❆▲●❖❘■❚▼❖❙ ❊❳❆❚❖❙ ❊ ❍❊❯❘❮❙❚■❈❖❙ P❆❘❆
P❘❖❇▲❊▼❆❙ ❙❊▲❊❚■❱❖❙ ❉❊ ❘❖❚❊❆▼❊◆❚❖ ❉❊
❘❆▼❖◆ P❊❘❊■❘❆ ▲❖P❊❙
❆▲●❖❘■❚▼❖❙ ❊❳❆❚❖❙ ❊ ❍❊❯❘❮❙❚■❈❖❙ P❆❘❆
P❘❖❇▲❊▼❆❙ ❙❊▲❊❚■❱❖❙ ❉❊ ❘❖❚❊❆▼❊◆❚❖ ❉❊
❱❊❮❈❯▲❖❙ ❈❖▼ ❘❊❙❚❘■➬Õ❊❙ ❉❊ ❈❖❇❊❘❚❯❘❆
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦ ❞♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡r✲ s✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ✲ ❉❡♣❛rt❛✲ ♠❡♥t♦ ❞❡ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦✳ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ❈✐ê♥❝✐❛ ❞❛ ❈♦♠♣✉t❛çã♦✳
❖r✐❡♥t❛❞♦r✿ ❆❧❡①❛♥❞r❡ ❙❛❧❧❡s ❞❛ ❈✉♥❤❛
❝
✷✵✶✷✱ ❘❛♠♦♥ P❡r❡✐r❛ ▲♦♣❡s✳ ❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳
▲♦♣❡s✱ ❘❛♠♦♥ P❡r❡✐r❛
▲✽✻✹❛ ❆❧❣♦r✐t♠♦s ❡①❛t♦s ❡ ❤❡✉ríst✐❝♦s ♣❛r❛ ♣r♦❜❧❡♠❛s s❡❧❡t✐✈♦s ❞❡ r♦t❡❛♠❡♥t♦ ❞❡ ✈❡í❝✉❧♦s ❝♦♠ r❡str✐çõ❡s ❞❡ ❝♦❜❡rt✉r❛ ✴ ❘❛♠♦♥ P❡r❡✐r❛ ▲♦♣❡s✳ ✖ ❇❡❧♦ ❍♦r✐③♦♥t❡✱ ✷✵✶✷
①①✱ ✻✷ ❢✳ ✿ ✐❧✳ ❀ ✷✾❝♠
❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✖ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ✲ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈✐ê♥❝✐❛ ❞❛
❈♦♠♣✉t❛çã♦✳
❖r✐❡♥t❛❞♦r✿ ❆❧❡①❛♥❞r❡ ❙❛❧❧❡s ❞❛ ❈✉♥❤❛
✶✳ ❈♦♠♣✉t❛çã♦ ✕ ❚❡s❡s✳ ✷✳ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛ ✕ ❚❡s❡s✳ ■✳ ❖r✐❡♥t❛❞♦r✳ ■■✳ ❚ít✉❧♦✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢✳ ❆❧❡①❛♥❞r❡ ❙❛❧❧❡s ❞❛ ❈✉♥❤❛✱ ❛❣r❛❞❡ç♦ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s✱ ❞❡❞✐❝❛çã♦✱ ✐♥❝❡♥t✐✈♦ ❡ ❜♦❛s r✐s❛❞❛s ❞✉r❛♥t❡ ❛s r❡✉♥✐õ❡s✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ♣♦r t❡r ❛❝r❡❞✐t❛❞♦ ♥♦ ♠❡✉ ♣♦t❡♥❝✐❛❧✱ t❡r ♠❡ ❣✉✐❛❞♦ ♥❡st❛ ❥♦r♥❛❞❛ ❡ t❡r ❛❥✉❞❛❞♦ ❛ ♠❡ t♦r♥❛r t❛♥t♦ ✉♠ ♣r♦✜ss✐♦♥❛❧ q✉❛♥t♦ ✉♠❛ ♣❡ss♦❛ ♠❡❧❤♦r✳ ❙✐♠♣❧❡s♠❡♥t❡✱ ♠✉✐t♦ ♦❜r✐❣❛❞♦✳
❆ ❈❛r♦❧✱ ♠✐♥❤❛ ♥♦✐✈❛✱ ❛❣r❛❞❡ç♦ ♣♦r s❡r ♠✐♥❤❛ ❝ú♠♣❧✐❝❡ ❛♦ ❧♦♥❣♦ ❞❡ss❡s ❛♥♦s ❡ ♣♦r t❡r s✉♣♦rt❛❞♦ ❡ss❡s ❛♥♦s ❞❡ ❞✐stâ♥❝✐❛✳ ❊st❡ é ♠❛✐s ✉♠ ♣❛ss♦ q✉❡ ❞❛♠♦s ❥✉♥t♦s ❞❡s❞❡ ❛ é♣♦❝❛ ❞♦ ❝♦❧❡❣✐❛❧✱ ❡ ♣❛rt❡ ❞❡st❛ ✈✐tór✐❛ é s✉❛✳
❆♦s ♠❡✉s ♣❛✐s✱ ❘❛✐♠✉♥❞♦ ❡ ❊❞♥❛✱ ✐r♠ã✱ ▲❛r✐ss❛✱ ❡ ❚♦✣ ❛❣r❛❞❡ç♦ ♣♦r s❡r❡♠ r❡s♣♦♥sá✈❡✐s ♣❡❧♦ q✉❡ s♦✉ ❤♦❥❡✳ ❱♦❝ês sã♦ ♦ ♠❡✉ ❛❧✐❝❡r❝❡✳
❆♦s Pr♦❢s✳ ◆✐♥❢❛ ❡ ❊✉❝❧✐♠❛r✱ ❛❣r❛❞❡ç♦ ♣❡❧♦ ❛t♦ ❞❡ ❣❡♥❡r♦s✐❞❛❞❡ q✉❡ ❢♦✐ ❢✉♥❞❛✲ ♠❡♥t❛❧ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆ ❲❡❧✐♥❣t♦♥ ❡ ❚❤✐❛❣♦✱ ❛❣r❛❞❡ç♦ ♣♦r s❡r❡♠ ❛ ♠✐♥❤❛ ❢❛♠í❧✐❛ ❡♠ ❇❡❧♦ ❍♦r✐③♦♥t❡✳ ❆ P❤✐❧❧✐♣♣❡ ❙❛♠❡r✱ ❛❣r❛❞❡ç♦ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ♣♦r t❡r ♠❡ ❛❥✉❞❛❞♦ ❡♠ ✉♠ ❞♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s q✉❡ ❡♥❢r❡♥t❡✐ ❞✉r❛♥t❡ ❡st❛ ❥♦r♥❛❞❛✳
❆♦ Pr♦❢✳ ❚❤✐❛❣♦ ◆♦r♦♥❤❛✱ ❱✐t♦r ❡ ❱✐♥í❝✐✉s✱ ❛❣r❛❞❡ç♦ ♣❡❧❛ ❡①♣❡r✐ê♥❝✐❛ ❞❡ t❡r tr❛❜❛❧❤❛❞♦ ❝♦♠ ✈♦❝ês ♥♦ ●♦♦❣❧❡ ❘❖❆❉❊❋✴❊❯❘❖ ❈❤❛❧❧❡♥❣❡ ✷✵✶✷✱ ❝✉❥♦ ❛♣r❡♥❞✐③❛❞♦ ❢♦✐ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆♦s ❛♠✐❣♦s ❞♦ ▲❛P❖✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❱✐t♦r✱ ❏❛♥✉❛r✐♦ ❡ ❆♠❛❞❡✉✱ ❛❣r❛❞❡ç♦ ♣❡❧❛s ❜♦❛s r✐s❛❞❛s ❡ ❢♦rç❛ ♥❛s ❤♦r❛s ❞❡ ❞✐✜❝✉❧❞❛❞❡✳
P♦r ✜♠✱ ❛❣r❛❞❡ç♦ à ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡✲ r✐♦r ✭❈❆P❊❙✮ q✉❡ ✜♥❛♥❝✐♦✉ ♠❡✉ ♠❡str❛❞♦ ❡ t♦❞♦s ❛q✉❡❧❡s q✉❡ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❘❡s✉♠♦
❊st❛ ❞✐ss❡rt❛çã♦ ❞❡❞✐❝❛✲s❡ ❛♦ ❡st✉❞♦ ❞❡ ❞♦✐s ♣r♦❜❧❡♠❛s ❞❡ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛✿ ♦ Pr♦❜❧❡♠❛ ❞❡ ❘♦t❡❛♠❡♥t♦ ❞❡ ❱❡í❝✉❧♦s ❙❡❧❡t✐✈♦ ▼✐♥✲▼❛① ❡ ♦ Pr♦❜❧❡♠❛ ❞❡ ❈♦❜❡rt✉r❛ ▼✉❧t✐✲❱❡í❝✉❧♦✳ ❖ ♣r✐♠❡✐r♦ s✉r❣❡ ❝♦♠♦ ♠♦❞❡❧♦ ♣❛r❛ ♣r♦❥❡t❛r r❡❞❡s ❞❡ s❡♥s♦r❡s ❡✜❝✐❡♥t❡s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❡♥❡r❣ét✐❝♦❀ ♦ s❡❣✉♥❞♦ s✉r❣❡ ❝♦♠♦ ♠♦❞❡❧♦ ♣❛r❛ ❞✐str✐❜✉✐r ✉♥✐❞❛❞❡s ♠ó✈❡✐s ❞❡ s❛ú❞❡✳ ❊st❡ tr❛❜❛❧❤♦ ♣r♦♣õ❡ ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s ❡ ❞✉❛s ❤❡✉ríst✐❝❛s ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞❡ ❘♦t❡❛♠❡♥t♦ ❞❡ ❱❡í❝✉❧♦s ❙❡❧❡t✐✈♦ ▼✐♥✲▼❛①✳ ❈♦♠ ❜❛s❡ ♥♦s ❡①♣❡r✐♠❡♥t♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s r❡❛❧✐③❛❞♦s✱ ♦s ♠ét♦❞♦s ♣r♦♣♦st♦s ♥ã♦ s❡ ♠♦str❛r❛♠ ❝♦♠♣❡t✐t✐✈♦s ❡♠ r❡❧❛çã♦ àq✉❡❧❡s ❡①✐st❡♥t❡s ♥❛ ❧✐t❡r❛t✉r❛✳ ◆♦ ❝♦♥t❡①t♦ ❞♦ Pr♦❜❧❡♠❛ ❞❡ ❈♦❜❡rt✉r❛ ▼✉❧t✐✲❱❡í❝✉❧♦✱ ❡st❡ tr❛❜❛❧❤♦ ♣r♦♣õ❡ ✉♠ ❛❧❣♦r✐t♠♦ ❇r❛♥❝❤✲❛♥❞✲Pr✐❝❡ ❡ ✉♠❛ ❤❡✉ríst✐❝❛✳ ◆♦s ❡①♣❡r✐♠❡♥t♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s r❡❛❧✐③❛❞♦s ♣❛r❛ ❛✈❛❧✐❛r ♦s ❛❧❣♦r✐t♠♦s ♣r♦✲ ♣♦st♦s✱ ♥♦✈❡ ❞❡♥tr❡ ❛s tr✐♥t❛ ✐♥stâ♥❝✐❛s ❝♦♥s✐❞❡r❛❞❛s ❢♦r❛♠ r❡s♦❧✈✐❞❛s ♥❛ ♦t✐♠❛❧✐❞❛❞❡✳ ❆♣❡s❛r ❞❡ ❛s té❝♥✐❝❛s ❡♠♣r❡❣❛❞❛s ♣❛r❛ ♦s ❞♦✐s ♣r♦❜❧❡♠❛s t❡r❡♠ ♦ ♠❡s♠♦ ❢✉♥❞❛♠❡♥t♦✱ ♥♦t❛✲s❡ q✉❡ ♦ ♠ét♦❞♦ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s q✉❛♥❞♦ ❛♣❧✐❝❛❞♦ ❛♦ s❡❣✉♥❞♦ ♣r♦❜❧❡♠❛ ♦❜t❡✈❡ ✉♠ ❞❡s❡♠♣❡♥❤♦ ♠❡❧❤♦r ❡♠ ❝♦♠♣❛r❛çã♦ ❛♦ ♣r✐♠❡✐r♦✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛✱ ❘♦t❡❛♠❡♥t♦ ❞❡ ❱❡í❝✉❧♦s✱ ❇r❛♥❝❤✲❛♥❞✲ Pr✐❝❡✱ ❍❡✉ríst✐❝❛s✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦✱ ✇❡ ✐♥✈❡st✐❣❛t❡ t✇♦ ❈♦♠❜✐♥❛t♦r✐❛❧ ❖♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✿ t❤❡ ▼✐♥✲▼❛① ❙❡❧❡❝t✐✈❡ ❱❡❤✐❝❧❡ ❘♦✉t✐♥❣ Pr♦❜❧❡♠ ❛♥❞ t❤❡ ▼✉❧t✐✲✈❡❤✐❝❧❡ ❈♦✈❡r✐♥❣ ❚♦✉r Pr♦❜❧❡♠✳ ❚❤❡ ❢♦r♠❡r ♠♦❞❡❧s t❤❡ ❞❡s✐❣♥ ♦❢ ❡♥❡r❣②✲❡✣❝✐❡♥t s❡♥s♦r ♥❡t✇♦r❦s❀ t❤❡ ❧❛tt❡r ♠♦❞❡❧s t❤❡ ❞❡❧✐✈❡r② ♦❢ ♠♦❜✐❧❡ ❤❡❛❧t❤✲❝❛r❡ ❢❛❝✐❧✐t✐❡s✳ ❲❡ ♣r♦♣♦s❡ ❛ ❈♦❧✉♠♥ ●❡♥❡r❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❛♥❞ t✇♦ ❤❡✉r✐st✐❝s ❢♦r t❤❡ ▼✐♥✲▼❛① ❙❡❧❡❝t✐✈❡ ❱❡❤✐❝❧❡ ❘♦✉t✐♥❣ Pr♦❜❧❡♠✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts ✇❡r❡ ♣❡r❢♦r♠❡❞ t♦ ❛ss❡ss t❤❡ q✉❛❧✐t② ♦❢ t❤❡ s♦❧✉t✐♦♥s ♦❜t❛✐♥❡❞ ❜② t❤❡s❡ ❤❡✉r✐st✐❝s ✐♥ ❝♦♠♣❛r✐s♦♥ t♦ t❤♦s❡ ❣✐✈❡♥ ❜② t❤❡ ♠❡t❤♦❞s ❢♦✉♥❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✳ ❚❤❡ ❡①♣❡r✐♠❡♥ts ✐♥❞✐❝❛t❡ t❤❛t t❤❡ ♠❡t❤♦❞s ♣r♦♣♦s❡❞ ❤❡r❡ ❢♦r t❤❡ ▼✐♥✲▼❛① ❙❡❧❡❝t✐✈❡ ❱❡❤✐❝❧❡ ❘♦✉t✐♥❣ Pr♦❜❧❡♠ ❛r❡ ♥♦t ❛s ❡✣❝✐❡♥t ❛♥❞ ❡✛❡❝t✐✈❡ ❛s t❤♦s❡ ❢♦✉♥❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❢♦r t❤❡ ♠❛❥♦r✐t② ♦❢ t❤❡ ❝♦♥s✐❞❡r❡❞ ✐♥st❛♥❝❡s✳ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ t❤❡ ▼✉❧t✐✲✈❡❤✐❝❧❡ ❈♦✈❡r✐♥❣ ❚♦✉r Pr♦❜❧❡♠✱ ✇❡ ♣r♦♣♦s❡ ❛ ❇r❛♥❝❤✲❛♥❞✲Pr✐❝❡ ❛❧❣♦r✐t❤♠ ❛♥❞ ❛ ❤❡✉r✐st✐❝✳ ❚❤❡ ❇r❛♥❝❤✲ ❛♥❞✲Pr✐❝❡ ❛❧❣♦r✐t❤♠ s♦❧✈❡❞ t♦ ♣r♦✈❡♥ ♦♣t✐♠❛❧✐t② ♥✐♥❡ ♦✉t ♦❢ t❤✐rt② ✐♥st❛♥❝❡s ❝♦♥s✐❞❡r❡❞ ✐♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡①♣❡r✐♠❡♥ts✳ ■♥ ❝♦♥❝❧✉s✐♦♥✱ ♦♥❡ ❝❛♥ ♦❜s❡r✈❡ t❤❛t t❤❡ ❛❧❣♦r✐t❤♠ ♣r♦♣♦s❡❞ ❢♦r t❤❡ s❡❝♦♥❞ ♣r♦❜❧❡♠ ❤❛❞ s✉♣❡r✐♦r ♣❡r❢♦r♠❛♥❝❡ ✇❤❡♥ ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ ❛❧❣♦r✐t❤♠ ♣r♦♣♦s❡❞ ❢♦r t❤❡ ✜rst ♦♥❡✱ ❛❧t❤♦✉❣❤ t❤❡ s❛♠❡ s❡t ♦❢ t❡❝❤♥✐q✉❡s ✇❡r❡ ❛♣♣❧✐❡❞ ❢♦r s♦❧✈✐♥❣ ❜♦t❤ ♣r♦❜❧❡♠s✳
❑❡②✇♦r❞s✿ ❈♦♠❜✐♥❛t♦r✐❛❧ ❖♣t✐♠✐③❛t✐♦♥✱ ❱❡❤✐❝❧❡ ❘♦✉t✐♥❣✱ ❇r❛♥❝❤✲❛♥❞✲Pr✐❝❡✱ ❍❡✉r✐s✲ t✐❝s✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ ■♥stâ♥❝✐❛ ❞♦ P❘❱❙▼▼ ❝♦♠ ✷ ✈❡í❝✉❧♦s ❡ ♣❛râ♠❡tr♦ ❘✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✷ P♦r❝❡♥t❛❣❡♠ ❞❡ ✈❛r✐á✈❡✐s ❜ás✐❝❛s ❞❡❣❡♥❡r❛❞❛s ❛♦ ❧♦♥❣♦ ❞❛s ✐t❡r❛çã♦❡s ♣❛r❛
❛ ✐♥stâ♥❝✐❛ ❡✐❧✺✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✸ P♦r❝❡♥t❛❣❡♠ ❞❡ ✈❛r✐á✈❡✐s ❜ás✐❝❛s ❞❡❣❡♥❡r❛❞❛s ❛♦ ❧♦♥❣♦ ❞❛s ✐t❡r❛çã♦❡s ♣❛r❛
❛ ✐♥stâ♥❝✐❛ st✼✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✹ P♦r❝❡♥t❛❣❡♠ ❞❡ ✈❛r✐á✈❡✐s ❜ás✐❝❛s ❞❡❣❡♥❡r❛❞❛s ❛♦ ❧♦♥❣♦ ❞❛s ✐t❡r❛çã♦❡s ♣❛r❛
❛ ✐♥stâ♥❝✐❛ ❡✐❧✼✻✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✺ P♦r❝❡♥t❛❣❡♠ ❞❡ ✈❛r✐á✈❡✐s ❜ás✐❝❛s ❞❡❣❡♥❡r❛❞❛s ❛♦ ❧♦♥❣♦ ❞❛s ✐t❡r❛çõ❡s ♣❛r❛
❛ ✐♥stâ♥❝✐❛ r❛t✾✾✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✻ P♦r❝❡♥t❛❣❡♠ ❞❡ ✈❛r✐á✈❡✐s ❜ás✐❝❛s ❞❡❣❡♥❡r❛❞❛s ❛♦ ❧♦♥❣♦ ❞❛s ✐t❡r❛çõ❡s ♣❛r❛
❛ ✐♥stâ♥❝✐❛ ❦r♦❆✶✵✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✼ P♦r❝❡♥t❛❣❡♠ ❞❡ ✈❛r✐á✈❡✐s ❜ás✐❝❛s ❞❡❣❡♥❡r❛❞❛s ❛♦ ❧♦♥❣♦ ❞❛s ✐t❡r❛çõ❡s ♣❛r❛
❛ ✐♥stâ♥❝✐❛ ❡✐❧✶✵✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸✳✶ P♦r❝❡♥t❛❣❡♠ ❞❡ ✈❛r✐á✈❡✐s ❜ás✐❝❛s ❞❡❣❡♥❡r❛❞❛s ❛♦ ❧♦♥❣♦ ❞❛s ✐t❡r❛çõ❡s q✉❛♥❞♦ |V|= 50 ❡ |W|= 50✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✸✳✷ P♦r❝❡♥t❛❣❡♠ ❞❡ ✈❛r✐á✈❡✐s ❜ás✐❝❛s ❞❡❣❡♥❡r❛❞❛s ❛♦ ❧♦♥❣♦ ❞❛s ✐t❡r❛çõ❡s q✉❛♥❞♦ |V|= 50 ❡ |W|= 100✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✸✳✸ P♦r❝❡♥t❛❣❡♠ ❞❡ ✈❛r✐á✈❡✐s ❜ás✐❝❛s ❞❡❣❡♥❡r❛❞❛s ❛♦ ❧♦♥❣♦ ❞❛s ✐t❡r❛çõ❡s q✉❛♥❞♦ |V|= 100 ❡|W|= 100✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
✸✳✹ P♦r❝❡♥t❛❣❡♠ ❞❡ ✈❛r✐á✈❡✐s ❜ás✐❝❛s ❞❡❣❡♥❡r❛❞❛s ❛♦ ❧♦♥❣♦ ❞❛s ✐t❡r❛çõ❡s q✉❛♥❞♦ |V|= 100 ❡|W|= 200✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
