ❖t✐♠✐③❛çã♦✳ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼❛♥♦❡❧ ❇❡③❡rr❛ ❈❛♠♣ê❧♦ ◆❡t♦
Dados Internacionais de Catalogação na Publicação Universidade Federal do Ceará
Biblioteca Universitária
Gerada automaticamente pelo módulo Catalog, mediante os dados fornecidos pelo(a) autor(a)
S656p Soares, Pablo Luiz Braga.
Problemas quadráticos binários : abordagem teórica e computacional / Pablo Luiz Braga Soares. – 2018. 125 f. : il. color.
Tese (doutorado) – Universidade Federal do Ceará, Centro de Ciências, Programa de Pós-Graduação em Ciência da Computação , Fortaleza, 2018.
Orientação: Prof. Dr. Manoel Bezerra Campêlo Neto.
1. t-Linearização. 2. Suavização Hiperbólica. 3. Programação Quadrática 0-1. 4. Desigualdades Válidas. I. Título.
P❆❇▲❖ ▲❯■❩ ❇❘❆●❆ ❙❖❆❘❊❙
P❘❖❇▲❊▼❆❙ ◗❯❆❉❘➪❚■❈❖❙ ❇■◆➪❘■❖❙✿ ❆❇❖❘❉❆●❊▼ ❚❊Ó❘■❈❆ ❊ ❈❖▼P❯❚❆❈■❖◆❆▲
❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ❣r❛❞✉❛çã♦ ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦ ❞♦ ❉❡✲ ♣❛rt❛♠❡♥t♦ ❞❡ ❈♦♠♣✉t❛çã♦ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉✲ t♦r ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦✳ ➪r❡❛ ❞❡ ❝♦♥✲ ❝❡♥tr❛çã♦✿ ❆❧❣♦r✐t♠♦s ❡ ❖t✐♠✐③❛çã♦✳
❆♣r♦✈♦❞❛ ❡♠✿ ❴❴ ✴ ❴❴ ✴ ✷✵✶✽✳
❇❆◆❈❆ ❊❳❆▼■◆❆❉❖❘❆
Pr♦❢✳ ❉r✳ ▼❛♥♦❡❧ ❇❡③❡rr❛ ❈❛♠♣ê❧♦ ◆❡t♦ ✭❖r✐❡♥t❛❞♦r✮ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá ✭❯❋❈✮
Pr♦❢✳ ❉r✳ ❘❛❢❛❡❧ ❈❛str♦ ❞❡ ❆♥❞r❛❞❡ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá ✭❯❋❈✮
Pr♦❢✳ ❉r✳ ❈❛r❧♦s ❉✐❡❣♦ ❘♦❞r✐❣✉❡s ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá ✭❯❋❈✮
Pr♦❢✳ ❉r✳ P❤✐❧✐♣♣❡ ❨✈❡s P❛✉❧ ▼✐❝❤❡❧♦♥ ❯♥✐✈❡rs✐té ❉✬❆✈✐❣♥♦♥
Pr♦❢✳ ❉r✳ ❆❞✐❧s♦♥ ❊❧✐❛s ❳❛✈✐❡r
✶
❆●❘❆❉❊❈■▼❊◆❚❖❙
➚ ❉❡✉s ♣♦r ❡stá s❡♠♣r❡ ❝✉✐❞❛♥❞♦ ❡ ❣✉✐❛♥❞♦ ♠✐♥❤❛s ❝❛♠✐♥❤❛❞❛s✱ ❛❧é♠ ❞❡ ❡s❝✉t❛r ❡ ❛❝♦♥s❡❧❤❛r ♠❡✉s ♣❡♥s❛♠❡♥t♦s ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❛ ♠✐♥❤❛ ✈✐❞❛✳
❆♦ ♠❡✉ ♣❛✐✱ ❋r❛♥❝✐s❝♦ ❙♦❛r❡s ✭❚✐❝ã♦ ❞❡ ❩é ❘❛✉❧✮✱ ♠❡✉ ❤❡ró✐✱ ♠❡✉ ❛♠✐❣♦✱ ♠❡✉ ❝♦♥✜❞❡♥t❡✱ s✐♠♣❧❡s♠❡♥t❡ ♠❡✉✳✳✳ q✉❡ ❛♣❡s❛r ❞❡ ❞✐st❛♥t❡ ♥✉♥❝❛ ❞❡✐①♦✉ ❢❛❧t❛r ♥❛❞❛ ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳
➚ ♠✐♥❤❛ ♠ã❡✱ ❘✐t❛ ❇r❛❣❛✱ q✉❡ ❛ss✉♠✐✉ ♦ ♣❛♣❡❧ ❞❡ ♠ã❡ ❡ ♣❛✐ ❛♦ ♠❡s♠♦ t❡♠♣♦ ❞❡s❞❡ q✉❡ ♣❛ss❛♠♦s ❛ ♠♦r❛r ❡♠ ▼♦ss♦ró✳ P❡❧♦ s❡✉ ❛♠♦r✱ ❝❛r✐♥❤♦✱ ❛♣♦✐♦ ❡ ❝♦♥✜❛♥ç❛✳ ❆ ❡❧❛ ❞❡❞✐❝♦ t♦❞♦ ♦ ♠ér✐t♦ ❞❛ ♠✐♥❤❛ ❡❞✉❝❛çã♦✱ ❢♦r♠❛çã♦ ❡ ❝❛rát❡r✳
➚ ♠✐♥❤❛ ✐r♠ã✱ ▲❛r✐ss❛ ❇r❛❣❛✱ ♣❡❧♦s ❝✉✐❞❛❞♦s ❡ q✉❛♥❞♦ ❝r✐❛♥ç❛ ❡ ♣❡❧❛s ♣r❡♦✲ ❝✉♣❛çõ❡s ❞❛ ❥✉✈❡♥t✉❞❡✳ ❱♦❝ê é ❣r❛♥❞❡ ❡ ✈❡♥❝❡rá✳
❆♦ ♠❡✉ ❛✈ô ❞❡ ❏❛♥❞✉ís✱ ❏♦sé ❙♦❛r❡s ✭❩é ❘❛✉❧✮✱ ♦ ❤♦♠❡♠ ♠❛✐s ❜♦♥❞♦s♦ ❡ sá❜✐♦ q✉❡ ❥á ❝♦♥❤❡❝✐✳ ❊s♣❡r♦ t❡r ❤❡r❞❛❞♦ ❡ss❛s s✉❛s ❝❛r❛❝t❡ríst✐❝❛s✱ ❛❧é♠ ❞❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ✈❡r s❡♠♣r❡ ♦ ❧❛❞♦ ❜♦♠ ❞❛s ❝♦✐s❛s ❛♣❡s❛r ❞❡ t✉❞♦✳
➚ ❋r❛♥❝✐s❝❛ ❙♦❛r❡s ✭◆❡♥é♠ ❘❛✉❧✮✱ ♠✐♥❤❛ ❛✈ó ❞❡ ❏❛♥❞✉ís✱ q✉❡ s❡♠♣r❡ ♠❡ r❡❝❡❜❡✉ ❞❡ ❜r❛ç♦s ❛❜❡rt♦s ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢✳ ▼❛♥♦❡❧ ❇❡③❡rr❛ ❈❛♠♣ê❧♦ ◆❡t♦✱ ♣❡❧❛ ❡①❝❡❧❡♥t❡ ♣❡s✲ s♦❛✱ ♣❡sq✉✐s❛❞♦r ❜r✐❧❤❛♥t❡✱ ♣r♦❢❡ss♦r ❡①❝❡♣❝✐♦♥❛❧✱ ❛ q✉❡♠ ❞❡✈♦ t♦❞❛ ♠✐♥❤❛ ❡✈♦❧✉çã♦ ❛❝❛✲ ❞ê♠✐❝❛ ❡ ❝♦♥q✉✐st❛✳ ◆✉♥❝❛ ❞❡✐①♦✉ ❞❡ ❛❝r❡❞✐t❛r q✉❡ ❞❛r✐❛ ❝❡rt♦✱ ♠❡s♠♦ ❡✉ ❥á ♥ã♦ ❛❝r❡❞✐✲ t❛♥❞♦ ♠❛✐s q✉❡ ❡r❛ ♣♦ssí✈❡❧✳ ❱♦❝ê ♠❡ ♦r✐❡♥t♦✉ ❝♦♠ ♠❛❡str✐❛ ❡ ♠❡ ❛❥✉❞♦✉ ❝♦♠♦ ♥❡♥❤✉♠ ♦✉tr♦✳ ❖❜r✐❣❛❞♦ ♠❡✉ ♠❡str❡ s❡r❡✐ s❡♠♣r❡ ❣r❛t♦ ♣❡❧♦s s❡✉s ❡♥s✐♥❛♠❡♥t♦s✳
➚ ♠✐♥❤❛ ❣r❛♥❞❡ ❛♠✐❣❛✱ Pr✐s❝✐❧❛ ❏á❝♦♠❡✱ q✉❡ s❡♠♣r❡ ♠❡ ❞❛r ♦s ♠❡❧❤♦r❡s ❝♦♥✲ s❡❧❤♦s ❛♠♦r♦s♦s ♠❛s q✉❡ ❡✉ ♥✉♥❝❛ s✐❣♦ ❤❛❤❛❤❛❤✳
➚ t♦❞♦s ♠❡✉s ❛♠✐❣♦s ❞♦ ❣r✉♣♦ P❛r●♦✱ ♣❡ss♦❛s ♠❛r❛✈✐❧❤♦s❛s q✉❡ ❞✐✈✐❞✐r❛♠ ♥❡ss❡s ❛♥♦s ♦s ❞✐❛s ❞❡ ❛♥s❡✐♦✱ ❞✐❛s ❞❡ ❧✉t❛✱ ✈✐❛❣❡♥s✱ s♦rr✐s♦s✱ ❥♦❣♦s✱ ❜❡❜❡❞❡✐r❛s✴❢❡st❛s ❡♠ ❡s♣❡❝✐❛❧ ♦s ❛♠✐❣♦s ❘❡♥♥❛♥ ❉❛♥t❛s ❡ ❘❛❢❛❡❧ ❚❡✐①❡✐r❛ q✉❡ ♠❡ ❛❝♦❧❤❡r❛♠ ❡♠ s✉❛s ❝❛s❛s q✉❛♥❞♦ ♣r❡❝✐s❡✐✳
➚ t♦❞♦s ♦s ♠❡✉s ❛❧✉♥♦s ❞❛ ❯❋❈ ❈❛♠♣✉s ❘✉ss❛s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ❞♦ ❣r✉♣♦ ▼✳❖✳❘✳❚✳❆✳▲ r❡♣r❡s❡♥t❛❞♦s ❛q✉✐ ♣♦r✿ ❈❛r❧♦s ❱✐❝t♦r ❆rr❅♠❜❛❞♦✭♠❡✉ ♣r✐♠❡✐r♦ ❛❧✉♥♦ ❞❡ ❚❈❈✮✱ ▼❛r❝✉s ❆r❛ú❥♦✭●r❛♥❞❡ ❛♠✐❣♦ q✉❡ ❡♥❝❛❜❡ç❛ ♠✐♥❤❛s ✐❞❡✐❛s ❡ ❢❛③ ❛s ❝♦✐s❛s ❛❝♦♥t❡✲ ❝❡r❡♠✮✱ ❈í❝❡r♦ ▼❛r❝❡❧♦✭▼❡✉ ❜♦❧s✐st❛ q✉❡ s❡♠♣r❡ s❡❣✉r♦✉ ❛ ♣❡t❡❝❛ q✉❛♥❞♦ ♣r❡❝✐s❡✐ ✜❝❛r ❛✉s❡♥t❡✮ ❡ ❉❛♥✐❡❧ ❘❡❜♦✉ç❛s✭▼❡✉ ❛❧✉♥♦ ❞❡ ✐♥✐❝✐❛çã♦ ❝✐❡♥tí✜❝❛ q✉❡ ❝♦♥tr✐❜✉✐✉ s✐❣♥✐✜❝❛t✐✲ ✈❛♠❡♥t❡ ♥❛ ♣❛rt❡ ✜♥❛❧ ❞❡ss❛ t❡s❡✮✳
✶
❘❊❙❯▼❖
❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦ é ♠♦str❛r ❛ ❛♣❧✐❝❛çã♦ ❞❡ ♠ét♦❞♦s ❞❛s ár❡❛s ❞❡ ♣r♦✲ ❣r❛♠❛çã♦ ❧✐♥❡❛r ❡ ♥ã♦ ❧✐♥❡❛r ♣❛r❛ tr❛t❛r ♣r♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛çã♦ ❝✉❥❛ ❢♦r♠✉❧❛çã♦ 0−1 ♥❛t✉r❛❧ t❡♠ ✉♠❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ q✉❛❞rát✐❝❛✳ P❛r❛ ❡ss❡ ✜♠✱ ❞❡s❡♥✈♦❧✈❡♠♦s ❛ ❡①t❡♥sã♦ ❞❛ té❝♥✐❝❛ ❞❛ t✕❧✐♥❡❛r✐③❛çã♦ ❡ ♠♦str❛♠♦s ❝♦♠♦ ❛♣❧✐❝á✲❧❛ ❛ ♣r♦❜❧❡♠❛s q✉❛❞rát✐❝♦s t❛✐s ❝♦♠♦ ❈♦rt❡ ▼á①✐♠♦✱ ❉✐✈❡rs✐❞❛❞❡ ▼á①✐♠❛ ❡ ❈❧✐q✉❡ ▼á①✐♠❛ P♦♥❞❡r❛❞❛ ❡♠ ❆r❡st❛✳ ❆❞✐❝✐♦✲ ♥❛❧♠❡♥t❡✱ r❡✈✐s❛♠♦s ❛ té❝♥✐❝❛ ❞❛ s✉❛✈✐③❛çã♦ ❤✐♣❡r❜ó❧✐❝❛ ❡ ♠♦str❛♠♦s ❝♦♠♦ ❛♣❧✐❝á✲❧❛ ❛♦ ♣r♦❜❧❡♠❛ ❞♦ ❝♦rt❡ ♠á①✐♠♦✳ ❘❡❛❧✐③❛♠♦s ✉♠ ❡st✉❞♦ t❡ór✐❝♦ ❞❡ ❝❛❞❛ ♣r♦❜❧❡♠❛✱ ♦♥❞❡ ❛♣r❡✲ s❡♥t❛♠♦s ♣❛rt✐❝✉❧❛r♠❡♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♥♦✈❛s ❞❡s✐❣✉❛❧❞❛❞❡s ✈á❧✐❞❛s ❡ ❤❡✉ríst✐❝❛s s✐♠♣❧❡s ♣❛r❛ ❝❛❞❛ ✉♠ ❞❡❧❡s✳ ▼♦str❛♠♦s ❝♦♠♦ ❣❡r❛r r❡str✐çõ❡s t✕❧✐♥❡❛r✐③❛❞❛s ♠❛✐s ❢♦rt❡s ♣❛r❛ ♦s ♣r♦❜❧❡♠❛s ❞❛ ❞✐✈❡rs✐❞❛❞❡ ♠á①✐♠❛ ❡ ❝❧✐q✉❡ ♠á①✐♠❛ ♣♦♥❞❡r❛❞❛ ❡♠ ❛r❡st❛✳ ❊①❡❝✉✲ t❛♠♦s ❡①♣❡r✐♠❡♥t♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛ ❛✈❛❧✐❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❞❡ ❝❛❞❛ té❝♥✐❝❛ ♠♦str❛❞❛ ❡♠ s❡♣❛r❛❞♦✱ ❛ss✐♠ ❝♦♠♦ ❝♦♠❜✐♥❛❞❛s ❝♦♠ ❛s ♥♦✈❛s r❡str✐çõ❡s ♣r♦♣♦st❛s✳ ❖s ❡①♣❡r✐♠❡♥t♦s ❛t❡st❛♠ ❛ q✉❛❧✐❞❛❞❡ ❞♦s ❧✐♠✐t❡s ♦❜t✐❞♦s ❜❡♠ ❝♦♠♦ ❛ ❢♦rç❛ ❞❛s ♥♦✈❛s r❡str✐çõ❡s✳ ❈♦♥s✲ tr✉í♠♦s ✉♠ ♠ét♦❞♦ ❡①❛t♦ ♣❛r❛ ❝❛❞❛ ♣r♦❜❧❡♠❛✱ ❜❛s❡❛❞♦ ♥❛s té❝♥✐❝❛s ❛♣r❡s❡♥t❛❞❛s✳ P❛r❛ ♦ ❝♦rt❡ ♠á①✐♠♦ ❡ ❝❧✐q✉❡ ♠á①✐♠❛ ♣♦♥❞❡r❛❞❛ ❡♠ ❛r❡st❛✱ ❝♦♠♣❛r❛♠♦s ♥♦ss♦ ♠ét♦❞♦ ❝♦♠ ❢♦r♠✉❧❛çõ❡s ❧✐♥❡❛r✐③❛❞❛s✳ ❏á ♣❛r❛ ❞✐✈❡rs✐❞❛❞❡ ♠á①✐♠❛✱ ❝♦♠♣❛r❛♠♦s ♥♦ss♦ ♠ét♦❞♦ ❝♦♠ ❢♦r♠✉❧❛çõ❡s ❧✐♥❡❛r✐③❛❞❛s ❡ ❝♦♠ ✉♠❛ ✐♠♣❧❡♠❡♥t❛çã♦ ♥♦ss❛ ❞♦ ♠❡❧❤♦r ♠ét♦❞♦ ❡①❛t♦✱ q✉❡ ♥ã♦ ✉t✐❧✐③❛ r❡s♦❧✈❡❞♦r ♠❛t❡♠át✐❝♦✱ ❝♦♥❤❡❝✐❞♦ ❞❛ ❧✐t❡r❛t✉r❛✳ ❊①♣❡r✐♠❡♥t♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❝♦♠ ✐♥stâ♥❝✐❛s ❞❛ ❧✐t❡r❛t✉r❛ ♠♦str❛♠ ✉♠ ❞❡s❡♠♣❡♥❤♦ s✉♣❡r✐♦r ❞♦ ♥♦ss♦ ♠ét♦❞♦✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ t✕▲✐♥❡❛r✐③❛çã♦✳ ❙✉❛✈✐③❛çã♦ ❍✐♣❡r❜ó❧✐❝❛✳ Pr♦❣r❛♠❛çã♦ ◗✉❛❞rát✐❝❛0−
❆❇❙❚❘❆❈❚
❚❤❡ ♠❛✐♥ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ s❤♦✇ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❧✐♥❡❛r ❛♥❞ ♥♦♥❧✐♥❡❛r ♣r♦✲ ❣r❛♠♠✐♥❣ ♠❡t❤♦❞s t♦ ❞❡❛❧ ✇✐t❤ ♣r♦❜❧❡♠s ✇❤♦s❡ ♥❛t✉r❛❧0−1❢♦r♠✉❧❛t✐♦♥ ❤❛s ❛ q✉❛❞r❛t✐❝ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥✳ ❚♦ t❤✐s ❡♥❞✱ ✇❡ ❞❡✈❡❧♦♣ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ t✲❧✐♥❡❛r✐③❛t✐♦♥ t❡❝❤♥✐q✉❡
❛♥❞ s❤♦✇ ❤♦✇ t♦ ❛♣♣❧② ✐t t♦ q✉❛❞r❛t✐❝ ♣r♦❜❧❡♠s s✉❝❤ ❛s ▼❛①✐♠✉♠ ❈✉t✱ ▼❛①✐♠✉♠ ❉✐✲ ✈❡rs✐t② ❛♥❞ ▼❛①✐♠✉♠ ❊❞❣❡✲❲❡✐❣❤t❡❞ ❈❧✐q✉❡✳ ■♥ ❛❞❞✐t✐♦♥✱ ✇❡ r❡✈✐❡✇ t❤❡ ❤②♣❡r❜♦❧✐❝ s♠♦✲ ♦t❤✐♥❣ t❡❝❤♥✐q✉❡ ❛♥❞ s❤♦✇ ❤♦✇ t♦ ❛♣♣❧② ✐t t♦ t❤❡ ♠❛①✐♠✉♠ ❝✉t ♣r♦❜❧❡♠✳ ❲❡ ❝❛rr② ♦✉t ❛ t❤❡♦r❡t✐❝❛❧ st✉❞② ♦❢ ❡❛❝❤ ♣r♦❜❧❡♠✱ ✇❤❡r❡ ✇❡ ♣❛rt✐❝✉❧❛r❧② ♣r❡s❡♥t t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ♥❡✇ ✈❛❧✐❞ ✐♥❡q✉❛❧✐t✐❡s ❢♦r ❡❛❝❤ ♦❢ t❤❡♠✳ ❲❡ ♣❡r❢♦r♠ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts t♦ ❡✈❛❧✉❛t❡ t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❡❛❝❤ t❡❝❤♥✐q✉❡ s❡♣❛r❛t❡❧②✱ ❛s ✇❡❧❧ ❛s ❝♦♠❜✐♥❡❞ ✇✐t❤ t❤❡ ♣r♦♣♦s❡❞ ♥❡✇ ❝♦♥str❛✐♥ts✳ ❚❤❡ ❡①♣❡r✐♠❡♥ts ❛tt❡st t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ❜♦✉♥❞s ♦❜t❛✐♥❡❞ ❛s ✇❡❧❧ ❛s t❤❡ str❡♥❣t❤ ♦❢ t❤❡ ♥❡✇ ✐♥❡q✉❛❧✐t✐❡s✳ ❲❡ ❞❡✈❡❧♦♣ ❛♥ ❡①❛❝t ♠❡t❤♦❞ ❢♦r ❡❛❝❤ ♣r♦❜❧❡♠✱ ❜❛s❡❞ ♦♥ t❤❡ ♣r❡s❡♥t❡❞ t❡❝❤♥✐q✉❡s✳ ❋♦r t❤❡ ♠❛①✐♠✉♠ ❝✉t ❛♥❞ ♠❛①✐♠✉♠ ❡❞❣❡✲✇❡✐❣❤t❡❞ ❝❧✐q✉❡✱ ✇❡ ❝♦♠♣❛r❡ ♦✉r ♠❡t❤♦❞ ✇✐t❤ ❧✐♥❡❛r✐③❡❞ ❢♦r♠✉❧❛t✐♦♥s✳ ❋♦r ♠❛①✐♠✉♠ ❞✐✈❡rs✐t②✱ ✇❡ ❝♦♠♣❛r❡ ♦✉r ♠❡t❤♦❞ ✇✐t❤ ❧✐♥❡❛r✐③❡❞ ❢♦r♠✉❧❛t✐♦♥s ❛♥❞ ✇✐t❤ ♦✉r ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❜❡st ❡①❛❝t ♠❡t❤♦❞✱ ✇❤✐❝❤ ❞♦❡s ♥♦t ✉s❡ ♠❛t❤❡♠❛t✐❝❛❧ s♦❧✈❡rs✱ ❦♥♦✇♥ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳ ❈♦♠♣✉t❛t✐♦✲ ♥❛❧ ❡①♣❡r✐♠❡♥ts ✇✐t❤ ✐♥st❛♥❝❡s ♦❢ t❤❡ ❧✐t❡r❛t✉r❡ s❤♦✇ s✉♣❡r✐♦r ♣❡r❢♦r♠❛♥❝❡ ♦❢ ♦✉r ♠❡t❤♦❞✳
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙
❋✐❣✉r❛ ✶ ✕ ❉✐❢❡r❡♥ç❛ ❡♥tr❡Θ :=PiPj<icπ(j)π(i)xˆπ(i)❡Θ′ :=Pi P
j<icπ′(j)π′(i)xˆπ′(i)✳
❊♠ ❛③✉❧ ❛s ♣♦s✐çõ❡s ❞✐❢❡r❡♥t❡s ❡♥tr❡ π ❡ π′✳ ❆s s❡t❛s ✐♥❞✐❝❛♠ ♦s ❝♦❡✲ ✜❝✐❡♥t❡s q✉❡ ❛♣❛r❡❝❡♠ Θ ♠❛s ♥ã♦ ❡♠ Θ′ ✭✜❣✉r❛ s✉♣❡r✐♦r✮ ❡ ✈✐❝❡✲✈❡rs❛ ✭✜❣✉r❛ ✐♥❢❡r✐♦r✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ❋✐❣✉r❛ ✷ ✕ ❈♦❡✜❝✐❡♥t❡sαq✉❡ ❛♣❛r❡❝❡♠ ❡♠ Θ(π, N0, N1;x)✳ ❆s s❡t❛s ✐♥❞✐❝❛♠ ❛ q✉❡
✈❛r✐á✈❡✐s ♦ ❝♦❡✜❝✐❡♥t❡ απ(j)π(i)✱ j < i✱ ❡stá ❛ss♦❝✐❛❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ❋✐❣✉r❛ ✸ ✕ ❈♦❡✜❝✐❡♥t❡sβ q✉❡ ❛♣❛r❡❝❡♠ ❡♠Θ(π, N0, N1;x)✳ ❆s s❡t❛s ✐♥❞✐❝❛♠ ❛ q✉❡
✈❛r✐á✈❡✐s ♦ ❝♦❡✜❝✐❡♥t❡ βπ(j)π(i)✱ j < i✱ ❡stá ❛ss♦❝✐❛❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ❋✐❣✉r❛ ✹ ✕ ❉✐❢❡r❡♥ç❛ ❡♥tr❡ Θ(π, N0, N1; ˆx)❡Θ(π′, N0, N1; ˆx)q✉❛♥❞♦ π(l+ 1)∈N1✳
❊♠ ❛③✉❧ ❛s ♣♦s✐çõ❡s ❞✐❢❡r❡♥t❡s ❡♥tr❡ π ❡ π′✳ ❆s s❡t❛s ✐♥❞✐❝❛♠ ♦s ❝♦✲
❡✜❝✐❡♥t❡s α ✭❋✐❣✳ ❛✮ ❡ β ✭❋✐❣✳ ❜✮ q✉❡ ❛♣❛r❡❝❡♠ Θ(π, N0, N1; ˆx) ♠❛s ♥ã♦ ❡♠ Θ(π′, N
0, N1; ˆx) ✭✜❣✉r❛ s✉♣❡r✐♦r✮ ❡ ✈✐❝❡✲✈❡rs❛ ✭✜❣✉r❛ ✐♥❢❡r✐♦r✮✱ ❛♣r❡s❡♥t❛♥❞♦ ❛s ✈❛r✐á✈❡✐s ❛ss♦❝✐❛❞❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ❋✐❣✉r❛ ✺ ✕ ❉✐❢❡r❡♥ç❛ ❡♥tr❡ Θ(π, N0, N1; ˆx)❡Θ(π′, N0, N1; ˆx)q✉❛♥❞♦ π(l+ 1)∈N0✳
❊♠ ❛③✉❧ ❛s ♣♦s✐çõ❡s ❞✐❢❡r❡♥t❡s ❡♥tr❡ π ❡ π′✳ ❆s s❡t❛s ✐♥❞✐❝❛♠ ♦s ❝♦✲ ❡✜❝✐❡♥t❡s α ✭❋✐❣✳ ❛✮ ❡ β ✭❋✐❣✳ ❜✮ q✉❡ ❛♣❛r❡❝❡♠ Θ(π, N0, N1; ˆx) ♠❛s ♥ã♦ ❡♠ Θ(π′, N0, N1; ˆx) ✭✜❣✉r❛ s✉♣❡r✐♦r✮ ❡ ✈✐❝❡✲✈❡rs❛ ✭✜❣✉r❛ ✐♥❢❡r✐♦r✮✱ ❛♣r❡s❡♥t❛♥❞♦ ❛s ✈❛r✐á✈❡✐s ❛ss♦❝✐❛❞❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ❋✐❣✉r❛ ✻ ✕ |u| ❡ θ(u, γ)✱ ♣❛r❛ γ = [1,0.8,0.5,0.3] ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ❋✐❣✉r❛ ✼ ✕ g(yi) ❡φ(λ, τ, yi)✱ ♣❛r❛ τ = [1,0.8,0.5,0.3] ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵
❋✐❣✉r❛ ✽ ✕ ❋✐①❛çã♦ xk = 1 ❡♠ ✉♠❛ s♦❧✉çã♦ ❞❡s❝❡♥❞❡♥t❡ ❞❡ xˆ✿ ❧✐♠✐t❡ ✐♥❢❡r✐♦r ✭❛✮
♠❛✐♦r q✉❡ ❧✐♠✐t❡ s✉♣❡r✐♦r ✭❜✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ❋✐❣✉r❛ ✾ ✕ ❋✐①❛çã♦ ❞❡ ✈❛r✐á✈❡✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ❋✐❣✉r❛ ✶✵ ✕ ❉❡❝✐sã♦ ❤❡✉ríst✐❝❛✳ ❈❛s♦s ❡♠ q✉❡ ∆(k∗,xˆ) > 0✱ ♦✉ s❡❥❛✱ |C
01(k∗,xˆ)| <
C2(k∗,xˆ)✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ❋✐❣✉r❛ ✶✶ ✕ ●❛♣ ♠é❞✐♦ ❝♦♠n ✜①♦ ❡ ✈❛r✐❛çã♦ ❞❡d ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
❋✐❣✉r❛ ✶✷ ✕ ●❛♣ ♠é❞✐♦ ❝♦♠d ✜①♦ ❡ ✈❛r✐❛çã♦ ❞❡ n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸
❋✐❣✉r❛ ✶✸ ✕ ■❧✉str❛çã♦ ❞❛s ❡①♣r❡ssõ❡s ❞❡ cmin(¯x, v) ❡ cmax(¯x, v)✳ ❆r❡st❛s ♣♦♥t✐❧❤❛✲ ❞❛s ❝♦♠♣õ❡♠cmin(¯x, v)✱ ❛r❡st❛s tr❛❝❡❥❛❞❛s ❝♦♠♣õ❡♠cmax(¯x, v)❡ ❛r❡st❛s ❝❤❡✐❛s ❝♦♠♣õ❡♠ ❛♠❜♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ❋✐❣✉r❛ ✶✹ ✕ ■❧✉str❛çã♦ ❞♦s t❡r♠♦s ❞❡δuv(¯x)✳ ❖s s✐♥❛✐s ♥❛s ❛r❡st❛s ✐♥❞✐❝❛♠ ❛ ❝♦♥tr✐✲
❜✉✐çã♦ ❞❡ ❝❛❞❛ ❝✉st♦ ♣❛r❛ ❛ ❡①♣r❡ssã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ❋✐❣✉r❛ ✶✺ ✕ ●❛♣ ♠é❞✐♦ ♦❜t✐❞♦ ♣❡❧❛s r❡❧❛①❛çõ❡sF2✱F3✱ (F1)πt✱ Fm ❡ (F1)πt +RF ✳ ✳ ✳ ✽✼
❋✐❣✉r❛ ✶✻ ✕ ●❛♣ ♠é❞✐♦ ♦❜t✐❞♦ ♣❡❧❛s r❡❧❛①❛çõ❡sF2✱F3✱ (F1)πt✱ Fm ❡ (F1)πt +RF ✳ ✳ ✳ ✽✽
❋✐❣✉r❛ ✶✼ ✕ ❋❧✉①♦❣r❛♠❛ ❞♦ ♠ét♦❞♦ ❡①❛t♦✳ (BB)π
t s❡❣✉❡ ❛s s❡t❛s ✈❡rt✐❝❛✐s✳ (F1)πt
❋✐❣✉r❛ ✶✽ ✕ ◆ú♠❡r♦ ♠é❞✐♦ ❞❡ r❡str✐çõ❡s ❣❡r❛❞❛s ♣♦r(F1)πt ❡(BB)πt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵
❋✐❣✉r❛ ✶✾ ✕ ❈♦♠♣❛r❛çã♦ ❞♦ ●❆P ♠é❞✐♦ ♦❜t✐❞♦ ♣♦rF CAm✱F CA2✱F CA3❡(F CA1)πt
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼ ❋✐❣✉r❛ ✷✵ ✕ ❈♦♠♣❛r❛çã♦ ❞♦Nrg ♠é❞✐♦ ♦❜t✐❞♦ ♣♦r (F CA1)πt ❡ F CA3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✽ ❋✐❣✉r❛ ✷✶ ✕ ●❛♣ ♠é❞✐♦✭30 ✐♥stâ♥❝✐❛s ♣♦s✐t✐✈❛s✮ ♦❜t✐❞♦ ♣❡❧❛s ❢♦r♠✉❧❛çõ❡s F CA1✱
F CAm✱ F CA2✱ F CA3 ❡ (BBCA)πt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✶
❋✐❣✉r❛ ✷✷ ✕ ❈♣✉ ♠é❞✐♦✭30 ✐♥stâ♥❝✐❛s ♣♦s✐t✐✈❛s✮ ♦❜t✐❞♦ ♣❡❧❛s ❢♦r♠✉❧❛çõ❡s F CA1✱
F CAm✱ F CA2✱ F CA3 ❡ (BBCA)πt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷
❋✐❣✉r❛ ✷✸ ✕ ●❛♣ ♠é❞✐♦✭30✐♥stâ♥❝✐❛s ♠✐st❛s✮ ♦❜t✐❞♦ ♣❡❧❛s ❢♦r♠✉❧❛çõ❡sF CA1✱F CAm✱
F CA2✱ F CA3 ❡(BBCA)πt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✸
❋✐❣✉r❛ ✷✹ ✕ ❈♣✉ ♠é❞✐♦✭30✐♥stâ♥❝✐❛s ♠✐st❛s✮ ♦❜t✐❞♦ ♣❡❧❛s ❢♦r♠✉❧❛çõ❡sF CA1✱F CAm✱
▲■❙❚❆ ❉❊ ❚❆❇❊▲❆❙
❚❛❜❡❧❛ ✶ ✕ ❱❛❧♦r❡s ✐♥❞✐✈✐❞✉❛✐s ❞❡ yji∗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
❚❛❜❡❧❛ ✷ ✕ P❡s♦ ❞♦ ❝♦rt❡ ❞❛ ♣❛rt✐çã♦(S, S)r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ ✈❡t♦r ❜✐♥ár✐♦x∈ {0,1}n ✹✹
❚❛❜❡❧❛ ✸ ✕ P❡s♦ ❞♦ ❝♦rt❡ ❞❛ (S, S) r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ ✈❡t♦r ❜✐♥ár✐♦ x∈ {−1,1}n ✳ ✳ ✹✺
❚❛❜❡❧❛ ✹ ✕ ❊q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡xi+xj −2xixj✱ |xi−xj| ❡ zij ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾
❚❛❜❡❧❛ ✺ ✕ ❈P❯✱ ❣❛♣✱ Rf ❡ Nrg ♦❜t✐❞♦s ♣❡❧❛ t✕❧✐♥❡❛r✐③❛çã♦✱ ❧✐♥❡❛r✐③❛çã♦ ❝❧áss✐❝❛ ❡
❧✐♥❡❛r✐③❛çã♦ ●❧♦✈❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽ ❚❛❜❡❧❛ ✻ ✕ ▲❇ ❡ ❈P❯ ♦❜t✐❞♦s ♣❡❧❛s ❤❡✉ríst✐❝❛s ❣✉❧♦s❛s✱SH✱SH+ ∆❡SH+ ∆ +BL ✻✾
❚❛❜❡❧❛ ✼ ✕ ❈♦♠♣❛r❛çã♦ ❡♥tr❡(F L)t ❡(F L)tGRV ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶
❚❛❜❡❧❛ ✽ ✕ ❈♦♠♣❛r❛çã♦ ❡♥tr❡(F L)tGRV ❡ (F L)tLRV ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
❚❛❜❡❧❛ ✾ ✕ Pr✐♥❝✐♣❛✐s ❛♣❧✐❝❛çõ❡s ❞♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐✈❡rs✐❞❛❞❡ ♠á①✐♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ❚❛❜❡❧❛ ✶✵ ✕●❆P✪✱ ❈P❯✭s✮ ❡Nrg ♠é❞✐♦ ♣❛r❛ ❛s r❡❧❛①❛çõ❡s (F1)πt ❡(F1)πt +RF ♥❛s
✐♥stâ♥❝✐❛s ❙✐❧✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ❚❛❜❡❧❛ ✶✶ ✕●❆P✪✱ ❈P❯✭s✮ ❡Nrg ♠é❞✐♦ ♣❛r❛ ❛s r❡❧❛①❛çõ❡s (F1)πt ❡(F1)πt +RF ♥❛s
✐♥stâ♥❝✐❛s ●❧♦✈❡r✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻ ❚❛❜❡❧❛ ✶✷ ✕●❆P✪✱ ❈P❯✭s✮✱Rf ❡Nrg ♠é❞✐♦ ♣❛r❛ ❛s r❡❧❛①❛çõ❡sF2✱F3✱Fm ❡(F1)πt+
RF ♥❛s ✐♥stâ♥❝✐❛s ❙✐❧✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼
❚❛❜❡❧❛ ✶✸ ✕●❆P✪✱ ❈P❯✭s✮✱Rf ❡Nrg ♠é❞✐♦ ♣❛r❛ ❛s r❡❧❛①❛çõ❡sF2✱F3✱Fm ❡(F1)πt+
RF ♥❛s ✐♥stâ♥❝✐❛s ●❧♦✈❡r✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼
❚❛❜❡❧❛ ✶✹ ✕❱❛❧♦r❡s ♠é❞✐♦s ❞❡ ●❆P✱ ❈P❯ ❡Nrg ❡ ★❖♣t ♦❜t✐❞♦s ♣♦r(F1)πt ❡(BB)πt ✾✵
❚❛❜❡❧❛ ✶✺ ✕❘❡s✉❧t❛❞♦s ❝♦♠ ❛s ✐♥stâ♥❝✐❛s ❙✐❧✈❛ ♦❜t✐❞♦s ♣❡❧❛s ❢♦r♠✉❧❛çõ❡s F1 ❡ Fm✱
❡ ♦s ♠ét♦❞♦s BBmax ❡ (BB)π
t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷
❚❛❜❡❧❛ ✶✻ ✕❘❡s✉❧t❛❞♦s ❝♦♠ ❛s ✐♥stâ♥❝✐❛s ●❧♦✈❡r✷ ♦❜t✐❞♦s ♣❡❧❛s ❢♦r♠✉❧❛çõ❡s F1 ❡
Fm✱ ❡ ♦s ♠ét♦❞♦sBBmax ❡ (BB)πt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸
❚❛❜❡❧❛ ✶✼ ✕❱❡rsõ❡s ❞♦ P❈▼P❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ❚❛❜❡❧❛ ✶✽ ✕●❆P✱Rf ❡Nrg♦❜t✐❞♦s ♣❡❧❛s r❡❧❛①❛çõ❡sF CAm✱F CA2✱F CA3❡(F CA1)πt
♥❛s ✐♥stâ♥❝✐❛s ♣♦s✐t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✼ ❚❛❜❡❧❛ ✶✾ ✕●❆P✱Rf ❡Nrg♦❜t✐❞♦s ♣❡❧❛s r❡❧❛①❛çõ❡sF CAm✱F CA2✱F CA3❡(F CA1)πt
♥❛s ✐♥stâ♥❝✐❛s ♠✐st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✽ ❚❛❜❡❧❛ ✷✵ ✕●❆P ❡ ❈P❯ ♦❜t✐❞♦s ♣❡❧❛s ❢♦r♠✉❧❛çõ❡sF CA1✱ F CAm✱ F CA2✱F CA3 ❡
(BBCA)π
t ♥❛s ✐♥stâ♥❝✐❛s ♣♦s✐t✐✈❛s ❝♦♠ m=⌊n/2⌋ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✶
❚❛❜❡❧❛ ✷✶ ✕●❆P ❡ ❈P❯ ♦❜t✐❞♦s ♣❡❧❛s ❢♦r♠✉❧❛çõ❡sF CA1✱ F CAm✱ F CA2✱F CA3 ❡ (BBCA)π
▲■❙❚❆ ❉❊ ❆❇❘❊❱■❆❚❯❘❆❙ ❊ ❙■●▲❆❙
❇✐q ▼❛❝ ❇✐♥❛r② ◗✉❛❞r❛t✐❝ ❛♥❞ ▼❛① ❈✉t ❙♦❧✈❡r ❉◆❆ ❉❡♦①②r✐❜♦♥✉❝❧❡✐❝ ❆❝✐❞
●❘❆❙P ●r❡❡❞② ❘❛♥❞♦♠✐③❡❞ ❆❞❛♣t✐✈❡ ❙❡❛r❝❤ Pr♦❝❡❞✉r❡ ■❇▼ ■♥t❡r♥❛t✐♦♥❛❧ ❇✉s✐♥❡ss ▼❛❝❤✐♥❡s
■❉❊ ■♥t❡❣r❛t❡s ❉❡✈❡❧♦♣♠❡♥t ❊♥✈✐r♦♥♠❡♥t ■❚❙ ■t❡r❛t✐✈❡ ❚❛❜✉ ❙❡❛r❝❤
▲❙✲❚P● ▲❛❣r❛♥❣✐❛♥ ❙♠♦♦t❤✐♥❣ ❚r✉♥❝❛t❡❞ Pr♦❥❡❝t ●r❛❞✐❡♥t ▲❙✲❚❋❲ ▲❛❣r❛❣✐❛♥ ❙♠♦♦t❤✐♥❣ ❚r✉♥❝❛t❡❞ ❋r❛♥❦✲❲♦❧❢❡ P❈▼P❆ Pr♦❜❧❡♠❛ ❞❛ ❈❧✐q✉❡ ▼á①✐♠❛ P♦♥❞❡r❛❞❛ ❡♠ ❆r❡st❛ P❉▼ Pr♦❜❧❡♠❛ ❞❛ ❉✐✈❡rs✐❞❛❞❡ ▼á①✐♠❛
P■ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ P■▼ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ ▼✐st❛ P▲ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r