✸✳✺ P♦r❝❡♥t❛❣❡♠ ❞❡ ✈❛r✐á✈❡✐s ❜ás✐❝❛s ❞❡❣❡♥❡r❛❞❛s ❛♦ ❧♦♥❣♦ ❞❛s ✐t❡r❛çõ❡s q✉❛♥❞♦ |V|= 200 ❡|W|= 200✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✷✳✶ ❘❡s✉❧t❛❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❝♦♠ ❞❡♥s✐❞❛❞❡ ✐❣✉❛❧ ❛ ✵✳✼✺✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✷ ❘❡s✉❧t❛❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❝♦♠ ❞❡♥s✐❞❛❞❡ ✐❣✉❛❧ ❛ ✶✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✸ ❘❡s✉❧t❛❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛ ❛s ✐♥stâ♥❝✐❛s P❘❱❙▼▼✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✸✳✶ ❘❡s✉❧t❛❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❝♦♠ p= 4✱ q= 200✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✸✳✷ ❘❡s✉❧t❛❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❝♦♠ p=∞ ❡ q=∞✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✸ ❘❡s✉❧t❛❞♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛ ❛s ✐♥stâ♥❝✐❛s P❈▼❱✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
▲✐st❛ ❞❡ ❙✐❣❧❛s
❇❈ ❇r❛♥❝❤✲❛♥❞✲❈✉t
❇P ❇r❛♥❝❤✲❛♥❞✲Pr✐❝❡
❈●❍ ❈♦❧✉♠♥ ●❡♥❡r❛t✐♦♥ ❍❡✉r✐st✐❝
❈■❆ ❈❤❡❛♣❡st ■♥s❡rt✐♦♥ ❆❧❣♦r✐t❤♠
●❘❆❙P ●r❡❡❞② ❘❛♥❞♦♠✐③❡❞ ❆❞❛♣t✐✈❡ ❙❡❛r❝❤ Pr♦❝❡❞✉r❡
●❙❊❈ ●❡♥❡r❛❧✐③❡❞ ❙✉❜t♦✉r ❊❧✐♠✐♥❛t✐♦♥ ❈♦♥str❛✐♥ts
❍●❈ ❍❡✉ríst✐❝❛ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s
❍▲ ❍❡✉ríst✐❝❛ ▲❛❣r❛❣❡❛♥❛
■▲❙ ■t❡r❛t❡❞ ▲♦❝❛❧ ❙❡❛r❝❤
▲❘❈ ▲✐st❛ ❘❡str✐t❛ ❞❡ ❈❛♥❞✐❞❛t♦s
▼❙ ▼ét♦❞♦ ❞❡ ❙✉❜❣r❛❞✐❡♥t❡
▼❙❉ ▼ét♦❞♦ ❞❡ ❙✉❜❣r❛❞✐❡♥t❡ ❉❡✢❡t✐❞♦
P❈▼❊❘❘ Pr♦❜❧❡♠❛ ❞❡ ❈❛♠✐♥❤♦ ▼í♥✐♠♦ ❊❧❡♠❡♥t❛r ❝♦♠ ❘❡str✐çõ❡s ❞❡ ❘❡❝✉rs♦
P❈▼❱ Pr♦❜❧❡♠❛ ❞❡ ❈♦❜❡rt✉r❛ ▼✉❧t✐✲❱❡í❝✉❧♦
P▼ Pr♦❜❧❡♠❛ ▼❡str❡
P▼▲ Pr♦❜❧❡♠❛ ▼❡str❡ ▲✐♥❡❛r
P▼▲❘ Pr♦❜❧❡♠❛ ▼❡str❡ ▲✐♥❡❛r ❘❡str✐t♦
PP▲ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r
P❘▲ Pr♦❜❧❡♠❛ ❞❡ ❘❡❧❛①❛çã♦ ▲❛❣r❛♥❣❡❛♥❛
P❘❱ Pr♦❜❧❡♠❛ ❞❡ ❘♦t❡❛♠❡♥t♦ ❞❡ ❱❡í❝✉❧♦s
P❘❱❙▼▼ Pr♦❜❧❡♠❛ ❞❡ ❘♦t❡❛♠❡♥t♦ ❞❡ ❱❡í❝✉❧♦s ❙❡❧❡t✐✈♦ ▼✐♥✲▼❛①
❘❙❙❋ ❘❡❞❡s ❞❡ ❙❡♥s♦r❡s s❡♠ ❋✐♦
❱◆❙ ❱❛r✐❛❜❧❡ ◆❡✐❣❤❜♦r❤♦♦❞ ❙❡❛r❝❤
❙✉♠ár✐♦
❆❣r❛❞❡❝✐♠❡♥t♦s ✈✐✐
❘❡s✉♠♦ ✐①
❆❜str❛❝t ①✐
▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✐✐✐
▲✐st❛ ❞❡ ❚❛❜❡❧❛s ①✈
▲✐st❛ ❞❡ ❙✐❣❧❛s ①✈✐✐
✶ ■♥tr♦❞✉çã♦ ✶
✷ ❆❧❣♦r✐t♠♦ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞❡ ❘♦t❡❛♠❡♥t♦
❞❡ ❱❡í❝✉❧♦s ❙❡❧❡t✐✈♦ ▼✐♥✲♠❛① ✺
✷✳✶ ❉❡✜♥✐çã♦ ❞♦ Pr♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷✳✷ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá✜❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✸ ❋♦r♠✉❧❛çõ❡s ▼❛t❡♠át✐❝❛s ♣❛r❛ ♦ P❘❱❙▼▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✸✳✶ ❋♦r♠✉❧❛çã♦ ●❙❊❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✸✳✷ ❘❡❢♦r♠✉❧❛çã♦ ♣♦r ❘❡❝♦❜r✐♠❡♥t♦ ❞❡ ❈♦♥❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✹ ❘❡s♦❧✈❡♥❞♦ ♦ Pr♦❜❧❡♠❛ ❞❡ Pr❡❝✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✹✳✶ ❆❧❣♦r✐t♠♦ ❊①❛t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✹✳✷ ❆❧❣♦r✐t♠♦ ❍❡✉ríst✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✹✳✸ P❛r❛❧❡❧✐③❛çã♦ ❞♦s ❆❧❣♦r✐t♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✺ ▼ét♦❞♦s ❞❡ ❊st❛❜✐❧✐③❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✺✳✶ ❊st❛❜✐❧✐③❛çã♦ ♣♦r P♦♥t♦s ■♥t❡r✐♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✺✳✷ ❊st❛❜✐❧✐③❛çã♦ ♣♦r ❘❡❧❛①❛çã♦ ▲❛❣r❛♥❣❡❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✻ ❍❡✉ríst✐❝❛s ♣❛r❛ ♦ P❘❱❙▼▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✻✳✶ ❍❡✉ríst✐❝❛ ▲❛❣r❛♥❣❡❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✷✳✻✳✷ ❍❡✉ríst✐❝❛ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✼ ❊①♣❡r✐♠❡♥t♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✼✳✶ ❆♠❜✐❡♥t❡ ❈♦♠♣✉t❛❝✐♦♥❛❧ ❡ ▼❡t♦❞♦❧♦❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✼✳✷ ❆✈❛❧✐❛çã♦ ❞❡ ▲✐♠✐t❡s Pr✐♠❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✼✳✸ ❉❡s❡♠♣❡♥❤♦ ❞♦ ❆❧❣♦r✐t♠♦ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✽ ❈♦♥❝❧✉sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸ ❆❧❣♦r✐t♠♦ ❇r❛♥❝❤✲❛♥❞✲Pr✐❝❡ ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞❡ ❈♦❜❡rt✉r❛ ▼✉❧t✐✲
❱❡í❝✉❧♦ ✹✶