P◗ Pr♦❜❧❡♠❛ ◗✉❛❞rát✐❝♦
P◗▼ Pr♦❜❧❡♠❛ ◗✉❛❞rát✐❝♦ ❞❛ ▼♦❝❤✐❧❛ P❘ P❛t❤✲❘❡❧✐♥❦✐♥❣
P❙❉ Pr♦❣r❛♠❛çã♦ ❙❡♠✐❞❡✜♥✐❞❛ ❘❆▼ ❘❛♥❞♦♠ ❆❝❝❡ss ▼❡♠♦r②
❙❇P❖ ❙✐♠♣ós✐♦ ❇r❛s✐❧❡✐r♦ ❞❡ P❡sq✉✐s❛ ❖♣❡r❛❝✐♦♥❛❧ ❙❍ ❙✉❛✈✐③❛çã♦ ❍✐♣❡r❜ó❧✐❝❛
❙❙ ❙❝❛tt❡r✲❙❡❛r❝❤
❯◗P ❯♥❝♦♥str❛✐♥❡❞ ◗✉❛❞r❛t✐❝ Pr♦❣r❛♠♠✐♥❣ ❱◆❙ ❱❛r✐❛❜❧❡ ◆❡✐❣❤❜♦r❤♦♦❞ ❙❡❛r❝❤
❙❯▼➪❘■❖
✶ ■◆❚❘❖❉❯➬➹❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷ ▼❊❚❖❉❖▲❖●■❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✶ ▲✐♥❡❛r✐③❛çã♦ ❈❧áss✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷ ▲✐♥❡❛r✐③❛çã♦ ❞❡ ●❧♦✈❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸ t✕▲✐♥❡❛r✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸✳✶ ❈♦❡✜❝✐❡♥t❡s ♥ã♦ ♥❡❣❛t✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸✳✷ ❊①t❡♥sã♦ ♣❛r❛ ❝♦❡✜❝✐❡♥t❡s ❛r❜✐trár✐♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✹ ❙✉❛✈✐③❛çã♦ ❍✐♣❡r❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸ ❈❖❘❚❊ ▼➪❳■▼❖ ✭▼❆❳✲❈❯❚✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✶ ❋♦r♠✉❧❛çõ❡s ▼❛t❡♠át✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✶✳✶ ❋♦r♠✉❧❛çã♦ ♣♦r r❡str✐çõ❡s tr✐❛♥❣✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✶✳✷ ❋♦r♠✉❧❛çã♦ ❜✐♥ár✐❛ {0,1} ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹
✸✳✶✳✸ ❋♦r♠✉❧❛çã♦ ❜✐♥ár✐❛ {−1,1} ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✶✹
✹✳✸ ❋♦rt❛❧❡❝✐♠❡♥t♦ ❞❛s ❘❡str✐çõ❡s t✕▲✐♥❡❛r✐③❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽ ✹✳✹ Pr♦♣r✐❡❞❛❞❡s ❞❛s ❙♦❧✉çõ❡s Ót✐♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✹✳✹✳✶ ❋✐①❛çã♦ ❞❡ ✈❛r✐á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✹✳✹✳✷ ❘❡str✐çõ❡s ✈á❧✐❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✹✳✺ ❍❡✉ríst✐❝❛ ❝♦♠ ❖r❞❡♥❛❞❛çã♦ ❡ ❇✉s❝❛ ▲♦❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✹✳✻ ❊①♣❡r✐♠❡♥t♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✹✳✻✳✶ ▲✐♠✐t❡ ❙✉♣❡r✐♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✹✳✻✳✷ ❯♠ ❛❧❣♦r✐t♠♦ ❜r❛♥❝❤ ❛♥❞ ❜♦✉♥❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ✹✳✼ ❈♦♥❝❧✉sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸
✺ ❈▲■◗❯❊ ▼➪❳■▼❆ P❖◆❉❊❘❆❉❆ ❊▼ ❆❘❊❙❚❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺
✺✳✶ ❋♦r♠✉❧❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✻ ✺✳✷ ❆♣❧✐❝❛çã♦ ❞❛ t✕▲✐♥❡❛r✐③❛çã♦ ❛♦ P❈▼P❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✼ ✺✳✸ ❋♦rt❛❧❡❝✐♠❡♥t♦ ❞❛s ❘❡str✐çõ❡s t✲▲✐♥❡❛r✐③❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾
✶✺
✶ ■◆❚❘❖❉❯➬➹❖
❆ ár❡❛ ❞❡ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛ ❡♥❣❧♦❜❛ ✉♠❛ ❣r❛♥❞❡ ❞✐✈❡rs✐❞❛❞❡ ❞❡ ♣r♦✲ ❜❧❡♠❛s ♣rát✐❝♦s q✉❡ ❜✉s❝❛♠ ♣♦r s♦❧✉çõ❡s q✉❡ ❢❛ç❛♠ ♠❡❧❤♦r ✉s♦ ❞♦s r❡❝✉rs♦s ❡♥✈♦❧✈✐❞♦s✳ ◆❡ss❡s ♣r♦❜❧❡♠❛s✱ ♦ ✐♥t❡r❡ss❡ é ❡♥❝♦♥tr❛r ❛ s♦❧✉çã♦ q✉❡ ♦t✐♠✐③❡ ♦s ♦❜❥❡t✐✈♦s ❞❡ ✐♥t❡r❡ss❡ ❡ r❡s♣❡✐t❡ ❛s r❡str✐çõ❡s ✐♠♣♦st❛s✳ ❊♠ ❣❡r❛❧✱ ♦s ♦❜❥❡t✐✈♦s ❝♦♥s✐st❡♠ ❡♠ ❛♣r♦✈❡✐t❛r ♠❡✲ ❧❤♦r ♦s ♠❛t❡r✐❛✐s ♥♦ ♣r♦❝❡ss♦ ❞❡ ♣r♦❞✉çã♦✱ ♦t✐♠✐③❛r ♦ t❡♠♣♦ ♣❛r❛ r❡❛❧✐③❛r ❝❡rt❛s ❛çõ❡s ❡ ♦♣❡r❛çõ❡s✱ tr❛♥s♣♦rt❛r ♠❛t❡r✐❛✐s ♣❡❧❛s ♠❡❧❤♦r❡s r♦t❛s✱ ❛✉♠❡♥t❛r ❧✉❝r♦s ❡ ❞✐♠✐♥✉✐r ❣❛st♦s ❡t❝ ✭▼✐②❛③❛✇❛ ❡ ❞❡ ❙♦✉③❛✱ ✷✵✶✺✮✳ P❛r❛ ✐❧✉str❛r ❛ ❞✐✈❡rs✐❞❛❞❡ ❞❛ ár❡❛✱ ✈ár✐♦s ♣r♦❜❧❡♠❛s ❡♥❝♦♥tr❛♠ ❛♣❧✐❝❛çõ❡s ❡♠ ❡s❝❛❧♦♥❛♠❡♥t♦ ❞❡ t❛r❡❢❛s ✭●✐✤❡r ❡ ❚❤♦♠♣s♦♥✱ ✶✾✻✵✮✱ ❧♦❝❛❧✐③❛✲ çã♦ ❞❡ ❝❡♥tr♦s ❞✐str✐❜✉✐❞♦r❡s ✭❈❤✉r❝❤ ❡ ❘❡✈❡❧❧❡✱ ✶✾✼✹✮✱ r♦t❡❛♠❡♥t♦ ❞❡ ✈❡í❝✉❧♦s ✭❚♦t❤ ❡ ❱✐❣♦✱ ✷✵✶✹✮✱ ♣r♦❥❡t♦ ❞❡ ❝✐r❝✉✐t♦s ❱▲❙■ ✭❇❛r❛❤♦♥❛ ❡t ❛❧✳✱ ✶✾✽✽✮✱ s❡q✉❡♥❝✐❛♠❡♥t♦ ❞❡ ❉◆❆ ✭❑❡❝❡❝✐♦❣❧✉ ❡ ▼②❡rs✱ ✶✾✾✺✮ ❡t❝✳
❖s ♣r♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛çã♦ ❝♦♠❜✐♥❛tór✐❛ ♣♦❞❡♠ s❡r ❞❡ ♠✐♥✐♠✐③❛çã♦ ♦✉ ❞❡ ♠❛①✐♠✐③❛çã♦ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ❛♣❧✐❝❛❞❛ ❛ ✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦✱ q✉❡ ❡♠ ❣❡r❛❧ é ❡♥✉♠❡rá✈❡❧✱ ♦♥❞❡ s❡ ❞❡s❡❥❛ ♦❜t❡r ❛ ♠❡❧❤♦r s♦❧✉çã♦ ♣♦ssí✈❡❧✳ ❆♣❡s❛r ❞❡ ✜♥✐t♦✱ ♠ét♦❞♦s ✐♥❣ê♥✉♦s ❞❡ ❡♥✉♠❡r❛r t♦❞❛s ❛s ♣♦ssí✈❡✐s s♦❧✉çõ❡s sã♦ ♣r❛t✐❝á✈❡✐s ❛♣❡♥❛s ♣❛r❛ t❛♠❛♥❤♦ ❞❡ ✐♥stâ♥❝✐❛s ❝♦♥s✐❞❡r❛❞❛s ♣❡q✉❡♥❛s ✭❲♦❧s❡②✱ ✶✾✾✽✮✳ P♦rt❛♥t♦✱ ♣❛r❛ ✐♥stâ♥❝✐❛s ❞❡ t❛♠❛♥❤♦s ♠♦❞❡r❛❞♦s ❡ ❣r❛♥❞❡s✱ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ♦ ✉s♦ ♠ét♦❞♦s ♠❛✐s ❡❧❛❜♦r❛❞♦s ♣❛r❛ ❡♥❝♦♥tr❛r ❛ ♠❡❧❤♦r s♦❧✉çã♦ ♣♦ssí✈❡❧✳
❯♠❛ ❞❛s ❢♦r♠❛s ❞❡ s❡ ❡♥t❡♥❞❡r ❡ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛çã♦ ❝♦♠❜✐♥❛✲ tór✐❛ é ❛tr❛✈és ❞♦ ✉s♦ ❞❡ ♠♦❞❡❧♦s ♠❛t❡♠át✐❝♦s✱ t❛✐s ❝♦♠♦ ❞❡ Pr♦❣r❛♠❛çã♦ ❙❡♠✐❞❡✜♥✐❞❛ ✭P❙❉✮✱ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ✭P▲✮✱ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ ✭P■✮✱ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ ▼✐st❛ ✭▲✐♥❡❛r✮ ✭P■▼✮✱ Pr♦❣r❛♠❛çã♦ ❇✐♥ár✐❛ ❡t❝✳ ❙❡♥❞♦ ♦ ♠♦❞❡❧♦ ♠❛✐s ❣❡r❛❧✱ ❛ ♣r♦❣r❛♠❛çã♦ s❡♠✐❞❡✜♥✐❞❛ ❝♦♥s✐st❡ ❡♠ ♠✐♥✐♠✐③❛r ♦✉ ♠❛①✐♠✐③❛r ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r s✉❥❡✐t♦ à r❡str✐çã♦ ❞❡ q✉❡ ♠❛tr✐③❡s ❡♥✈♦❧✈✐❞❛s ♥❛ ❢♦r♠✉❧❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ s❡❥❛♠ s✐♠étr✐❝❛s ❡ s❡♠✐❞❡✜♥✐❞❛s ♣♦s✐t✐✈❛s ✭❱❛♥❞❡♥❜❡r❣❤❡ ❡ ❇♦②❞✱ ✶✾✾✻✮✳ ❖ ♠♦❞❡❧♦ ♠❛✐s ♣❛rt✐❝✉❧❛r✱ ❛ ♣r♦❣r❛♠❛çã♦ ❧✐♥❡❛r✱ ❝♦♥s✐st❡ ❡♠ ♠❛①✐♠✐③❛r ♦✉ ♠✐♥✐♠✐③❛r ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r ❞❡✜♥✐❞❛ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s✱ ❛s q✉❛✐s ❞❡✈❡♠ s❛t✐s❢❛③❡r ✉♠ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❡ ✐♥❡q✉❛çõ❡s ❧✐♥❡❛r❡s✳ ❏á ❛ ♣r♦❣r❛♠❛çã♦ ✐♥t❡✐r❛✱ ✐♥t❡✐r❛ ♠✐st❛ ❡ ✐♥t❡✐r❛ ❜✐♥ár✐❛✱ ❝♦♥s✐st❡♠ ❜❛s✐❝❛♠❡♥t❡ ❞❛ ♣r♦❣r❛♠❛✲ çã♦ ❧✐♥❡❛r ❝♦♠ ❛❝rés❝✐♠♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞❛ r❡str✐çã♦ ❞❡ q✉❡ ❛❧❣✉♠❛s ✈❛r✐á✈❡✐s s❡❥❛♠ ✐♥t❡✐r❛s✱ t♦❞❛s s❡❥❛♠ ✐♥t❡✐r❛s ❡ t♦❞❛s ✐♥t❡✐r❛s ❞♦ t✐♣♦0−1 ✭❋❡rr❡✐r❛ ❡t ❛❧✳✱ ✷✵✵✶✮✳