✸✳✶ ❉❡✜♥✐çã♦ ❞♦ Pr♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✷ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá✜❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✸✳✸ ❘❡❢♦r♠✉❧❛çã♦ ♣♦r ❘❡❝♦❜r✐♠❡♥t♦ ❞❡ ❈♦♥❥✉♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✹ ❘❡s♦❧✈❡♥❞♦ ♦ Pr♦❜❧❡♠❛ ❞❡ Pr❡❝✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✹✳✶ ❆❧❣♦r✐t♠♦ ❊①❛t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✹✳✷ ❆❧❣♦r✐t♠♦ ❍❡✉ríst✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✺ ❍❡✉ríst✐❝❛ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✻ ❆❧❣♦r✐t♠♦ ❇r❛♥❝❤✲❛♥❞✲Pr✐❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✸✳✼ ❊①♣❡r✐♠❡♥t♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✼✳✶ ❆♠❜✐❡♥t❡ ❈♦♠♣✉t❛❝✐♦♥❛❧ ❡ ▼❡t♦❞♦❧♦❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✼✳✷ ❘❡s✉❧t❛❞♦s ❈♦♠♣✉t❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✸✳✽ ❈♦♥❝❧✉sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
✹ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✺✼
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✾
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❖ Pr♦❜❧❡♠❛ ❞❡ ❘♦t❡❛♠❡♥t♦ ❞❡ ❱❡í❝✉❧♦s ✭P❘❱✮ ❢♦✐ ✐♥tr♦❞✉③✐❞♦ ❡♠ ❉❛♥t③✐❣ ✫ ❘❛♠s❡r ❬✶✾✺✾❪ ❡ é ✉♠ ♣r♦❜❧❡♠❛ ❝❧áss✐❝♦ ❞❡ ❖t✐♠✐③❛çã♦ ❈♦♠❜✐♥❛tór✐❛✳ ◆❛q✉❡❧❡ ♣r♦❜❧❡♠❛✱ ❞❡✈❡✲ s❡ ❞❡t❡r♠✐♥❛r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ r♦t❛s✱ ❣❡r❛❧♠❡♥t❡ ❝♦♠♣❛rt✐❧❤❛♥❞♦ ❛ ♠❡s♠❛ ♦r✐❣❡♠ ♦✉ ❞❡♣ós✐t♦✱ q✉❡ ♠✐♥✐♠✐③❡ ♦ ❝✉st♦ ❞❡ ❞❡s❧♦❝❛♠❡♥t♦ ❞❛ ❢r♦t❛ ❞❡ ✈❡í❝✉❧♦s✳
❉❡✈✐❞♦ à s✉❛ r❡❧❡✈â♥❝✐❛ t❛♥t♦ ♣rát✐❝❛ q✉❛♥t♦ t❡ór✐❝❛✱ sã♦ ❡♥❝♦♥tr❛❞❛s ❞✐✈❡rs❛s ✈❛r✐❛çõ❡s ❞♦ P❘❱ ♥❛ ❧✐t❡r❛t✉r❛✳ ❉❡♥tr❡ ❛s ♣r✐♥❝✐♣❛✐s✱ ❞❡st❛❝❛♠✲s❡✿ ✭✐✮ P❘❱ ❝♦♠ r❡✲ str✐çã♦ ❞❡ ❞✐stâ♥❝✐❛ ❬▲❛♣♦rt❡ ❡t ❛❧✳✱ ✶✾✽✹❪✱ ❡♠ q✉❡ ❝❛❞❛ r♦t❛ ♥ã♦ ❞❡✈❡ ❡①❝❡❞❡r ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❧✐♠✐t❡ ❞❡ ❞✐stâ♥❝✐❛❀ ✭✐✐✮ P❘❱ ❝♦♠ ❥❛♥❡❧❛ ❞❡ t❡♠♣♦ ❬❇rä②s② ✫ ●❡♥❞r❡❛✉✱ ✷✵✵✺❪✱ ❡♠ q✉❡ ❝❛❞❛ ❝❧✐❡♥t❡ ❞❡✈❡ s❡r ✈✐s✐t❛❞♦ ❞❡♥tr♦ ❞❡ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ❡s♣❡❝í✜❝♦✱ ✭✐✐✐✮ P❘❱ ❝♦♠ ❝♦❧❡t❛ ❡ ❡♥tr❡❣❛ ❬❙❛✈❡❧s❜❡r❣❤ ✫ ❙♦❧✱ ✶✾✾✺❪✱ ❡♠ q✉❡ ❝❛❞❛ ✈❡í❝✉❧♦ ❞❡✈❡ r❡❛❧✐③❛r ❝♦❧❡t❛s ❡ ❡♥tr❡❣❛s ❞❡ ♠❡r❝❛❞♦r✐❛s ❡♠ ♣♦♥t♦s ❡s♣❡❝í✜❝♦s✱ ✭✐✈✮ P❘❱ ❝♦♠ ❇❛❝❦✲ ❤❛✉❧s ❬●♦❡ts❝❤❛❧❝❦① ✫ ❏❛❝♦❜s✲❇❧❡❝❤❛✱ ✶✾✽✾❪✱ q✉❡ ♣♦❞❡ s❡r ✈✐st♦ ❝♦♠♦ ✉♠ P❘❱ ❝♦♠ ❝♦❧❡t❛ ❡ ❡♥tr❡❣❛ ❡♠ q✉❡ t♦❞❛s ❛s ❡♥tr❡❣❛s ❞❡✈❡♠ s❡r r❡❛❧✐③❛❞❛s ❛♥t❡s ❞❡ q✉❛❧q✉❡r ❝♦❧❡t❛✱ ✭✈✮ P❘❱ ❝♦♠ s❛t❡❧❧✐t❡s ❬❇❛r❞ ❡t ❛❧✳✱ ✶✾✾✽❪✱ ❡♠ q✉❡ ❝❛❞❛ ✈❡í❝✉❧♦ ♣♦❞❡ ✈✐s✐t❛r ♣♦♥t♦s ❡s♣❡❝í✜❝♦s✱ ❞❡♥♦♠✐♥❛❞♦s s❛t❡❧❧✐t❡s✱ ♣❛r❛ r❡❛❜❛st❡❝❡r s✉❛ ❝❛r❣❛ ❡♠ ✈❡③ ❞❡ r❡t♦r♥❛r ❛♦ ❞❡♣ós✐t♦✱ ❡ ✭✈✐✮ P❘❱ ❡st♦❝ást✐❝♦ ❬▲❛♣♦rt❡ ❡t ❛❧✳✱ ✷✵✵✷❪✱ ❡♠ q✉❡ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛ sã♦ ❡st♦❝ást✐❝♦s ❡ s❡❣✉❡♠ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ❞✐str✐❜✉✐çã♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡✳
◆❡st❡ tr❛❜❛❧❤♦✱ ❞✉❛s ✈❛r✐❛çõ❡s ❞♦ P❘❱ sã♦ ❡st✉❞❛❞❛s✳ ❆s ❞✉❛s ♣♦ss✉❡♠ ♥❛t✉r❡③❛ s❡❧❡t✐✈❛✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ♥❡♠ t♦❞♦s ♦s ✈ért✐❝❡s ♣r❡❝✐s❛♠ s❡r ✈✐s✐t❛❞♦s✱ ❡ r❡str✐çõ❡s ❞❡ ❝♦❜❡rt✉r❛✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ✉♠ ✈ért✐❝❡ ♥ã♦ ✈✐s✐t❛❞♦ ❞❡✈❡ ❡st❛r ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ♣ré✲ ❞❡✜♥✐❞❛ ❞❡ ❛❧❣✉♠ ✈ért✐❝❡ q✉❡ ♦ s❡❥❛✳ ❊ss❛s ❞✉❛s ❝❛r❛❝t❡ríst✐❝❛s ♥ã♦ sã♦ ♠✉✐t♦ ❝♦♠✉♥s ❞❡♥tr❡ ❛s ✈❛r✐❛çõ❡s ❞♦ P❘❱ ❡①✐st❡♥t❡s ♥❛ ❧✐t❡r❛t✉r❛✱ ❡♠❜♦r❛ r❡♣r❡s❡♥t❡♠ s✐t✉❛çõ❡s ♠✉✐t❛s ✈❡③❡s ✈❡r✐✜❝❛❞❛s ❡♠ ❛♣❧✐❝❛çõ❡s✳
❆ ♣r✐♠❡✐r❛ ✈❛r✐❛çã♦✱ ❞❡♥♦♠✐♥❛❞❛ Pr♦❜❧❡♠❛ ❞❡ ❘♦t❡❛♠❡♥t♦ ❞❡ ❱❡í❝✉❧♦s ❙❡❧❡t✐✈♦ ▼✐♥✲▼❛① ✭P❘❱❙▼▼✮✱ ❞✐❢❡r❡ ❞❛s ❞❡♠❛✐s ❞❛ ❧✐t❡r❛t✉r❛ ♣♦r✱ ❛❧é♠ ❞❡ ♣♦ss✉✐r r❡str✐çõ❡s
✷ ❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦
❞❡ ❝♦❜❡rt✉r❛ ❡ ♥❛t✉r❡③❛ s❡❧❡t✐✈❛✱ t❡r ✉♠❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ q✉❡ ❝♦♥s✐st❡ ❡♠ ♠✐♥✐♠✐③❛r ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♠❛✐♦r r♦t❛✳ ❙✉❛ ❛♣❧✐❝❛çã♦ ❡stá r❡❧❛❝✐♦♥❛❞❛ à ❡✜❝✐ê♥❝✐❛ ❡♥❡r❣ét✐❝❛ ♥♦ ♣r♦❥❡t♦ ❞❡ r❡❞❡s ❞❡ s❡♥s♦r❡s✱ ❛s q✉❛✐s tê♠ s✐❞♦ t❡♠❛ ❞❡ ✐♥t❡♥s❛ ♣❡sq✉✐s❛ ♥❛s ú❧t✐♠❛s ❞é❝❛❞❛s ❬❆❦②✐❧❞✐③ ❡t ❛❧✳✱ ✷✵✵✷❪✳
❉✐❢❡r❡♥t❡♠❡♥t❡ ❞❛s ✐♥ú♠❡r❛s ✈❛r✐❛çõ❡s ❞♦ P❘❱ ❡♥❝♦♥tr❛❞❛s ♥❛ ❧✐t❡r❛t✉r❛✱ ♣❛r❛ ❛s q✉❛✐s sã♦ ❡♥❝♦♥tr❛❞♦s tr❛❜❛❧❤♦s q✉❡ ❞❛t❛♠ ❞♦ ✜♥❛❧ ❞❛ ❞é❝❛❞❛ ❞❡ ✺✵ ❬❉❛♥t③✐❣ ✫ ❘❛♠s❡r✱ ✶✾✺✾❪✱ ♦ P❘❱❙▼▼ ❢♦✐ r❡❝❡♥t❡♠❡♥t❡ ♣r♦♣♦st♦ ❡♠ ❱❛❧❧❡ ❡t ❛❧✳ ❬✷✵✵✾❪✳ ❈♦♠ ❜❛s❡ ♥♦s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ❡♠ ❬❱❛❧❧❡ ❡t ❛❧✳✱ ✷✵✵✾✱ ✷✵✶✶❪✱ ❛ ♦t✐♠❛❧✐❞❛❞❡ ❞❡ ✐♥stâ♥❝✐❛s ❞❡ t❛♠❛♥❤♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣❡q✉❡♥♦ ♣ô❞❡ s❡r ❛t❡st❛❞❛❀ ♣❛r❛ ✐♥stâ♥❝✐❛s ♠❛✐♦r❡s✱ ❤á ❣❛♣ ❞❡ ❞✉❛❧✐❞❛❞❡ ❞❡ ♠❛✐s ❞❡ ✹✵✪✱ ♠❡s♠♦ ♦s ❛❧❣♦r✐t♠♦s ♣r♦♣♦st♦s t❡♥❞♦ s✐❞♦ ❡①❡❝✉t❛❞♦s ♣♦r ❛♣r♦①✐♠❛❞❛♠❡♥t❡ q✉❛tr♦ ❤♦r❛s✳ ❊♠ ❱❛❧❧❡ ❡t ❛❧✳ ❬✷✵✶✶❪✱ ❢♦✐ ❛♣♦♥t❛❞❛ ❛ ❞✐✜❝✉❧❞❛❞❡ ❡♠ r❡s♦❧✈❡r ✐♥stâ♥❝✐❛s ♠❛✐♦r❡s ❞♦ P❘❱❙▼▼ ♥❛ ♦t✐♠❛❧✐❞❛❞❡✱ ❝♦♠♦ t❛♠❜é♠ ❢♦✐ s✉❣❡r✐❞♦ q✉❡ ♦✉tr❛s ❛❜♦r❞❛❣❡♥s ❢♦ss❡♠ ✐♥✈❡st✐❣❛❞❛s ❛ ✜♠ ❞❡ t♦r♥❛r ❡st❡ ♣r♦❜❧❡♠❛ ♠❡❧❤♦r r❡s♦❧✈✐❞♦ ♥❛ ♣rát✐❝❛✳ ❈♦♠ ❜❛s❡ ♥✐ss♦✱ ❤á ✉♠❛ ❝❛rê♥❝✐❛ ❞❡ ♠ét♦❞♦s ♠❡♥♦s ❝✉st♦s♦s ♣❛r❛ ♦ P❘❱❙▼▼✳
❆ s❡❣✉♥❞❛ ✈❛r✐❛çã♦✱ ❞❡♥♦♠✐♥❛❞❛ Pr♦❜❧❡♠❛ ❞❡ ❈♦❜❡rt✉r❛ ▼✉❧t✐✲❱❡í❝✉❧♦ ✭P❈▼❱✮✱ ❞✐❢❡r❡ ❞❛s ❞❡♠❛✐s ✈❛r✐❛çõ❡s ❡♥❝♦♥tr❛❞❛s ♥❛ ❧✐t❡r❛t✉r❛ ♣♦rq✉❡ ✭✐✮ ❝❛❞❛ ✈ért✐❝❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❡s♣❡❝í✜❝♦ ♥ã♦ ♣♦❞❡ s❡r ✈✐s✐t❛❞♦✱ ♠❛s ❞❡✈❡ s❡r ❝♦❜❡rt♦ ❡ ✭✐✐✮ ❝❛❞❛ ✈ért✐❝❡ ❞❡ ✉♠ ♦✉tr♦ ❝♦♥❥✉♥t♦ ❡s♣❡❝í✜❝♦ ❞❡✈❡ s❡r ♦❜r✐❣❛t♦r✐❛♠❡♥t❡ ✈✐s✐t❛❞♦ ♣♦r ❛❧❣✉♠❛ r♦t❛✱ ❡♥q✉❛♥t♦ ♦s ❞❡♠❛✐s ✈ért✐❝❡s ♣♦❞❡♠ ♦✉ ♥ã♦ s❡r ✈✐s✐t❛❞♦s✳ ❙✉❛ ❛♣❧✐❝❛çã♦ ❡stá r❡❧❛❝✐♦♥❛❞❛ ❝♦♠ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ ✉♥✐❞❛❞❡s ♠ó✈❡✐s ❞❡ s❛ú❞❡✱ ❞❡ ♠♦❞♦ q✉❡ ✉♠❛ r❡❣✐ã♦ ♥ã♦ ✈✐s✐t❛❞❛ ❞❡✈❡ ❡st❛r ❛ ✉♠❛ ❞✐stâ♥❝✐❛ ♣ré✲❞❡✜♥✐❞❛ ❞❡ ❛❧❣✉♠❛ r❡❣✐ã♦ q✉❡ ❛ s❡❥❛✳ ❖ P❈▼❱ ❢♦✐ ♣r♦♣♦st♦ ❡♠ ❬❍❛❝❤✐❝❤❛ ❡t ❛❧✳✱ ✷✵✵✵❪✱ ♥♦ q✉❛❧ t❛♠❜é♠ ❢♦r❛♠ ❛♣r❡s❡♥t❛❞❛s três ❤❡✉ríst✐❝❛s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✳ ❈♦♠ ❜❛s❡ ♥❛ r❡✈✐sã♦ ❜✐❜❧✐♦❣rá✜❝❛ r❡❛❧✐③❛❞❛✱ é ❞❡s❝♦♥❤❡❝✐❞❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❛❧❣♦r✐t♠♦ ❡①❛t♦ ♣❛r❛ r❡s♦❧✈❡r ✐♥stâ♥❝✐❛s ❞♦ P❈▼❱ ♥❛ ♦t✐♠❛❧✐❞❛❞❡✳
❆❧❣♦r✐t♠♦s ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s ❡ ❇r❛♥❝❤✲❛♥❞✲Pr✐❝❡ ✭❇P✮ tê♠ s✐❞♦ ❛♣❧✐❝❛✲ ❞♦s ❝♦♠ s✉❝❡ss♦ ❛ ❞✐✈❡rs♦s ♣r♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛çã♦✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ♥♦ ❝♦♥t❡①t♦ ❞♦ P❘❱ ❬❇❛r♥❤❛rt ❡t ❛❧✳✱ ✶✾✾✽❪✳ P♦r ❝♦♥t❛ ❞✐ss♦✱ ♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✐♥✈❡st✐❣❛r s❡✱ ❛ss✐♠ ❝♦♠♦ ♣❛r❛ ♦✉tr❛s ✈❛r✐❛çõ❡s ❞♦ P❘❱✱ ❛❧❣♦r✐t♠♦s ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s ❡ ❇P ♣r♦❞✉③❡♠ ❜♦♥s r❡s✉❧t❛❞♦s ♥♦ ❝♦♥t❡①t♦ ❞♦ P❘❱❙▼▼ ❡ ❞♦ P❈▼❱✳
✸
❡①❛t❛ ❡ ✭✐✈✮ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠❛ ❤❡✉ríst✐❝❛ ❜❛s❡❛❞❛ ♥♦ ❛❧❣♦r✐t♠♦ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s✳
❈❛♣ít✉❧♦ ✷
❆❧❣♦r✐t♠♦ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s
♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞❡ ❘♦t❡❛♠❡♥t♦ ❞❡
❱❡í❝✉❧♦s ❙❡❧❡t✐✈♦ ▼✐♥✲♠❛①
◆❡st❡ ❝❛♣ít✉❧♦✱ ♦ Pr♦❜❧❡♠❛ ❞❡ ❘♦t❡❛♠❡♥t♦ ❞❡ ❱❡í❝✉❧♦s ❙❡❧❡t✐✈♦ ▼✐♥✲▼❛① é ❡st✉❞❛❞♦✳ ❯♠❛ r❡❢♦r♠✉❧❛çã♦ ♣♦r ❘❡❝♦❜r✐♠❡♥t♦ ❞❡ ❈♦♥❥✉♥t♦s✱ ✉♠ ❛❧❣♦r✐t♠♦ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✲ ✉♥❛s ❡ ❞✉❛s ❤❡✉ríst✐❝❛s sã♦ ❛♣r❡s❡♥t❛❞❛s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✳ P♦r ✜♠✱ r❡s✉❧t❛❞♦s ❞♦s ❡①♣❡r✐♠❡♥t♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❝♦♥❞✉③✐❞♦s ♣❛r❛ ❛✈❛❧✐❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❡ ❛ q✉❛❧✐❞❛❞❡ ❞❛ s♦❧✉çõ❡s ❞♦s ❛❧❣♦r✐t♠♦s ♣r♦♣♦st♦s sã♦ ❛♣r❡s❡♥t❛❞♦s✳
✷✳✶ ❉❡✜♥✐çã♦ ❞♦ Pr♦❜❧❡♠❛
❙❡❥❛K={1, ..., K}✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❡í❝✉❧♦s ✐❞ê♥t✐❝♦s ❝♦♠ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❝❛r❣❛ ❝♦♥s✐❞✲ ❡r❛❞❛ ✐❧✐♠✐t❛❞❛✳ ❙❡❥❛ G= (V, E)✉♠ ❣r❛❢♦ s✐♠♣❧❡s✱ ❝♦♠♣❧❡t♦✱ ✜♥✐t♦ ❡ ♥ã♦✲❞✐r❡❝✐♦♥❛❞♦
❝♦♠ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s V = {1, ..., n} ❡ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s E ={{i, j}: i, j ∈V}✱ s❡♥❞♦ m = |E|✳ ❖ ✈ért✐❝❡ 1 ∈ V ❞❡♥♦t❛ ♦ ❞❡♣ós✐t♦✱ ♦ q✉❛❧ é ♦ ♣♦♥t♦ ❞❡ ♦r✐❣❡♠ ❞❡ t♦❞♦s ♦s ✈❡í❝✉❧♦s✱ ❡♥q✉❛♥t♦E ❞❡♥♦t❛ t♦❞♦s ♦s ♣♦ssí✈❡✐s ♠♦✈✐♠❡♥t♦s ❞♦s ✈❡í❝✉❧♦s ❡♥tr❡ ♣❛r❡s ❞❡ ✈ért✐❝❡s✳ ❯♠❛ ♠❛tr✐③ ❞❡ ❞✐stâ♥❝✐❛ D= (dij) ❡♥tr❡ t♦❞♦ ♣❛r ❞❡ ✈ért✐❝❡s ❞❡V✱
♦❜❡❞❡❝❡♥❞♦ à ♣r♦♣r✐❡❞❛❞❡ ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✱ é ❞❡✜♥✐❞❛ s♦❜r❡ E❀ ❛ss✉♠❡✲s❡ q✉❡ dij = dji ❡ dii = 0✱ ∀i, j ∈ V✳ P♦r ✜♠✱ s❡❥❛♠ R ≥0 ✉♠ ✈❛❧♦r r❡❛❧ ♥ã♦ ♥❡❣❛t✐✈♦ ❡
ω(i) ={j ∈ V : dij ≤ R},∀i ∈V✱ ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s ✈ért✐❝❡s q✉❡ sã♦ ❝♦❜❡rt♦s
♣❡❧♦ ✈ért✐❝❡ i✱ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ❛ ❞✐stâ♥❝✐❛ ❡✉❝❧✐❞✐❛♥❛ ❡♥tr❡ ♦ ✈ért✐❝❡ i ❡ ❛q✉❡❧❡s ❡♠ ω(i) ♥ã♦ ❡①❝❡❞❡ R✳
❈♦♠ ❜❛s❡ ♥♦ q✉❡ ❢♦✐ ❡①♣♦st♦✱ ♦ P❘❱❙▼▼ ❝♦♥s✐st❡ ❡♠ ❡♥❝♦♥tr❛r ✉♠ ❝♦♥❥✉♥t♦
✻ ❈❛♣ít✉❧♦ ✷✳ ❆❧❣♦r✐t♠♦ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞❡❘♦t❡❛♠❡♥t♦ ❞❡ ❱❡í❝✉❧♦s ❙❡❧❡t✐✈♦ ▼✐♥✲♠❛①