✶✻
✷✵✶✵✮ ❡t❝✳
❆ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡ ❡♠ ♣r♦❜❧❡♠❛s ❞❡ ♣r♦❣r❛♠❛çã♦ ✐♥t❡✐r❛ é ♥♦r♠❛❧♠❡♥t❡ tr❛t❛❞❛ ❝♦♠ ♦ ✉s♦ ❞❡ té❝♥✐❝❛s ❡♥✈♦❧✈❡♥❞♦ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❧✐♥❡❛r ♣♦r ♣❛rt❡s ✭❉❛♥t③✐❣✱ ✶✾✻✸❀ ❍✉✱ ✶✾✻✾✮ ♦✉ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❛ ❢✉♥çã♦ ♥ã♦ ❧✐♥❡❛r ❡♠ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧✱ ❛ s❡❣✉✐r ❝♦♥✈❡rt✐❞❛ ❡♠ ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r ❞❡ ✈❛r✐á✈❡✐s 0−1 ✭❇❛❧❛s✱ ✶✾✻✹❀ ❍❛♠♠❡r ❡ ❘✉❞❡❛♥✉✱ ✶✾✻✽✮✳ ◆❛ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s✱ ❛ ♥ã♦ ❧✐♥❡❛r✐❞❛❞❡ ❡♠ ♣r♦❜❧❡♠❛s ❞❡ ♣r♦❣r❛♠❛çã♦ ✐♥t❡✐r❛ ❛♣❛r❡❝❡ ❥á ♥❛ ❢♦r♠❛ ♣♦❧✐♥♦♠✐❛❧✱ s❡♥❞♦ q✉❡ ✉♠ ♥ú♠❡r♦ s✐❣♥✐✜❝❛t✐✈♦ ❞♦s ❝❛s♦s ❡♥✈♦❧✈❡ ❛♣❡♥❛s t❡r♠♦s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ✭●❧♦✈❡r✱ ✶✾✼✺✮✳
◆❛ ❧✐t❡r❛t✉r❛ ❞❡ ♣r♦❣r❛♠❛çã♦ ♠❛t❡♠át✐❝❛✱ ❝♦st✉♠❛✲s❡ ❝❤❛♠❛r ❞❡ ♣r♦❣r❛♠❛✲ çã♦ q✉❛❞rát✐❝❛ ❛ s✉❜ár❡❛ q✉❡ ❡♥❣❧♦❜❛ ♣r♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛çã♦ ❝♦♥s✐st✐♥❞♦ ♥❛ ♠❛①✐♠✐✲ ③❛çã♦✴♠✐♥✐♠✐③❛çã♦ ❞❡ ✉♠❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ s✉❥❡✐t❛ ❛ r❡str✐çõ❡s ❡①♣r❡ss❛s ♣♦r ✐❣✉❛❧✲ ❞❛❞❡s✴❞❡s✐❣✉❛❧❞❛❞❡s ❧✐♥❡❛r❡s✳ P❛rt✐❝✉❧❛r♠❡♥t❡✱ q✉❛♥❞♦ ❛s ✈❛r✐á✈❡✐s ❞♦ ♣r♦❜❧❡♠❛ ❡stã♦ r❡str✐t❛s ❛ ✈❛❧♦r❡s 0−1✱ ❡♥❝♦♥tr❛♠♦✲♥♦s ♥♦ ❡s❝♦♣♦ ❞❛ ♣r♦❣r❛♠❛çã♦ q✉❛❞rát✐❝❛ ❜✐♥ár✐❛✳
❊ss❛ ár❡❛ ❡♥❣❧♦❜❛ ♠✉✐t♦s ❞♦s ♣r♦❜❧❡♠❛s ◆P✲❉✐❢í❝❡✐s ♠❛✐s ❞❡s❛✜❛❞♦r❡s ❡♠ ♦t✐♠✐③❛çã♦ ❝♦♠❜✐♥❛tór✐❛ ✭●❛r② ❡ ❏♦❤♥s♦♥✱ ✶✾✼✾✮✳
❉❡✈✐❞♦ à ❛♠♣❧❛ ❣❛♠❛ ❞❡ ♣r♦❜❧❡♠❛s q✉❛❞rát✐❝♦s ❜✐♥ár✐♦s ❞❛ ❧✐t❡r❛t✉r❛ ❡ à ❞✐✲ ✜❝✉❧❞❛❞❡ ❡♠ r❡s♦❧✈ê✲❧♦s ❞❡ ❢♦r♠❛ ❡①❛t❛✱ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ❛ ♦❜t❡♥çã♦ ❞❡ ❧✐♠✐t❡s ✭✐♥❢❡r✐♦r ❡ s✉♣❡r✐♦r✮ q✉❡ ♣♦ss❛♠ ❛✉①✐❧✐❛r ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♠ét♦❞♦s ❡①❛t♦s✳ ●✉❡②❡ ❡ ▼✐❝❤❡✲ ❧♦♥ ✭✷✵✵✾✮ ❞✐✈✐❞✐r❛♠ ❡ss❡s ❧✐♠✐t❡s ❡♠ 04 ❣r✉♣♦s✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s té❝♥✐❝❛s ✉s❛❞❛s ♣❛r❛ ♦❜tê✲❧♦s✿ ♣r♦❣r❛♠❛çã♦ s❡♠✐❞❡✜♥✐❞❛ ✭●♦❡♠❛♥s ❡ ❲✐❧❧✐❛♠s♦♥✱ ✶✾✾✺❀ ❘❡♥❞❧✱ ❘✐♥❛❧❞✐✱ ❡ ❲✐❡❣❡❧❡✱ ✷✵✶✵✮✱ ❞❡❝♦♠♣♦s✐çã♦ ❧❛❣r❛♥❣❡❛♥❛ ✭❈❤❛r❞❛✐r❡ ❡ ❙✉tt❡r✱ ✶✾✾✺❀ ❊❧❧♦✉♠✐✱ ❋❛②❡✱ ❡ ❙♦✉t✐❢✱ ✷✵✵✵✮✱ ♠ét♦❞♦s ♣♦s✐❢♦r♠ ✭❇♦r♦s ❡ ❍❛♠♠❡r✱ ✷✵✵✷❀ ❇✐❧❧✐♦♥♥❡t ❡ ❙✉tt❡r✱ ✶✾✾✹✮ ❡ té❝♥✐❝❛s ❞❡ ❧✐♥❡❛r✐③❛çã♦ ✭❋♦rt❡t✱ ✶✾✺✾❜✱❛❀ ❘♦❞r✐❣✉❡s✱ ✷✵✶✵✮✳ ❆❝r❡s❝❡♥t❛♠♦s ❛ ❡ss❡ ❣r✉♣♦ ❛ s✉❛✈✐③❛çã♦ ❤✐♣❡r❜ó❧✐❝❛✱ té❝♥✐❝❛ ❛♠♣❧❛♠❡♥t❡ ❛♣❧✐❝❛❞❛ ❛ ♣r♦❜❧❡♠❛s ❝♦♥tí♥✉♦s ♥ã♦ ❞✐❢❡r❡♥✲ ❝✐á✈❡✐s ✭❳❛✈✐❡r✱ ✷✵✶✵✮✱ ✭❳❛✈✐❡r ❡ ❳❛✈✐❡r✱ ✷✵✶✶✮✱ ✭❳❛✈✐❡r ❡t ❛❧✳✱ ✷✵✶✹✮✱ ✭❙♦✉③❛ ❡t ❛❧✳✱ ✷✵✶✶✮✱ ✭❇❛❣✐r♦✈ ❡t ❛❧✳✱ ✷✵✶✺✮✱ ✭❳❛✈✐❡r ❡ ❳❛✈✐❡r✱ ✷✵✶✻✮✱ ♠❛s ❝♦♠ ♣♦t❡♥❝✐❛❧ ❞❡ ❛♣❧✐❝❛çã♦ t❛♠❜é♠ ❛ ♣r♦❜❧❡♠❛s q✉❛❞rát✐❝♦s ❜✐♥ár✐♦s✳
❉❡♥tr❡ ❛s té❝♥✐❝❛s ❞❡ ❧✐♥❡❛r✐③❛çã♦ ♣❛r❛ ♣r♦❜❧❡♠❛s q✉❛❞rát✐❝♦s ❜✐♥ár✐♦s✱ q✉❡ ❝♦♥s✐st❡♠ ❡♠ tr❛♥s❢♦r♠❛r ❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❡♠ ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r✱ ❘♦❞r✐❣✉❡s ❡t ❛❧✳ ✭✷✵✶✷✮ ❞❡st❛❝❛♠ ❞✉❛s ❛❜♦r❞❛❣❡♥s✿ ✭✐✮ ❧✐♥❡❛r✐③❛çã♦ ❝♦♠ ❛❝rés❝✐♠♦ ❞❡ ✉♠ ♥ú♠❡r♦ ♣♦❧✐♥♦✲ ♠✐❛❧ ❞❡ ✈❛r✐á✈❡✐s ❡ r❡str✐çõ❡s ❧✐♥❡❛r❡s❀ ✭✐✐✮ ❧✐♥❡❛r✐③❛çã♦ s❡♠ ❛❞✐çã♦ ❞❡ ✈❛r✐á✈❡✐s✱ ♠❛s ❝♦♠ ✉♠ ♥ú♠❡r♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ r❡str✐çõ❡s ❧✐♥❡❛r❡s✳ ◆♦ ♣r✐♠❡✐r♦ ❣r✉♣♦✱ ❡♥❝♦♥tr❛✲s❡ ♦ ♣r♦❝❡❞✐✲ ♠❡♥t♦ ❝❧áss✐❝♦ ❞❡ ❧✐♥❡❛r✐③❛çã♦ ✭❋♦rt❡t✱ ✶✾✺✾❜✮ ❡ ❛ ❧✐♥❡❛r✐③❛çã♦ ♣r♦♣♦st❛ ♣♦r ●❧♦✈❡r ✭✶✾✼✺✮❀ ♥♦ s❡❣✉♥❞♦ ❣r✉♣♦✱ ❡stá ❛ t✲❧✐♥❡❛r✐③❛çã♦✱ ♣r♦♣♦st❛ ♥♦ ❝♦♥t❡①t♦ ❞♦ Pr♦❜❧❡♠❛ ◗✉❛❞rát✐❝♦
❞❛ ▼♦❝❤✐❧❛ ✭P◗▼✮ ✭❘♦❞r✐❣✉❡s✱ ✷✵✶✵❀ ❘♦❞r✐❣✉❡s ❡t ❛❧✳✱ ✷✵✶✷✮✳
❆ t✲❧✐♥❡❛r✐③❛çã♦ é ✉♠❛ ❛❜♦r❞❛❣❡♠ r❡❝❡♥t❡ ❡ ❛✐♥❞❛ ♣♦✉❝♦ ❡st✉❞❛❞❛✱ q✉❡ ❝♦♥s✐st❡ ❜❛s✐❝❛♠❡♥t❡ ❞❡ ❞✉❛s ❡t❛♣❛s✳ Pr✐♠❡✐r♦✱ s✉❜st✐t✉✐✲s❡ ♦ t❡r♠♦ q✉❛❞rát✐❝♦ ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ♣♦r ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧ t✱ q✉❡ é ❧✐♠✐t❛❞❛ s✉♣❡r✐♦r♠❡♥t❡ ♣❡❧❛ ❡①♣r❡ssã♦ q✉❛❞rát✐❝❛✱ ❝♦♠ ❛
✶✼
✉♠ ❝♦♥❥✉♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ r❡str✐çõ❡s ❧✐♥❡❛r❡s✱ ❡q✉✐✈❛❧❡♥t❡s q✉❛♥❞♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞♦ t❡r♠♦ q✉❛❞rát✐❝♦ sã♦ ♥ã♦ ♥❡❣❛t✐✈♦s ✭❘♦❞r✐❣✉❡s ❡t ❛❧✳✱ ✷✵✶✷✮✳
❆ s✉❛✈✐③❛çã♦ ❤✐♣❡r❜ó❧✐❝❛ é ✉♠❛ té❝♥✐❝❛ ❞❡ ♣❡♥❛❧✐③❛çã♦ ♣r♦♣♦st❛ ♣❛r❛ r❡s♦❧✲ ✈❡r ♣r♦❜❧❡♠❛s ♥ã♦ ❞✐❢❡r❡♥❝✐á✈❡✐s q✉❡ t❡♠ ♦❜t✐❞♦ ❜♦♥s r❡s✉❧t❛❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ✭❳❛✈✐❡r✱ ✷✵✶✵❀ ❳❛✈✐❡r ❡ ❳❛✈✐❡r✱ ✷✵✶✻❀ ❇❛❣✐r♦✈✱ ❆❧ ◆✉❛✐♠❛t✱ ❡ ❙✉❧t❛♥♦✈❛✱ ✷✵✶✸❀ ❳❛✈✐❡r ❡ ❳❛✈✐❡r✱ ✷✵✶✶✮✳ ❊❧❛ ❝♦♥s✐st❡ ❡♠ s✉❜st✐t✉✐r ❛ r❡s♦❧✉çã♦ ❞✐r❡t❛ ❞♦ ♣r♦❜❧❡♠❛ ♥ã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ ♣❡❧❛ r❡s♦❧✉çã♦ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♣r♦❜❧❡♠❛s ❞✐❢❡r❡♥❝✐á✈❡✐s✱ ❡♠ ♣r✐♥❝í♣✐♦ ♠❛✐s ❢á❝❡✐s✱ ❡ q✉❡ ❣r❛❞❛t✐✈❛♠❡♥t❡ s❡ ❛♣r♦①✐♠❛♠ ❞♦ ♣r♦❜❧❡♠❛ ♦r✐❣✐♥❛❧ ✭▲❛✈♦r✱ ✷✵✵✹✮✳ ❈❛❞❛ s✉❜♣r♦❜❧❡♠❛✱ ♣♦r s❡r ✐♥✜♥✐t❛♠❡♥t❡ ❞✐❢❡r❡♥❝✐á✈❡❧✱ t♦r♥❛ ✈✐á✈❡❧ s✉❛ r❡s♦❧✉çã♦ ❛tr❛✈és ❞❡ ♠ét♦❞♦s ❞❡ ♦t✐✲ ♠✐③❛çã♦ ♠❛✐s r♦❜✉st♦s✱ q✉❡ sã♦ ❢✉♥❞❛♠❡♥t❛❞♦s ❡♠ ✐♥❢♦r♠❛çõ❡s ❞♦ ❣r❛❞✐❡♥t❡✱ t❛✐s ❝♦♠♦ ♠ét♦❞♦ ❞♦ ❣r❛❞✐❡♥t❡ ❝♦♥❥✉❣❛❞♦✱ q✉❛s❡✲◆❡✇t♦♥ ♦✉ ◆❡✇t♦♥ ✭❳❛✈✐❡r ❡ ❳❛✈✐❡r✱ ✷✵✶✻✮✳