❞❡K r♦t❛s ✭✉♠❛ ♣❛r❛ ❝❛❞❛ ✈❡í❝✉❧♦✮ ❞❡ ♠♦❞♦ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♠❛✐♦r ❞❡❧❛s s❡❥❛ ♠✐♥✐♠✐③❛❞♦✳ ◆❡st❡ ♣r♦❜❧❡♠❛✱ ✉♠❛ r♦t❛ é ✉♠ ❝✐❝❧♦ ❝♦♠ ♦r✐❣❡♠ ❡ ❞❡st✐♥♦ ♥♦ ❞❡♣ós✐t♦✱ ✈✐s✐t❛♥❞♦ ♥♦ ♠í♥✐♠♦ ❞♦✐s ♦✉tr♦s ✈ért✐❝❡s ❛❧é♠ ❞♦ ❞❡♣ós✐t♦✳ ❖ s✉❜❣r❛❢♦ ❞❡G ✐♥❞✉③✐❞♦ ♣♦r ✉♠ ✈❡í❝✉❧♦ k é ❞❡♥♦t❛❞♦ ♣♦r Hk = (Vk, Ek)✱ ♥♦ q✉❛❧ Vk ❞❡♥♦t❛ ♦ ❝♦♥❥✉♥t♦ ❞♦s
✈ért✐❝❡s ✈✐s✐t❛❞♦s ❡Ek♦ ❝♦♥❥✉♥t♦ ❞❛s ❛r❡st❛s ♣❡r❝♦rr✐❞❛s ♣❡❧♦ ✈❡í❝✉❧♦k✳ ◆♦ P❘❱❙▼▼✱
♥❡♠ t♦❞♦s ♦s ✈ért✐❝❡s ♣r❡❝✐s❛♠ s❡r ✈✐s✐t❛❞♦s ♣♦r ❛❧❣✉♠❛ r♦t❛✳ ◆♦ ❡♥t❛♥t♦✱ ✉♠ ✈ért✐❝❡ i∈V q✉❡ ♥ã♦ é ✈✐s✐t❛❞♦ ❞❡✈❡ s❡r ❝♦❜❡rt♦ ♣♦r ❛❧❣✉♠ ✈ért✐❝❡ ✈✐s✐t❛❞♦ ♣♦r ❛❧❣✉♠❛ r♦t❛✱ ✐st♦ é✱ ♣❛r❛ ❛❧❣✉♠ k ∈ K✱ ❞❡✈❡ ❤❛✈❡r ✉♠ ✈ért✐❝❡ j ∈ Vk t❛❧ q✉❡ i ∈ ω(j)✳ ❈♦♠
❡①❝❡çã♦ ❞♦ ❞❡♣ós✐t♦✱ ✉♠ ✈ért✐❝❡ só ♣♦❞❡ s❡r ✈✐s✐t❛❞♦ ♣♦r ♥♦ ♠á①✐♠♦ ✉♠❛ ú♥✐❝❛ r♦t❛✳ ❖ P❘❱❙▼▼ é N P✲❉✐❢í❝✐❧✱ ❥á q✉❡ ♦ Pr♦❜❧❡♠❛ ❞♦ ❈❛✐①❡✐r♦ ❱✐❛❥❛♥t❡ ❬❉❛♥t③✐❣ ❡t ❛❧✳✱ ✶✾✺✹❪ é ✉♠ ❞❡ s❡✉s ❝❛s♦s ♣❛rt✐❝✉❧❛r❡s q✉❛♥❞♦ K = 1 ❡ ω(i) = {i},∀i∈V✳
✷✳✷ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá✜❝❛
❖ P❘❱❙▼▼ ❢♦✐ r❡❝❡♥t❡♠❡♥t❡ ♣r♦♣♦st♦ ❡♠ ❬❱❛❧❧❡ ❡t ❛❧✳✱ ✷✵✵✽❪ ❝♦♠♦ ♠♦❞❡❧♦ ♣❛r❛ ♣r♦✲ ❥❡t❛r ❘❡❞❡s ❞❡ ❙❡♥s♦r❡s s❡♠ ❋✐♦ ✭❘❙❙❋✮ ❡✜❝✐❡♥t❡s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❡♥❡r❣ét✐❝♦✳ ❖ ♠♦❞❡❧♦ ♣r♦♣♦st♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♠✐♥✐♠✐③❛r ♦ t❡♠♣♦ ❞❡ ❡♥tr❡❣❛ ✭❞♦ ✐♥❣❧ês✱ ❞❡❧❛②✮ ❞❛s ♠❡♥s❛❣❡♥s ❡ ♦ ❝♦♥s✉♠♦ ❞❡ ❡♥❡r❣✐❛ ❞❛ r❡❞❡✱ ♦s q✉❛✐s sã♦ ❢❛t♦r❡s ❝rít✐❝♦s ♥♦ ♣r♦❥❡t♦ ❞❡ r❡❞❡s ❞❡ s❡♥s♦r❡s ❬❆❦②✐❧❞✐③ ❡t ❛❧✳✱ ✷✵✵✷❪✳ P❛r❛ ❡ss❡ ✜♠✱ ❛❧❣✉♥s ♥ós ❡s♣❡❝✐❛✐s✱ ❞❡✲ ♥♦♠✐♥❛❞♦s s♦r✈❡❞♦✉r♦s ✭❞♦ ✐♥❣❧ês✱ s✐♥❦s✮✱ s❡ ❞❡s❧♦❝❛♠ ♣❡❧❛ r❡❣✐ã♦ ♠♦♥✐t♦r❛❞❛ ❛ ✜♠ ❞❡ ❝♦❧❡t❛r ♦s ❞❛❞♦s s❡♥s♦r✐❛❞♦s ♣❡❧♦s ♥ós ❞❛ r❡❞❡✳ ❊ss❡ ❞❡s❧♦❝❛♠❡♥t♦ é ❢❡✐t♦ ❞❡ ✉♠❛ ❢♦r♠❛ t❛❧ q✉❡✱ ❡♠ ✈❡③ ❞❡ ✈✐s✐t❛r t♦❞♦s ♦s ♥ós ❞❛ r❡❞❡✱ ❝❛❞❛ s♦r✈❡❞♦✉r♦ ✈✐s✐t❛ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ ♥ós✳ ➚q✉❡❧❡ ♥ó ✈✐s✐t❛❞♦ ♣♦r ❛❧❣✉♠ s♦r✈❡❞♦✉r♦ é ❞❛❞♦ ♦ ♥♦♠❡ ❝❧✉st❡r ❤❡❛❞✳ ❈❛❞❛ ❝❧✉st❡r ❤❡❛❞ ❡♥❣❧♦❜❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♥ós q✉❡ ❞❡❧❡ ❞✐st❛♠ ♥ã♦ ♠❛✐s ❞❡R✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡ ❛ ❝♦♠✉♥✐❝❛çã♦ ❞✐r❡t❛ ❡♥tr❡ ♥ós s❡♥s♦r❡s é ♣r♦✐❜✐❞❛❀ ❛♣❡♥❛s ❛ ❝♦♠✉♥✐❝❛çã♦ ❡♥tr❡ s♦r✈❡❞♦✉r♦s ❡ ♥ós s❡♥s♦r❡s ♣♦❞❡ ♦❝♦rr❡r✱ ❡ ❡st❛ ♦❝♦rr❡ q✉❛♥❞♦ ♦ ❝❧✉st❡r ❤❡❛❞ ❛ss♦❝✐❛❞♦ é ✈✐s✐t❛❞♦ ♣♦r ❛❧❣✉♠ s♦r✈❡❞♦✉r♦✳ ❯♠❛ ✐❧✉str❛çã♦ ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ P❘❱❙▼▼ ♥♦ ♣r♦❥❡t♦ ❞❡ ✉♠❛ r❡❞❡ ❞❡ s❡♥s♦r❡s é ❛♣r❡s❡♥t❛❞❛ ❛ s❡❣✉✐r✳
✷✳✷✳ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá❢✐❝❛ ✼
❋✐❣✉r❛ ✷✳✶✿ ■♥stâ♥❝✐❛ ❞♦ P❘❱❙▼▼ ❝♦♠ ✷ ✈❡í❝✉❧♦s ❡ ♣❛râ♠❡tr♦ ❘✳
q✉❛tr♦✱ ❡ ♦s ❞❡♠❛✐s ♥ós ♣♦ss✉❡♠ ❣r❛✉ ♦✉ ③❡r♦ ♦✉ ❞♦✐s ♥❛ s♦❧✉çã♦✳ ❆❧é♠ ❞✐ss♦✱ ❝❛❞❛ r♦t❛ ✈✐s✐t❛ ♥♦ ♠í♥✐♠♦ ❞♦✐s ♦✉tr♦s ✈ért✐❝❡s ❛❧é♠ ❞♦ ✈ért✐❝❡ ❞❡ ♦r✐❣❡♠✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ♦s r❡s✉❧t❛❞♦s ❞❛s s✐♠✉❧❛çõ❡s r❡❛❧✐③❛❞❛s ❡♠ ✉♠ s✐♠✉❧❛❞♦r ❞❡ r❡✲ ❞❡s✱ ❛♣r❡s❡♥t❛❞♦s ❡♠ ❬❱❛❧❧❡ ❡t ❛❧✳✱ ✷✵✵✽❪✱ ♦ ♠♦❞❡❧♦ ❞❡ ♦r❣❛♥✐③❛çã♦ ❞❡ ❘❙❙❋ ♣r♦♣♦st♦ s❡ ♠♦str♦✉ s✉♣❡r✐♦r ❛♦s tr❛❜❛❧❤♦s ❝♦♠♣❛r❛❞♦s ❡♠ t♦❞❛s ❛s ♠étr✐❝❛s ❛✈❛❧✐❛❞❛s✳ ❆ ❛❜♦r✲ ❞❛❣❡♠ ❞❡ s♦❧✉çã♦ ♣r♦♣♦st❛ ♥❛q✉❡❧❡ tr❛❜❛❧❤♦ é ❜❛s❡❛❞❛ ❡♠ ✉♠❛ ♠❡t❛❤❡✉ríst✐❝❛ ●r❡❡❞② ❘❛♥❞♦♠✐③❡❞ ❆❞❛♣t✐✈❡ ❙❡❛r❝❤ Pr♦❝❡❞✉r❡ ✭●❘❆❙P✮ ❬❋❡♦ ✫ ❘❡s❡♥❞❡✱ ✶✾✾✺❪✱ s❡♥❞♦ ❝♦♠✲ ♣♦st❛ ♣♦r ❞✉❛s ❢❛s❡s✿ ❛ ❢❛s❡ ❞❡ ❝♦♥str✉çã♦ ❞❛ s♦❧✉çã♦ ✐♥✐❝✐❛❧ ❡ ❛ ❢❛s❡ ❞❡ ❜✉s❝❛ ❧♦❝❛❧✳
❆ ❢❛s❡ ❞❡ ❝♦♥str✉çã♦ ❞❛ s♦❧✉çã♦ ✐♥✐❝✐❛❧ ❞♦ ♠ét♦❞♦ ♣r♦♣♦st♦ é ✉♠❛ ❛❞❛t❛♣çã♦ ❞♦ ❛❧❣♦r✐t♠♦ ❈❤❡❛♣❡st ■♥s❡rt✐♦♥ ❆❧❣♦r✐t❤♠ ✭❈■❆✮ ❬❏✉❧str♦♠✱ ✶✾✾✾❪✳ ❆ ❛❞❛♣t❛çã♦ ❡♠ q✉❡stã♦ ❝♦♥s✐st❡ ❜❛s✐❝❛♠❡♥t❡ ❡♠ ❛❞✐❝✐♦♥❛r à r♦t❛ ♦ ✈ért✐❝❡ q✉❡ r❡s✉❧t❛ ♥♦ ♠❡♥♦r ✐♥✲ ❝r❡♠❡♥t♦ ❡♠ t❡r♠♦s ❞❡ ❞✐stâ♥❝✐❛ ❞❡♥tr❡ ❛q✉❡❧❡s ❞✐s♣♦♥í✈❡✐s✱ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s q✉❡ ♣♦❞❡♠ s❡r ❝♦❜❡rt♦s ❝❛s♦ ♦ ✈ért✐❝❡ ❡♠ q✉❡stã♦ ✈❡♥❤❛ ❛ s❡r ✐♥s❡r✐❞♦✳ ❖s ❛✉t♦r❡s ♣r♦♣✉s❡r❛♠ três ✈❡rsõ❡s ❞✐❢❡r❡♥t❡s ❞♦ ●❘❆❙P✱ ❛s q✉❛✐s ❛♣❧✐❝❛♠ ♦ ♠❡s♠♦ ♠ét♦❞♦ ♥❛ ❢❛s❡ ❞❡ ❝♦♥str✉çã♦ ❞❛ s♦❧✉çã♦ ✐♥✐❝✐❛❧✱ ♠❛s s❡ ❞✐❢❡r❡♥❝✐❛♠ ♥❛ ❢❛s❡ ❞❡ ❜✉s❝❛ ❧♦❝❛❧✳
✽ ❈❛♣ít✉❧♦ ✷✳ ❆❧❣♦r✐t♠♦ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞❡❘♦t❡❛♠❡♥t♦ ❞❡ ❱❡í❝✉❧♦s ❙❡❧❡t✐✈♦ ▼✐♥✲♠❛①
❞❡ r♦t❛s ❞✐st✐♥t❛s✳ ❆ ❢❛s❡ ❞❡ ❜✉s❝❛ ❧♦❝❛❧✱ ♣♦r s✉❛ ✈❡③✱ ❝♦♥s✐st❡ ♥❛ ❜✉s❝❛ ❧♦❝❛❧ ✷✲❖P❚ ❝✐t❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡✳ P♦r ✜♠✱ ❛ t❡r❝❡✐r❛ ✈❡rsã♦ ❞♦ ●❘❆❙P ✉t✐❧✐③❛✲s❡ ❞❛ ♠❡t❛❤❡✉ríst✐❝❛ ❱❛r✐❛❜❧❡ ◆❡✐❣❤❜♦r❤♦♦❞ ❙❡❛r❝❤ ✭❱◆❙✮ ❬▼❧❛❞❡♥♦✈✐❝ ✫ ❍❛♥s❡♥✱ ✶✾✾✼❪ ♥❛ ❢❛s❡ ❞❡ ❜✉s❝❛ ❧♦❝❛❧✳ ❖ ❱◆❙ é ❝♦♠♣♦st♦ ♣❡❧❛s ❜✉s❝❛s ❧♦❝❛✐s ✷✲❖P❚ ❡ ❙❲❆P ❥á ❞✐s❝✉t✐❞❛s✳
❊♠ ❬❱❛❧❧❡ ❡t ❛❧✳✱ ✷✵✵✾❪✱ ❢♦✐ ♣r♦♣♦st❛ ❛ ♣r✐♠❡✐r❛ ❢♦r♠✉❧❛çã♦ ♠❛t❡♠át✐❝❛ ♣❛r❛ ♦ P❘❱❙▼▼✳ ◆♦ tr❛❜❛❧❤♦✱ ✉♠❛ ❢♦r♠✉❧❛çã♦ ❞❡ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ ❝♦♠ ✉♠ ♥ú♠❡r♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ r❡str✐çõ❡s ❞♦ t✐♣♦ ●❡♥❡r❛❧✐③❡❞ ❙✉❜t♦✉r ❊❧✐♠✐♥❛t✐♦♥ ❈♦♥str❛✐♥ts ✭●❙❊❈✮ ❡ ❞✉❛s ❛❜♦r❞❛❣❡♥s ❡①❛t❛s ❢♦r❛♠ ♣r♦♣♦st❛s✳ ❯♠❛ ❞❛s ❛❜♦r❞❛❣❡♥s ❝♦♥s✐st❡ ❡♠ ✉♠ ❛❧❣♦r✐t♠♦ ❇r❛♥❝❤✲❛♥❞✲❈✉t ✭❇❈✮✱ ♥♦ q✉❛❧ ❞❡s✐❣✉❛❧❞❛❞❡s ❇❧♦ss♦♠ ❬❊❞♠♦♥❞s✱ ✶✾✻✺❪ ❡ ●❙❊❈ sã♦ s❡♣❛r❛❞❛s✳ ❆ ♦✉tr❛ ❛❜♦r❞❛❣❡♠✱ ♣♦r s✉❛ ✈❡③✱ ❝♦♥s✐st❡ ❡♠ ✉♠ ❛❧❣♦r✐t♠♦ ▲♦❝❛❧ ❇r❛♥❝❤✐♥❣ ❬❋✐s❝❤❡tt✐ ✫ ▲♦❞✐✱ ✷✵✵✸❪ q✉❡ r❡❝❡❜❡ ❝♦♠♦ ❡♥tr❛❞❛ ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❡ ✉t✐❧✐③❛ ♦ ❇❈ ♣r♦♣♦st♦ ❝♦♠♦ r❡s♦❧✈❡❞♦r ❞❡ ❝❛❞❛ s✉❜♣r♦❜❧❡♠❛✳ ❚❡st❡s ❝♦♠♣✉t❛❝✐♦♥❛✐s ❢♦r❛♠ ❝♦♥❞✉③✐❞♦s ❡♠ ✶✶ ✐♥stâ♥❝✐❛s ❞❛ ❜✐❜❧✐♦t❡❝❛ ❚❙P▲■❇ ❬❚❙P✱ ✷✵✶✷❪✳ ◆♦s ❡①♣❡r✐♠❡♥t♦s r❡♣♦rt❛❞♦s✱ ❛ ♦t✐♠❛❧✐❞❛❞❡ ❞❡ ♣♦✉❝❛s ✐♥stâ♥❝✐❛s ❢♦✐ ❛t❡st❛❞❛✱ s❡♥❞♦ q✉❡✱ ♣❛r❛ ❛q✉❡❧❛s ❡♠ q✉❡ ♥ã♦ ❢♦✐ ♣♦ssí✈❡❧ ♦❜t❡r ✉♠ ❝❡rt✐✜❝❛❞♦ ❞❡ ♦t✐♠❛❧✐❞❛❞❡✱ ❤♦✉✈❡ ❣❛♣ ❞❡ ❞✉❛❧✐❞❛❞❡ ❞❡ ❛té ✹✵✪✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡✱ ♣❛r❛ ❛s ✐♥stâ♥❝✐❛s ❛♥❛❧✐s❛❞❛s✱ ♦ ❣❛♣ ❞❡ ❞✉❛❧✐❞❛❞❡ ❝r❡s❝✐❛ à ♠❡❞✐❞❛ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ✈❡í❝✉❧♦s ❛✉♠❡♥t❛✈❛✱ ♦ q✉❡ t❡r♠✐♥❛ ♣♦r s❡r ✉♠ ❣❛r❣❛❧♦ ❞♦ ❛❧❣♦r✐t♠♦ ♣r♦♣♦st♦✳
❘❡❝❡♥t❡♠❡♥t❡✱ ❱❛❧❧❡ ❡t ❛❧✳ ❬✷✵✶✶❪ ❛♣r❡s❡♥t❛r❛♠ ♥♦✈❛s ❝♦♥tr✐❜✉✐çõ❡s ❡ ♥♦✈♦s r❡✲ s✉❧t❛❞♦s ♣❛r❛ ♦ P❘❱❙▼▼✱ ❛❧é♠ ❞♦s ♠ét♦❞♦s ♣r♦♣♦st♦s ❡♠ ❬❱❛❧❧❡ ❡t ❛❧✳✱ ✷✵✵✾❪✳ ◆❛q✉❡❧❡ tr❛❜❛❧❤♦✱ é ❛♣r❡s❡♥t❛❞❛ ✉♠❛ ♠❡t❛❤❡✉ríst✐❝❛ ❤í❜r✐❞❛ ❜❛s❡❛❞❛ ❡♠ ●❘❆❙P ❡ ❡♠ ■▲❙✱ ❝♦♠♦ t❛♠❜é♠ ✉♠ ♠ét♦❞♦ ❤❡✉ríst✐❝♦ ❞❡♥♦♠✐♥❛❞♦ ❈♦❧✉♠♥ ●❡♥❡r❛t✐♦♥ ❍❡✉r✐s✲ t✐❝ ✭❈●❍✮✳ ❖ ●❘❆❙P ❤í❜r✐❞♦ ♣r♦♣♦st♦ ✉t✐❧✐③❛ ♦ ❛❧❣♦r✐t♠♦ ❈■❆ ♥❛ ❢❛s❡ ❝♦♥str✉t✐✈❛✱ ❡♥q✉❛♥t♦ ❛ ❢❛s❡ ❞❡ ❜✉s❝❛ ❧♦❝❛❧ ✉t✐❧✐③❛ ✉♠ ■▲❙✳ ◆♦ ■▲❙ ♣r♦♣♦st♦✱ ♦ ♠❡❝❛♥✐s♠♦ ❞❡ ♣❡r✲ t✉r❜❛çã♦ ❝♦♥s✐st❡ ♥❛ r❡♠♦çã♦ ✭❝♦♠ ✉♠❛ ❝❡rt❛ ♣r♦❜❛❜✐❧✐❞❛❞❡✮ ❞❡ ✉♠ ✈ért✐❝❡ ❞❡ ✉♠❛ r♦t❛✱ ❡♥q✉❛♥t♦ q✉❡ ♦s ♣r♦❝❡❞✐♠❡♥t♦s ✷✲❖P❚ ❡ ❙❲❆P sã♦ ❛♣❧✐❝❛❞♦s ♥❛ ❢❛s❡ ❞❡ ❜✉s❝❛ ❧♦❝❛❧✱ ❞❡ ♠♦❞♦ ❛ ❝♦♠♣♦r ✉♠ ❱◆❙✳ ❆ ❤❡✉ríst✐❝❛ ❈●❍ é ❞❡s❝r✐t❛ ❛ s❡❣✉✐r✳
❆ ❤❡✉ríst✐❝❛ ❈●❍ é ❢✉♥❞❛♠❡♥t❛❞❛ ♥❛ r❡s♦❧✉çã♦ ❞❡ ✉♠ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❛ r❡❢♦r♠✉❧❛çã♦ ♣♦r r❡❝♦❜r✐♠❡♥t♦ ❞❡ ❝♦♥❥✉♥t♦s ♣❛r❛ ♦ P❘❱❙▼▼✳ ❆ss✉♠❛ q✉❡ Q ❞❡♥♦t❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s r♦t❛s ✈✐á✈❡✐s ♣❛r❛ ♦ P❘❱❙▼▼✳ ❆ ❤❡✉ríst✐❝❛ ❜❛s❡✐❛✲s❡ ♥❛ ✐❞❡✐❛ ❞❡ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❛ss♦❝✐❛❞♦ à r❡❢♦r♠✉❧❛çã♦ ❝♦♠ ❜❛s❡ ❡♠ ✉♠ ❝♦♥❥✉♥t♦ r❡str✐t♦ ❞❡ ❝♦❧✉♥❛s Qr ⊂ Q✱ t❛❧ q✉❡ |Qr| ≪ |Q|✱ s♦❜ ❛ ❝♦♥❞✐çã♦ ❞❡
q✉❡ ✉♠❛ s♦❧✉çã♦ ✈✐á✈❡❧ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ♣♦ss❛ s❡r ❡♥❝♦♥tr❛❞❛ ♣❡❧❛ ❝♦♠❜✐♥❛çã♦ ❞❡K r♦t❛s ❞❡♥tr❡ ❛q✉❡❧❛s ❡♠Qr✳ ❆ ❤❡✉ríst✐❝❛ é ❝♦♠♣♦st❛ ♣♦r ❞✉❛s ❡t❛♣❛s q✉❡ sã♦ ❞❡s❝r✐t❛s
❛ s❡❣✉✐r✳
✷✳✷✳ ❘❡✈✐sã♦ ❇✐❜❧✐♦❣rá❢✐❝❛ ✾
♦ ●❘❆❙P sã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ❣❡r❛r ♥♦✈❛s r♦t❛s ❛ ♣❛rt✐r ❞❛ s♦❧✉çã♦ ♦❜t✐❞❛ ♥❛ ♣r✐♠❡✐r❛ ❡t❛♣❛✳ ❆s ♥♦✈❛s r♦t❛s ♣❛ss❛♠ ❛ ❝♦♠♣♦r ✉♠ ♥♦✈♦ ❝♦♥❥✉♥t♦ Qr✱ ❝♦♠ ❜❛s❡ ♥♦ q✉❛❧ ✉♠
♥♦✈♦ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ é ❢♦r♠✉❧❛❞♦ ❡ r❡s♦❧✈✐❞♦ ♥❛ ♦t✐♠❛❧✐❞❛❞❡✳ ❚♦❞♦ ♦ ♣r♦❝❡ss♦ é r❡♣❡t✐❞♦ ❡♥q✉❛♥t♦ ❛ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❛♣r✐♠♦r❛r ❛ ♠❡❧❤♦r s♦❧✉çã♦ ❝♦♥❤❡❝✐❞❛✱ q✉❡ é ❝♦♥st❛♥t❡♠❡♥t❡ ❛t✉❛❧✐③❛❞❛ ❛♦ ❧♦♥❣♦ ❞♦ ❛❧❣♦r✐t♠♦✳ ❆♣❡s❛r ❞❛s ♥♦✈❛s ❝♦♥tr✐❜✉✐çõ❡s✱ ♦s ❛✉t♦r❡s ❛♣♦♥t❛r❛♠ ❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ r❡s♦❧✈❡r ♦ P❘❱❙▼▼ ❝♦♠ ❜❛s❡ ♥♦s ❡①♣❡r✐♠❡♥t♦s ❝♦♥❞✉③✐❞♦s✳