◆❡st❡ tr❛❜❛❧❤♦✱ ❝♦♥❝❡♥tr❛♠♦✲♥♦s ❡♠ ✉t✐❧✐③❛r té❝♥✐❝❛s ❛✐♥❞❛ ♣♦✉❝♦ ❡①♣❧♦r❛❞❛s ♣❛r❛ tr❛t❛r ♣r♦❜❧❡♠❛s q✉❛❞rát✐❝♦s 0−1✱ ❡♠ ♣❛rt✐❝✉❧❛r t✕❧✐♥❡❛r✐③❛çã♦ ❡ s✉❛✈✐③❛çã♦ ❤✐✲
♣❡r❜ó❧✐❝❛✳ ❖ ♣r♦❜❧❡♠❛ q✉❛❞rát✐❝♦ ✭P◗✮✱ ❜❛s❡ ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ♦✉tr♦s ♣r♦❜❧❡♠❛s ❞❡ss❛ ❝❧❛ss❡✱ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
(P Q) max
n−1
X
i=1 n X
j=i+1
cijxixj + n X
i=1
bixi
s.a x∈ {0,1}n,
♦♥❞❡ cij✱ 1 ≤ i < j ≤ n✱ sã♦ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s ❛r❜✐trár✐♦s✳ P♦r ❝♦♥✈❡♥✐ê♥❝✐❛✱ ❞❡✜♥✐♠♦s
♦s ❝♦❡✜❝✐❡♥t❡s ❛❞✐❝✐♦♥❛✐s cij = cji ❡ cii = 0✱ 1 ≤ j ≤ i ≤ n✳ ❆ ❡ss❡ ♣r♦❜❧❡♠❛ ❜❛s❡
❝♦♥s✐❞❡r❛♠♦s ♦ ❛❝rés❝✐♠♦ ❞❡ r❡str✐çõ❡s ❧✐♥❡❛r❡s✱ ❡♠ ♣❛rt✐❝✉❧❛r r❡str✐çõ❡s ❞❡ ❝❛r❞✐♥❛❧✐❞❛❞❡✳ ❖ r❡st❛♥t❡ ❞❡ss❡ tr❛❜❛❧❤♦ ❝♦♥té♠ 5 ❝❛♣ít✉❧♦s ❡ ❡stá ♦r❣❛♥✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳ ◆♦ ❈❛♣ít✉❧♦ 2 ✐r❡♠♦s ❛❜♦r❞❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❧✐♥❡❛r✐③❛çã♦ ✭❈❧áss✐❝❛✱ ●❧♦✈❡r ❡ t✕ ❧✐♥❡❛r✐③❛çã♦✮ ❡♠ ♣r♦❜❧❡♠❛s q✉❛❞rát✐❝♦s ❜✐♥ár✐♦s✳ ❱❛♠♦s ❛♣r❡s❡♥t❛r ♣r♦✈❛s ❛❧t❡r♥❛t✐✈❛s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ r❡❛❧✐③❛❞♦ ♣♦r ❘♦❞r✐❣✉❡s ✭✷✵✶✵✮ ♣❛r❛ ✭P◗✮ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♥ã♦ ♥❡❣❛t✐✈♦s✱ q✉❡ s❡rã♦ ❜❛s❡ ♣❛r❛ ❛ ❡①t❡♥sã♦ ❞♦ ♠ét♦❞♦ ❞❛ t✕❧✐♥❡❛r✐③❛çã♦ ♣❛r❛ ✭P◗✮ q✉❛♥❞♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ sã♦ ❛r❜✐trár✐♦s✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ ❣❡♥❡r❛❧✐③❛♠♦s ♦s r❡✲ s✉❧t❛❞♦s ♦❜t✐❞♦s ❡♠ ✭❘♦❞r✐❣✉❡s✱ ✷✵✶✵✮ ❡ ✭❘♦❞r✐❣✉❡s ❡t ❛❧✳✱ ✷✵✶✷✮✳ ❱❛♠♦s ♠♦str❛r q✉❡ ❛ t✕❧✐♥❡❛r✐③❛çã♦ ♦❢❡r❡❝❡ ♠❡❧❤♦r ❧✐♠✐t❡ s✉♣❡r✐♦r ❞❡♥tr❡ ❛s ❧✐♥❡❛r✐③❛çõ❡s ❛q✉✐ ❡st✉❞❛❞❛s✳ ❆✐♥❞❛ ♥♦ ❈❛♣ít✉❧♦ 2 ✈❛♠♦s r❡✈✐s❛r ♦ ❛r❝❛❜♦✉ç♦ ❞❛ s✉❛✈✐③❛çã♦ ❤✐♣❡r❜ó❧✐❝❛✱ ✉s❛❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ✭❳❛✈✐❡r ❡ ❳❛✈✐❡r✱ ✷✵✶✻✮✳
✶✽
❞❛ t✕❧✐♥❡❛r✐③❛çã♦ q✉❛♥❞♦ ❝♦♥s✐❞❡r❛♠♦s ❛ ❢♦r♠✉❧❛çã♦ q✉❛❞rát✐❝❛ ❞♦ ♣r♦❜❧❡♠❛ ❞♦ ❝♦rt❡ ♠á①✐♠♦✳ ❊①♣❡r✐♠❡♥t♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♠♦str❛♠ q✉❡ ❛ t✕❧✐♥❡❛r✐③❛çã♦ r❡q✉❡r ♠❡♥♦s r❡s✲ tr✐çõ❡s ♣❛r❛ ❛❧❝❛♥ç❛r ♦ ♠❡s♠♦ ❧✐♠✐t❡ ♦❜t✐❞♦ ♣❡❧❛s ❧✐♥❡❛r✐③❛çõ❡s ❝❧áss✐❝❛ ❡ ❣❧♦✈❡r✳ ❆✐♥❞❛ ♥❡ss❡ ❝❛♣ít✉❧♦✱ ✈❛♠♦s ♠♦str❛r ❝♦♠♦ ♦❜t❡r ✉♠❛ ✈❡rsã♦ s✉❛✈✐③❛❞❛ ❞♦ ♠♦❞❡❧♦ ✐rr❡str✐t♦ ❞❡✜♥✐❞♦ s♦❜r❡ ✈❛r✐á✈❡✐s ✐♥❞❡①❛❞❛s ♣❡❧♦s ✈ért✐❝❡s✱ ♣♦ré♠ ❢♦rt❛❧❡❝✐❞♦ ♣❡❧❛s r❡str✐çõ❡s tr✐✲ ❛♥❣✉❧❛r❡s✱ q✉❡ sã♦ ❡①♣r❡ss❛s ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❡♠ ❢✉♥çã♦ ❞❛s ✈❛r✐á✈❡✐s ✐♥❞❡①❛❞❛s ♣❡❧❛s ❛r❡st❛s✳ ❊①♣❡r✐♠❡♥t♦s ❛t❡st❛♠ ♠❡❧❤♦r❡s ❧✐♠✐t❡s q✉❛♥❞♦ ❛ s✉❛✈✐③❛çã♦ é ❢♦rt❛❧❡❝✐❞❛ ♣❡❧❛s r❡str✐çõ❡s tr✐❛♥❣✉❧❛r❡s ❡ ✐♥❝♦r♣♦r❛ ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ ❜✉s❝❛ ❧♦❝❛❧✳ ❉❡r✐✈❛♠♦s ❛✐♥❞❛ ❝♦♥✲ ❞✐çõ❡s ❞❡ ♦t✐♠❛❧✐❞❛❞❡ q✉❡ ♣❡r♠✐t❡♠ ❛ ❡①♣r❡ssã♦ ❞❡ ♥♦✈❛s ❞❡s✐❣✉❛❧❞❛❞❡s ✈á❧✐❞❛s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✱ ♦ ❡st❛❜❡❧❡❝✐♠❡♥t♦ ❞❡ t❡♦r❡♠❛s ♣❛r❛ ✜①❛çã♦ ❞❡ ✈❛r✐á✈❡✐s✱ ❛❧é♠ ❞❛ ♣r♦♣♦s✐çã♦ ❞❡ ❤❡✉ríst✐❝❛s ❣✉❧♦s❛s s✐♠♣❧❡s✳ ❚♦❞♦s ❡ss❡s ✐♥❣r❡❞✐❡♥t❡ q✉❡ ♣r♦♣♦♠♦s sã♦ ❝♦♠❜✐♥❛❞♦s ❞❡ ❞✐❢❡r❡♥t❡s ❢♦r♠❛s ♥❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ♠ét♦❞♦ ❞❡ s♦❧✉çã♦ ❡①❛t❛ ♣❛r❛ ♦ ♠❛①✲❝✉t✳ P♦r ✜♠✱ ❢❡❝❤❛♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ❝♦♠❡♥tár✐♦s ❡ ❝♦♥❝❧✉sõ❡s✳
❖ ❈❛♣ít✉❧♦ 4é ❞❡❞✐❝❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ❞❛ ❞✐✈❡rs✐❞❛❞❡ ♠á①✐♠❛ ✭P❉▼✮✳ ❱❛♠♦s r❡✈✐s❛r s✉❛s ♣r✐♥❝✐♣❛✐s ❢♦r♠✉❧❛çõ❡s ❡✱ ❡♠ s❡❣✉✐❞❛✱ ♠♦str❛r ❛ ❛♣❧✐❝❛çã♦ ❞❛ t✕❧✐♥❡❛r✐③❛çã♦ ❛♦ ♣r♦❜❧❡♠❛✳ ■♥s♣✐r❛❞♦s ❡♠ ❘♦❞r✐❣✉❡s ✭✷✵✶✵✮✱ ❞❡♠♦♥str❛♠♦s ✉♠ t❡♦r❡♠❛ ❞❡ ❧✐❢t✐♥❣ ♣❛r❛ ❢♦rt❛❧❡❝❡r ❛s r❡str✐çõ❡s ❞❛ t✕❧✐♥❡❛r✐③❛çã♦✱ ✉s❛♥❞♦ ❛ r❡str✐çã♦ ❞❡ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞♦ ♣r♦❜❧❡♠❛ ✭P❉▼✮✳ ❊①♣❡r✐♠❡♥t♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❝♦♠ ❝❧❛ss❡s ❞❡ ✐♥stâ♥❝✐❛s ❞❛ ❧✐t❡r❛t✉r❛ ❛t❡st❛♠ q✉❡ ❛s r❡str✐çõ❡s t✲❧✐♥❡❛r✐③❛❞❛s ❢♦rt❛❧❡❝✐❞❛s ♦❜tê♠ ♦s ♠❡❧❤♦r❡s ❧✐♠✐t❡s ❞❡♥tr❡ ❛s ❧✐♥❡❛r✐③❛çõ❡s ❝♦♠♣❛r❛❞❛s✳ ❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ♥♦✈❛s r❡str✐çõ❡s ✈á❧✐❞❛s✱ ❛ ♣❛r✲ t✐r ❞❡ ❣❡♥❡r❛❧✐③❛çõ❡s ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♣♦❞❛ ✐♥✐❝✐❛❧♠❡♥t❡ ♠♦str❛❞❛s ❡♠ ▼❛rtí✱ ●❛❧❧❡❣♦✱ ❡ ❉✉❛rt❡ ✭✷✵✶✵✮✳ ❆✐♥❞❛ ♥❡ss❡ ❝❛♣ít✉❧♦✱ ❡♠♣r❡❣❛♠♦s t❛❧ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ t❡ór✐❝♦ ♣❛r❛ ♣r♦♣♦s✐çã♦ ❞❡ ✉♠ ♠ét♦❞♦ q✉❡ r❡s♦❧✈❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❢♦r♠❛ ❡①❛t❛ ❡ r❡❛❧✐③❛♠♦s ♦✉tr♦s ❡①♣❡r✐♠❡♥t♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s✱ ❝♦♠♣❛r❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❝♦♠ ❛q✉❡❧❡s ❞♦ ♠❡❧❤♦r ♠ét♦❞♦ ❡①❛t♦ ❞✐s♣♦♥í✈❡❧✳ ❋❡❝❤❛♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ❝♦♠❡♥tár✐♦s ❡ ❝♦♥❝❧✉sõ❡s✳
✶✾
✷ ▼❊❚❖❉❖▲❖●■❆
◆❡st❡ ❝❛♣ít✉❧♦✱ ✈❛♠♦s ❛❜♦r❞❛r ♦ ♣r♦❜❧❡♠❛ q✉❛❞rát✐❝♦ ❜✐♥ár✐♦
(P Q) max
n−1
X
i=1 n X
j=i+1
cijxixj + n X
i=1
bixi = n X
i=1 i−1
X
j=1
cjixixj+ n X
i=1
bixi
s.a x∈ {0,1}n,
✭✶✮
♦♥❞❡ cij✱ 1≤i < j ≤n✱ sã♦ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s ❛r❜✐trár✐♦s✳ P♦r ❝♦♥✈❡♥✐ê♥❝✐❛✱ ❞❡✜♥✐♠♦s ♦s
❝♦❡✜❝✐❡♥t❡s ❛❞✐❝✐♦♥❛✐scij =cji ❡cii= 0✱1≤j ≤i≤n✳
◆❛s três ♣ró①✐♠❛s s✉❜s❡çõ❡s✱ ✈❛♠♦s ❛♣r❡s❡♥t❛r três ♠❛♥❡✐r❛s ❞❡ ♦❜t❡r ✉♠❛ ❢♦r♠✉❧❛çã♦ ❧✐♥❡❛r ❡q✉✐✈❛❧❡♥t❡✱ ❞✉❛s ❞❡❧❛s ❜❛st❛♥t❡ ❝♦♥❤❡❝✐❞❛s ♥❛ ❧✐t❡r❛t✉r❛✳ ❆ ♣r✐♠❡✐r❛✱ ❛ q✉❡ ♥♦s r❡❢❡r✐r❡♠♦s ❝♦♠♦ ❧✐♥❡❛r✐③❛çã♦ ❝❧áss✐❝❛✱ ♦❜té♠ ✉♠ ♠♦❞❡❧♦ ❧✐♥❡❛r 0− 1✱ ❝♦♠ ✉♠ ♥ú♠❡r♦ q✉❛❞rát✐❝♦ ✭❡♠ n✮ ❞❡ ✈❛r✐á✈❡✐s ❡ r❡str✐çõ❡s❀ ❛ s❡❣✉♥❞❛✱ ♣r♦♣♦st❛ ♣♦r ●❧♦✈❡r
✭✶✾✼✺✮✱ ❧❡✈❛ ❛ ✉♠ ♣r♦❜❧❡♠❛ ❧✐♥❡❛r 0−1 ♠✐st♦✱ ❞❡✜♥✐❞♦ ❛ ♣❛rt✐r ❞❛s ♠❡s♠❛s ✈❛r✐á✈❡✐s ❜✐♥ár✐❛s ❞❡ ✭✶✮✱ ❥✉♥t♦ ❝♦♠ ✉♠ ♥ú♠❡r♦ ❧✐♥❡❛r ❞❡ ✈❛r✐á✈❡✐s ❝♦♥tí♥✉❛s ❡ r❡str✐çõ❡s✳ ❆ t❡r❝❡✐r❛✱ ❝❤❛♠❛❞❛t✲❧✐♥❡❛r✐③❛çã♦✱ ❢♦✐ ♣r♦♣♦st❛ ♠❛✐s r❡❝❡♥t❡♠❡♥t❡ ❡ ✉s❛✱ ❛❧é♠ ❞❛s ✈❛r✐á✈❡✐s ❜✐♥ár✐❛s
♦r✐❣✐♥❛✐s✱ ❛♣❡♥❛s ✉♠❛ ✈❛r✐á✈❡❧ ❝♦♥tí♥✉❛❀ ♣♦ré♠✱ ❞❡♠❛♥❞❛ ✉♠ ♥ú♠❡r♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ r❡str✐çõ❡s✱ q✉❡ ♣♦❞❡♠ s❡r s❡♣❛r❛❞❛s ♣♦❧✐♥♦♠✐❛❧♠❡♥t❡✳ ❊❧❛ ❢♦✐ ♠♦str❛❞❛ ❝♦rr❡t❛ q✉❛♥❞♦ ♦s ❝♦❡✜❝✐❡♥t❡s cij sã♦ ♥ã♦ ♥❡❣❛t✐✈♦s✳ ❆q✉✐ ♠♦str❛♠♦s ❝♦♠♦ ❛♣❧✐❝á✲❧❛ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s
❛r❜✐trár✐♦s✱ ❣❡♥❡r❛❧✐③❛♥❞♦ r❡s✉❧t❛❞♦s ❞❛ ❧✐t❡r❛t✉r❛✱ q✉❡ ✐♥❝❧✉❡♠ t❛♥t♦ ❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❝♦♠ ♦ ♠♦❞❡❧♦ ♦r✐❣✐♥❛❧ ✭✶✮✱ ❝♦♠♦ t❛♠❜é♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❛❧❣♦r✐t♠♦ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛ s❡♣❛r❛çã♦ ❞❛s r❡str✐çõ❡s✳ P❛rt❡ ❞❡ss❡s r❡s✉❧t❛❞♦s ❝♦♠♣õ❡ ✉♠ ❛rt✐❣♦ ♣✉❜❧✐❝❛❞♦ ♥♦s ❛♥❛✐s ♥♦ ❙✐♠♣ós✐♦ ❇r❛s✐❧❡✐r♦ ❞❡ P❡sq✉✐s❛ ❖♣❡r❛❝✐♦♥❛❧ ✲ ❙❇P❖ ✷✵✶✼ ✭❙♦❛r❡s ❡t ❛❧✳✱ ✷✵✶✼✮✳
◆❛ ú❧t✐♠❛ s❡çã♦✱ ❞❡s❝r❡✈❡♠♦s ❜r❡✈❡♠❡♥t❡ ♦ ♠ét♦❞♦ ❞❡ s✉❛✈✐③❛çã♦ ❤✐♣❡r❜ó❧✐❝❛✱ ✐♥✐❝✐❛❧♠❡♥t❡ ♣r♦♣♦st♦ ♣❛r❛ s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❝♦♥tí♥✉♦s ♥ã♦ ❞✐❢❡r❡♥❝✐á✈❡✐s✳ ❊♠ ♥♦ss♦ tr❛❜❛❧❤♦✱ ❡❧❡ s❡rá ✉s❛❞♦ ♣❛r❛ ♦❜t❡r ❜♦❛s s♦❧✉çõ❡s ❞❡ ♣r♦❜❧❡♠❛s q✉❛❞rát✐❝♦s ❜✐♥ár✐♦s✳
✷✳✶ ▲✐♥❡❛r✐③❛çã♦ ❈❧áss✐❝❛
❆ té❝♥✐❝❛ ❞❡ ❧✐♥❡❛r✐③❛çã♦ ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ q✉❛❞rát✐❝♦ ❜✐♥ár✐♦ ❝♦♥s✐st❡ ❡♠ tr❛♥s❢♦r♠❛r ❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❡♠ ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r✱ ♣♦❞❡♥❞♦ s❡r ❞✐✈✐❞✐❞❛ ❡♠ ❞✉❛s ❛❜♦r❞❛❣❡♥s✱ s❡♥❞♦ ❡❧❛s✿ ✭✐✮ ❧✐♥❡❛r✐③❛çã♦ s❡♠ ❛❞✐çã♦ ❞❡ ✈❛r✐á✈❡✐s✱ ♠❛s ❝♦♠ ✉♠ ♥ú♠❡r♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ r❡str✐çõ❡s❀ ✭✐✐✮ ❧✐♥❡❛r✐③❛çã♦ ❝♦♠ ✉♠ ♥ú♠❡r♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡ ✈❛r✐á✈❡✐s ❡ r❡s✲ tr✐çõ❡s ✭❘♦❞r✐❣✉❡s ❡t ❛❧✳✱ ✷✵✶✷✮✳ ◆♦ ❝♦♥t❡①t♦ ❞❡ss❛ s❡❣✉♥❞❛ ❛❜♦r❞❛❣❡♠✱ ❡♠ ❋♦rt❡t ✭✶✾✺✾❜✮ ❡ ❋♦rt❡t ✭✶✾✺✾❛✮✱ ♦ ❛✉t♦r ♠♦str❛ ❝♦♠♦ ❝♦♥✈❡rt❡r ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♣r♦❣r❛♠❛çã♦ ♣♦❧✐♥♦✲ ♠✐❛❧ 0−1 ❡♠ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♣r♦❣r❛♠❛çã♦ ❧✐♥❡❛r 0−1✳ ❊ss❡ ♣r♦❝❡❞✐♠❡♥t♦✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❧✐♥❡❛r✐③❛çã♦ ❝❧áss✐❝❛✱ ❝♦♥s✐st❡ ❡♠ s✉❜st✐t✉✐r ❝❛❞❛ ♣r♦❞✉t♦xixj ♣♦r ✉♠❛ ✈❛r✐á✈❡❧ ♥ã♦
♥❡❣❛t✐✈❛yij ❡ ❛❞✐❝✐♦♥❛r r❡str✐çõ❡syij ≤xi, yij ≤xj, yij ≥xi+xj −1✳
✷✵
✭✶✮ ❡ r❡❡s❝r❡✈ê✲❧♦ ❝♦♠♦✿
(P Q)lc max n−1
X
i=1 n X
j=i+1
cijyij + n X
i=1
bixi
s.a yij ≤xi, ∀ 1≤i < j≤n,
yij ≤xj, ∀1≤i < j ≤n,
yij ≥xi+xj−1 ∀ 1≤i < j ≤n,
yij ∈ {0,1} ∀ 1≤i < j ≤n,
x∈ {0,1}n.
✭✷✮
◆♦t❡ q✉❡ ♣♦❞❡♠♦s s✉❜st✐t✉✐ryij ∈ {0,1} ♣♦r0≤yij ≤1 ❡♠ ✭✷✮✳
P❛r❛ ✉♠ ú♥✐❝♦ t❡r♠♦ q✉❛❞rát✐❝♦✱ ❛ ❧✐♥❡❛r✐③❛çã♦ ❝❧áss✐❝❛ é ❛♣❡rt❛❞❛✱ ❝♦♠♦ ♠♦str❛ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✳
Pr♦♣♦s✐çã♦ ✷✳✶✳ ❙❡ A=conv{(x1, x2, y)∈B3 :y=x1x2} ❡ B ={(x1, x2, y)∈R3 :y≤
x1, y ≤x2, y ≥x1+x2−1, y ≥0, x1 ≤1, x2 ≤1}✱ ❡♥tã♦ A=B✳
❉❡♠♦♥str❛çã♦✳ ❈❧❛r❛♠❡♥t❡ A ⊆ B✳ ❈♦♥s✐❞❡r❡ ❛❣♦r❛ (¯x1,x¯2,y¯) ∈ B✳ ❚❡♠♦s ❡♥tã♦ (¯x1,x¯2,y¯) = (¯x1 −y¯)(1,0,0) + (¯x2 −y¯)(0,1,0) + ¯y(1,1,1) + (1 + ¯y −x¯1 −x¯2)(0,0,0)✱ ♠♦str❛♥❞♦ q✉❡(¯x1,x¯2,y¯)∈A✳
❊♥tr❡t❛♥t♦✱ é s❛❜✐❞♦ q✉❡ ❛ r❡❧❛①❛çã♦ ❧✐♥❡❛r ❞❡ ✭✷✮ t♦r♥❛✲s❡ ❢r❛❝❛ q✉❛♥❞♦ ❤á ♠✉✐t♦s t❡r♠♦s q✉❛❞rát✐❝♦s✳
✷✳✷ ▲✐♥❡❛r✐③❛çã♦ ❞❡ ●❧♦✈❡r
●❧♦✈❡r ❡ ❲♦♦❧s❡② ✭✶✾✼✹✮ ♣r♦♣õ❡♠ ✉♠❛ ♦✉tr❛ ❧✐♥❡❛r✐③❛çã♦ ♣❛r❛ ✭P◗✮✳ ❆ ✐❞❡✐❛ é ❞❡✜♥✐r ✈❛r✐á✈❡✐s ❝♦♥tí♥✉❛su∈Rn−1 t❛✐s q✉❡
ui =xi n X
j=i+1
cijxj ∀ i∈ {1, . . . , n−1}. ✭✸✮
P❛r❛ ✐ss♦✱ s❡❥❛
Ui = n X
j=i+1
c+ij ❡ Ui =
n X
j=i+1
c−ij,
♦♥❞❡ c+ij = max(0, cij)❡ c−ij = min(0, cij)✳ ◆♦t❡ q✉❡ Ui ≤ n X
j=i+1
cijxj ≤Ui ❡ Ui ≤ui ≤Ui✳
✷✶
(P Q)lg max n−1
X
i=1
ui+ n X
i=1
bixi ✭✹✮
s.a ui ≤Uixi, 1≤i≤n−1, ✭✺✮
ui ≤ n X
j=i+1
cijxj−Ui(1−xi), 1≤i≤n−1, ✭✻✮
x∈ {0,1}n. ✭✼✮
◗✉❛♥❞♦xi = 1✱ ❛ r❡str✐çã♦ ✭✺✮ t♦r♥❛✲s❡ r❡❞✉♥❞❛♥t❡ ❡♥q✉❛♥t♦ ✭✻✮ ❧❡✈❛ ❛ui ≤ Pn
j=i+1cijxj✳
◆♦ ót✐♠♦✱ t❡♠♦s ❛ ✐❣✉❛❧❞❛❞❡✱ ♦ q✉❡ ❡q✉✐✈❛❧❡ ❛ ✭✸✮ ♣❛r❛ xi = 1✳ ◗✉❛♥❞♦ xi = 0✱ ✭✺✮
❡st❛❜❡❧❡❝❡ q✉❡ ui ≤ 0✱ ❧❡✈❛♥❞♦ ✭✻✮ ❛ s❡ t♦r♥❛r r❡❞✉♥❞❛♥t❡✳ ❆❧é♠ ❞✐ss♦✱ ❛ ❢✉♥çã♦ ✭✹✮
❣❛r❛♥t❡ui = 0 ♥♦ ót✐♠♦✳ P♦rt❛♥t♦ ✭✺✮✕✭✼✮ ❣❛r❛♥t❡♠ ❛ ✐❣✉❛❧❞❛❞❡ ✭✸✮ ♥♦ ót✐♠♦✳
❖ ♠♦❞❡❧♦ ❧✐♥❡❛r✐③❛❞♦ ✭✹✮✲✭✼✮ ♣♦ss✉✐ ♠❡♥♦s ✈❛r✐á✈❡✐s q✉❡ ✭✷✮✱ ❡♥tr❡t❛♥t♦ s✉❛ r❡❧❛①❛çã♦ ♣♦❞❡ s❡r ♠❛✐s ❢r❛❝❛✳
❚❡♦r❡♠❛ ✷✳✶✳ ❖ ✈❛❧♦r ❞❛ r❡❧❛①❛çã♦ ❞❡ ✭✹✮✲✭✼✮ é ♣❡❧♦ ♠❡♥♦s ❛q✉❡❧❡ ❞❛ r❡❧❛①❛çã♦ ❞❡ ✭✷✮✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ (x, y) ✈✐á✈❡❧ ♣❛r❛ ❛ r❡❧❛①❛çã♦ ❞❡ ✭✷✮✳ ❉❡✜♥❛ ui = Pnj=i+1cijyij✳ ❖
✈❛❧♦r ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ✭✹✮ ❡♠(x, u)é tr✐✈✐❛❧♠❡♥t❡ ✐❣✉❛❧ ❛♦ ✈❛❧♦r ❞❡ ♦❜❥❡t✐✈♦ ❞❡ ✭✷✮ ❡♠ (x, y)✳ ❊♥tã♦✱ é s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ (x, u) s❛t✐s❢❛③ ✭✺✮✲✭✻✮✳ P♦r ✉♠ ❧❛❞♦✱ t❡♠♦s q✉❡
ui = n X
j=i+1
cijyij ≤ n X
j=i+1
c+ijyij ≤ n X
j=i+1
c+ijxi.
P♦r ♦✉tr♦ ❧❛❞♦✱
ui = n X
j=i+1
cijyij = n X
j=i+1
c+ ijyij +
n X
j=i+1
c−ijyij
≤
n X
j=i+1
c+ijxj + n X
j=i+1
c−ij(xi+xj−1) = n X
j=i+1
cijxj − n X
j=i+1
c−ij(1−xi).
✷✳✸ t✕▲✐♥❡❛r✐③❛çã♦
✷✷
❧✐♥❡❛r❡s✱ ❞❡r✐✈❛❞❛s ❞❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦ ❢❡❝❤♦ ❝♦♥✈❡①♦ ❞♦ ❝♦♥❥✉♥t♦ ❞❡✜♥✐❞♦ ♣❡❧❛ r❡str✐çã♦ q✉❛❞rát✐❝❛ ❡ r❡str✐çõ❡s ❞❡ ✐♥t❡❣r❛❧✐❞❛❞❡ ✭❘♦❞r✐❣✉❡s✱ ✷✵✶✵✮✳ ◆♦ ❝❛s♦ ❞♦ ✭P◗▼✮✱ ♦s ❝♦❡✜✲ ❝✐❡♥t❡s ❞♦ t❡r♠♦ q✉❛❞rát✐❝♦ sã♦ ♥ã♦ ♥❡❣❛t✐✈♦s✳ ❊ss❛ ♣r♦♣r✐❡❞❛❞❡ é ❢♦rt❡♠❡♥t❡ ✉s❛❞❛ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ❘♦❞r✐❣✉❡s ❡t ❛❧✳ ✭✷✵✶✷✮✳
❱❛♠♦s ❡st❡♥❞❡r ❛ t❡♦r✐❛ ❞❛ t✲❧✐♥❡❛r✐③❛çã♦ ❡ ❛♣❧✐❝á✲❧❛ ❛♦ ♣r♦❜❧❡♠❛ ✭✶✮ ❝♦♠ ❝♦✲ ❡✜❝✐❡♥t❡s ❛r❜✐trár✐♦s ❛ss♦❝✐❛❞♦s ❛♦s t❡r♠♦s q✉❛❞rát✐❝♦s✳ P❛r❛ ❢❛❝✐❧✐t❛r ♦ ❡♥t❡♥❞✐♠❡♥t♦✱ ♣r✐♠❡✐r♦ ❛♣r❡s❡♥t❛♠♦s✱ ♥❛ ❙✉❜s❡çã♦ ✭✷✳✸✳✶✮✱ ❛ t✲❧✐♥❡❛r✐③❛çã♦ ♣❛r❛ ♦ ❝❛s♦ ♦♥❞❡ ♦s ❝♦❡✲ ✜❝✐❡♥t❡s ❞♦ t❡r♠♦ q✉❛❞rát✐❝♦ sã♦ ♥ã♦ ♥❡❣❛t✐✈♦s✱ s❡❣✉✐♥❞♦ ❘♦❞r✐❣✉❡s ❡t ❛❧✳ ✭✷✵✶✷✮✳ ◆❛ s✉❜s❡çã♦ s❡❣✉✐♥t❡✱ ❡st❡♥❞❡♠♦s ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ♣❛r❛ ❝✉st♦s ❛r❜✐trár✐♦s✳
✷✳✸✳✶ ❈♦❡✜❝✐❡♥t❡s ♥ã♦ ♥❡❣❛t✐✈♦s
❆♦ ❧♦♥❣♦ ❞❡st❛ s✉❜s❡çã♦✱ ❝♦♥s✐❞❡r❛♠♦s ♦ ♣r♦❜❧❡♠❛ ✭P◗✮ ❡ ❛❞♠✐t✐♠♦s q✉❡ cij = cji ≥
0 ∀1 ≤ i ≤ j ≤ n✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣r✐♠❡✐r❛ ❡t❛♣❛ ❞❡ ❘♦❞r✐❣✉❡s ✭✷✵✶✵✮✱ r❡❡s❝r❡✈❡♠♦s
✭✶✮ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ t✱ ❣❡r❛♥❞♦
(P Q)t max t+ n X
i=1
bixi
s.a t≤
n X
i=1 i−1
X
j=1
cjixixj,
(x, t)∈Bn×R.
P❛r❛ ❞❡s❝r❡✈❡r ❛ s❡❣✉♥❞❛ ❡t❛♣❛✱ s❡❥❛♠
Z =
(
(x, t)∈Bn×R:t≤
n X
i=1 i−1
X
j=1
cjixixj )
♦ ❝♦♥❥✉♥t♦ ✈✐á✈❡❧ ❞❡ (P Q)t ❡ Sn ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ♣❡r♠✉t❛çõ❡s ❞❡ {1, . . . , n}✳ ❖
❝♦♥❥✉♥t♦ Z ♣♦❞❡ s❡r ❡①♣r❡ss♦ ♣♦r r❡str✐çõ❡s ❧✐♥❡❛r❡s✱ ❝♦♠♦ s❡❣✉❡✿
❚❡♦r❡♠❛ ✷✳✷✳ ❙❡ cij ≥0✱ ♣❛r❛ t♦❞♦ 1≤i, j ≤n✱ ❡♥tã♦ Z =Z′✱ ♦♥❞❡
Z′ =
(
(x, t)∈Bn×R:t≤
n X
i=1 i−1
X
j=1
cπ(j)π(i)xπ(i) ∀π∈Sn )
.
❉❡♠♦♥str❛çã♦✳ Pr✐♠❡✐r♦✱ ♥♦t❡ q✉❡
n X
i=1 i−1
X
j=1
cjixixj = n X
i=1 i−1
X
j=i
cπ(j)π(i)xπ(i)xπ(j) ∀π∈Sn. ✭✽✮
✷✸
n X
i=1 i−1
X
j=1
cπ(j)π(i)xπ(i)xπ(j)≤ n X
i=1 i−1
X
j=1
cπ(j)π(i)xπ(i).
❙✉♣♦♥❤❛ ❛❣♦r❛ (x, t) ∈ Z′ ❡ s❡❥❛ π ∈ Sn ❛ ♣❡r♠✉t❛çã♦ q✉❡ ♦r❞❡♥❛ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ x
❡♠ ♦r❞❡♠ ♥ã♦ ❝r❡s❝❡♥t❡✱ ♦✉ s❡❥❛✱ xπ(j) ≥ xπ(i) s❡ j ≤ i✳ ❖❜s❡r✈❡ q✉❡ (xπ(1), . . . , xπ(n)) = (1, . . . ,1,0, . . . ,0)✳ ❈♦♠♦(x, t)∈Z′✱ t❡♠♦s q✉❡
t≤
n X
i=1 i−1
X
j=1
cπ(j)π(i)xπ(i)= n X
i=1 i−1
X
j=1
cπ(j)π(i)xπ(i)xπ(j),
♦♥❞❡ ❛ ✐❣✉❛❧❞❛❞❡ ❞❡❝♦rr❡ ❞❛ ♦r❞❡♥❛çã♦ ❞❛s ❡♥tr❛❞❛s ❞❡x ♣♦r π✳
P❡❧♦ ❚❡♦r❡♠❛ ✭✽✮✱ ♣♦❞❡♠♦s s✉❜st✐t✉✐rZ♣♦rZ′ ♥❛ ❞❡s❝r✐çã♦ ❞❡(P Q)
t✱ ♦❜t❡♥❞♦
(P Q)π
t max t+ n X
i=1
bixi
s.a t ≤
n X
i=1 i−1
X
j=1
cπ(j)π(i)xπ(i) ∀π∈Sn,
(x, t)∈Bn×R,
✭✾✮
❡ ❝♦♥❝❧✉✐♥❞♦ ❛ss✐♠ ❛ s❡❣✉♥❞❛ ❡t❛♣❛✳ ◆♦t❡ q✉❡ (P Q)π
t é ✉♠ ♠♦❞❡❧♦ ❧✐♥❡❛r ✐♥t❡✐r♦ ❝♦♠ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❛s ♠❡s♠❛s
✈❛r✐á✈❡✐s q✉❡ (P Q)✱ ♠❛s ❝♦♠ ✉♠ ♥ú♠❡r♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ r❡str✐çõ❡s✳ ❋❡❧✐③♠❡♥t❡✱ ❛ s❡♣❛✲ r❛çã♦ ❞❛s r❡str✐çõ❡s ❧✐♥❡❛r❡s q✉❡ ❞❡✜♥❡♠ Z′ ♣♦❞❡ s❡r ❢❡✐t❛ ❡♠ t❡♠♣♦ ♣♦❧✐♥♦♠✐❛❧✱ ❝♦♠♦ ♠♦str❛r❡♠♦s ❛ s❡❣✉✐r✳
❚❡♦r❡♠❛ ✷✳✸✳ ❙❡❥❛ xˆ∈[0,1]n✳ ❙❡ c
ij ≥0✱ ♣❛r❛ t♦❞♦ 1≤i, j ≤n✱ ❡♥tã♦ ✉♠❛ s♦❧✉çã♦ ❞♦
♣r♦❜❧❡♠❛
min
π∈Sn
X
i X
j<i
cπ(j)π(i)xˆπ(i) ✭✶✵✮
♦❝♦rr❡ ♥❛ ♣❡r♠✉t❛çã♦ ˆπ ∈Sn q✉❡ ♦r❞❡♥❛ ❛s ❝♦♠♣♦♥❡♥t❡s ❞❡ xˆ ❡♠ ♦r❞❡♠ ♥ã♦ ❝r❡s❝❡♥t❡✱
♦✉ s❡❥❛✱ t❛❧ q✉❡
ˆ
xπˆ(i) ≥xˆπˆ(j) ∀i, j i ≤j ≤n. ✭✶✶✮
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛π ∈Sn ✉♠❛ s♦❧✉çã♦ ót✐♠❛ ♣❛r❛ ✭✶✵✮✳ ❙✉♣♦♥❤❛ q✉❡ π ♥ã♦ s❛t✐s❢❛③ ❛
♣r♦♣r✐❡❞❛❞❡ ✭✶✶✮✳ ❙❡❥❛ ❡♥tã♦ l < n ♦ ♠❛✐♦r ✐♥t❡✐r♦ t❛❧ q✉❡ ❛s ♣r✐♠❡✐r❛s l ♣♦s✐çõ❡s ❞❡ xˆ✱ s❡❣✉♥❞♦π✱ ❡stã♦ ♦r❞❡♥❛❞❛s✱ ♦✉ s❡❥❛✱
ˆ
xπ(1) ≥xˆπ(2)≥. . .≥xˆπ(l)<xˆπ(l+1). ❙❡❥❛k ≤l ♦ ♠❡♥♦r ✐♥t❡✐r♦ s❛t✐s❢❛③❡♥❞♦ xˆπ(l+1) >xˆπ(k)✱ ❞❡ ♠♦❞♦ q✉❡
ˆ