❱❛❧❧❡ ❡t ❛❧✳ ❬✷✵✶✶❪ r❡♣♦rt❛r❛♠ ❡①♣❡r✐♠❡♥t♦s ❝♦♠ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈✐♥t❡ ✐♥stâ♥✲ ❝✐❛s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ❛✈❛❧✐❛r ❛s ❛❜♦r❞❛❣❡♥s ❡①❛t❛s ❡ ❤❡✉ríst✐❝❛s ♣r♦♣♦st❛s ♣❛r❛ ♦ P❘❱❙▼▼✳ P❛r❛ ♦ tr❛t❛♠❡♥t♦ ❡①❛t♦✱ ♦ ❇❈✱ ❞❡♥tr♦ ❞♦ ❧✐♠✐t❡ ❞❡ ✹ ❤♦r❛s ❞❡ ❡①❡❝✉çã♦✱ ❢♦✐ ❝❛♣❛③ ❞❡ ❛t❡st❛r ❛ ♦t✐♠❛❧✐❞❛❞❡ ❞❡ três ✐♥stâ♥❝✐❛s✱ ❡♥q✉❛♥t♦ q✉❡✱ ♣❛r❛ ❛q✉❡❧❛s ❡♠ q✉❡ ♥ã♦ ❢♦✐ ♣♦ssí✈❡❧ ❢❛③ê✲❧♦✱ ♦s ❣❛♣s ❞❡ ❞✉❛❧✐❞❛❞❡ ♦❜t✐❞♦s ✈❛r✐❛r❛♠ ❡♥tr❡ ✵✳✼✪ ❡ ✹✼✳✸✪ ♣❛r❛ ♦s ❝❡♥ár✐♦s ❛✈❛❧✐❛❞♦s✳ P♦r s✉❛ ✈❡③✱ ♦ ▲♦❝❛❧ ❇r❛♥❝❤✐♥❣ ❢♦✐ ❝❛♣❛③ ❞❡ ♠❡❧❤♦r❛r ❛ s♦❧✉çã♦ ✐♥✐❝✐❛❧ ❞❡ ❡♥tr❛❞❛ ♥❛ ♠❛✐♦r✐❛ ❞♦s ❝❛s♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ à ♠❡❞✐❞❛ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ✈❡í❝✉❧♦s ❛✉♠❡♥t❛✈❛✳ P❛r❛ ♦ tr❛t❛♠❡♥t♦ ❤❡✉ríst✐❝♦✱ ❛ ❤❡✉ríst✐❝❛ ❈●❍ ♥ã♦ ❛♣❡♥❛s ❢♦r♥❡❝❡✉ s♦❧✉çõ❡s ♠❡❧❤♦r❡s q✉❡ ♦s ❧✐♠✐t❡s s✉♣❡r✐♦r❡s ❢♦r♥❡❝✐❞♦s ♣❡❧♦ ❇❈ ✭q✉❛♥❞♦ ❡st❡ ♥ã♦ ❢♦✐ ❝❛♣❛③ ❞❡ ❛t❡st❛r ❛ ♦t✐♠❛❧✐❞❛❞❡✮ ❡ ♣❡❧♦ ▲♦❝❛❧ ❇r❛♥❝❤✐♥❣✱ ❝♦♠♦ t❛♠❜é♠ ♦❜t❡✈❡ t❛✐s s♦❧✉çõ❡s ❡♠ ♠❡♥♦s t❡♠♣♦✳ ❆ s❛❜❡r✱ ❛ ❤❡✉ríst✐❝❛ ❈●❍ ❞❡♠♦r♦✉ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✉♠ ❞é❝✐♠♦ ❞♦ t❡♠♣♦ ♥❡❝❡ssár✐♦ ♣❡❧❛s ♦✉tr❛s ❞✉❛s ❛❜♦r❞❛❣❡♥s✳ ❊♠ ❝♦♥❝❧✉sã♦✱ ♦s ❛✉t♦r❡s ❛♣♦♥t❛r❛♠ ❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ r❡s♦❧✈❡r ✐♥stâ♥❝✐❛s ❞♦ P❘❱❙▼▼ ♥❛ ♦t✐♠❛❧✐❞❛❞❡✱ ♣❡r♠❛♥❡❝❡♥❞♦ ❞✐✈❡rs❛s ✐♥stâ♥❝✐❛s s❡♠ s❡✉ ✈❛❧♦r ót✐♠♦ ❝♦♥❤❡❝✐❞♦✳ ◆ã♦ ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ❡st✉❞♦ ❛❧❣✉♠ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s ❛❧❣♦r✐t♠♦s ♣r♦♣♦st♦s à ♠❡❞✐❞❛ q✉❡ ❛ ❞❡♥s✐❞❛❞❡ ❞❛ r❡❞❡ ✈❛r✐❛✳
❯♠ ♣r♦❜❧❡♠❛ r❡❧❛❝✐♦♥❛❞♦ ❛♦ P❘❱❙▼▼✱ ✐♥t✐t✉❧❛❞♦ ▼✐♥✲♠❛① ❑✲❱❡❤✐❝❧❡s ❲✐♥❞② ❘✉r❛❧ P♦st♠❛♥ Pr♦❜❧❡♠ ✭▼▼ ❑✲❲❘PP✮✱ ❢♦✐ ♣r♦♣♦st♦ ❡♠ ❇❡♥❛✈❡♥t ❡t ❛❧✳ ❬✷✵✵✾❪✳ ❙❡❥❛ ✉♠ ❣r❛❢♦ G = (V, E) ❝♦♠ ❝✉st♦s ♥ã♦✲♥❡❣❛t✐✈♦s cij ❡ cji ❛tr✐❜✉í❞♦s ♣❛r❛ ❝❛❞❛ ❛r❡st❛
{i, j} ∈ E✳ ❉❛❞♦s ✉♠ ✈ért✐❝❡ ❝♦♠♦ ❞❡♣ós✐t♦✱ ✉♠ s✉❜❝♦♥❥✉♥t♦ Er ⊆ E ❞❡ ❛r❡st❛s ❡
✉♠ ♥ú♠❡r♦ K ❞❡ ✈❡í❝✉❧♦s ❝♦♠ ❝❛♣❛❝✐❞❛❞❡ ❝♦♥s✐❞❡r❛❞❛ ✐❧✐♠✐t❛❞❛✱ ♥♦ ▼▼ ❑✲❲❘PP✱ ❜✉s❝❛✲s❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ K r♦t❛s ✭✉♠❛ ♣❛r❛ ❝❛❞❛ ✈❡í❝✉❧♦✮✱ ❞❡ ♠♦❞♦ q✉❡ ❝❛❞❛ r♦t❛ ✐♥✐❝✐❡ ❡ t❡r♠✐♥❡ ♥♦ ❞❡♣ós✐t♦ ❡ ❝❛❞❛ ❛r❡st❛ ❡♠ Er s❡❥❛ ♣❡r❝♦rr✐❞❛ ♣♦r ✉♠ ú♥✐❝♦ ✈❡í❝✉❧♦✳
❖ ♣r♦❜❧❡♠❛ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♠✐♥✐♠✐③❛r ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♠❛✐♦r r♦t❛ ❛ ✜♠ ❞❡ ❡♥❝♦♥✲ tr❛r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ r♦t❛s ❜❛❧❛♥❝❡❛❞❛s ❡♠ t❡r♠♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦✳ ◆❛q✉❡❧❡ tr❛❜❛❧❤♦✱ ♦s ❛✉t♦r❡s ♣r♦♣✉s❡r❛♠ ✉♠❛ ❢♦r♠✉❧❛çã♦ ❞❡ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ ❝♦♠ ✉♠ ♥ú♠❡r♦ ❡①✲ ♣♦♥❡♥❝✐❛❧ ❞❡ r❡str✐çõ❡s ❡ ✉♠ ❛❧❣♦r✐t♠♦ ❇❈✳ ❖ P❘❱❙▼▼ ❞✐❢❡r❡ ❞♦ ▼▼ ❑✲❲❘PP ♣♦r ❞♦✐s ❛s♣❡❝t♦s✿ ❤á r❡str✐çõ❡s ❞❡ ❝♦❜❡rt✉r❛ ❡ ♥ã♦ ❤á r❡str✐çã♦ q✉❡ ❞❡t❡r♠✐♥❛❞❛s ❛r❡st❛s ❞❡✈❛♠ s❡r ✈✐s✐t❛❞❛s✳
✶✵ ❈❛♣ít✉❧♦ ✷✳ ❆❧❣♦r✐t♠♦ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞❡❘♦t❡❛♠❡♥t♦ ❞❡ ❱❡í❝✉❧♦s ❙❡❧❡t✐✈♦ ▼✐♥✲♠❛①
♠✐♥✲♠❛① ❢♦✐ ♣r♦♣♦st❛ ❡♠ ❬●❧❛❛❜✱ ✷✵✵✷❪✳ ◆♦ ♣r♦❜❧❡♠❛✱ ✐♥t✐t✉❧❛❞♦ ▲❛s❡r ▼✉❧t✐✲❙❝❛♥♥❡r Pr♦❜❧❡♠ ✭▲▼❙P✮✱ ❜✉s❝❛✲s❡ ❞❡✜♥✐r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ m r♦t❛s✱ ✉♠❛ ♣❛r❛ ❝❛❞❛ ❧❛s❡r✱ ❞❡ ♠♦❞♦ ❛ ♠✐♥✐♠✐③❛r ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♠❛✐♦r r♦t❛✳ ❉✐❢❡r❡♥t❡♠❡♥t❡ ❞♦ P❘❱❙▼▼✱ ♥❛q✉❡❧❡ ♣r♦❜❧❡♠❛✱ t♦❞♦s ♦s ✈ért✐❝❡s ❞❡✈❡♠ s❡r ✈✐s✐t❛❞♦s ♣♦r ❛❧❣✉♠ ✈❡í❝✉❧♦ ✭❧❛s❡r✮ ❡ ❝❛❞❛ ✈❡í❝✉❧♦ ♣♦❞❡ t❡r ✉♠ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ❞✐❢❡r❡♥t❡✳ ❖s ❛✉t♦r❡s ♣r♦♣✉s❡r❛♠ ❞✉❛s ❤❡✉ríst✐❝❛s✿ ✉♠❛ ❤❡✉ríst✐❝❛ ❞❡ ✐♥s❡rçã♦ ❡ ✉♠❛ ❤❡✉ríst✐❝❛ ❣✉❧♦s❛ ❞❡ ❛r❝♦s✳ ❆ ✜♠ ❞❡ ❛✈❛❧✐❛r ❛ q✉❛❧✐❞❛❞❡ ❞❛s ❤❡✉ríst✐❝❛s ♣r♦♣♦st❛s✱ ♦s ❛✉t♦r❡s ♣r♦♣✉s❡r❛♠ ❞✉❛s r❡❧❛①❛çõ❡s✿ ✉♠❛ r❡❧❛①❛çã♦ ❝♦♠❜✐♥❛tór✐❛ ❞❡♥♦♠✐♥❛❞❛ m−f orest ❡ ♦✉tr❛ q✉❡ r❡❧❛①❛ ❛s r❡str✐çõ❡s ❞♦ ♥ú♠❡r♦ ❞❡ r♦t❛s ❡ ❞❛ ✐♥❡①✐stê♥❝✐❛ ❞❡ s✉❜❝✐❝❧♦✳
❯♠ ♦✉tr♦ ♣r♦❜❧❡♠❛ ❝♦♠ ♥❛t✉r❡③❛ ♠✐♥✲♠❛① ❢♦✐ ♣r♦♣♦st♦ ❡♠ ❬❨❛❦✐❝✐ ✫ ❑❛r❛s❛❦❛❧✱ ✷✵✶✷❪✳ ◆♦ ♣r♦❜❧❡♠❛✱ ✐♥t✐t✉❧❛❞♦ ❱❡❤✐❝❧❡ ❘♦✉t✐♥❣ Pr♦❜❧❡♠ ✇✐t❤ ❙♣❧✐t ❉❡❧✐✈❡r② ❛♥❞ ❍❡t✲ ❡r♦❣❡♥❡♦✉s ❉❡♠❛♥❞ ✭❱❘P❙❉❍❉✮✱ ❞❡✈❡✲s❡ ❞❡✜♥✐r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ K r♦t❛s✱ ✉♠❛ ♣❛r❛ ❝❛❞❛ ✈❡í❝✉❧♦✱ ❞❡ ♠♦❞♦ q✉❡ ❝❛❞❛ ❝♦♥s✉♠✐❞♦r t❡♥❤❛ s✉❛ ❞❡♠❛♥❞❛ ❛t❡♥❞✐❞❛✳ ❖ ♣r♦❜❧❡♠❛ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♠✐♥✐♠✐③❛r ♦ ♠❛✐♦r t❡♠♣♦ ❣❛st♦ ♣❡❧❛s r♦t❛s✳ ❖ t❡♠♣♦ ❣❛st♦ ♣♦r ✉♠❛ r♦t❛ ❝♦♥s✐st❡ ♥❛ s♦♠❛ ❡♥tr❡ ♦ t❡♠♣♦ ❞✐s♣❡♥❞✐❞♦ ❝♦♠ ♦ ❞❡s❧♦❝❛♠❡♥t♦ ❞♦ ✈❡í❝✉❧♦ ❛ss♦❝✐❛❞♦ ❡ ❛q✉❡❧❡ ❞✐s♣❡♥❞✐❞♦ ♥♦ ❛t❡♥❞✐♠❡♥t♦ ❞❡ ❝❛❞❛ ❝❧✐❡♥t❡ ✈✐s✐t❛❞♦✳ ❱❛❧❡ r❡ss❛❧t❛r q✉❡ ♦s ✈❡í❝✉❧♦s ♣♦❞❡♠ ♣♦ss✉✐r ✈❡❧♦❝✐❞❛❞❡ ❞❡ ❞❡s❧♦❝❛♠❡♥t♦ ❞✐❢❡r❡♥t❡ ❡ ♦ ❝✉st♦ ❞❡ ❛t❡♥❞✐✲ ♠❡♥t♦ ❞❡ ✉♠ ❝❧✐❡♥t❡ ♣♦r ✈❡í❝✉❧♦s ❞✐st✐♥t♦s ♣♦❞❡ ✈❛r✐❛r✳ ◆❛q✉❡❧❡ tr❛❜❛❧❤♦✱ ♦s ❛✉t♦r❡s ♣r♦♣✉s❡r❛♠ ✉♠❛ ❢♦r♠✉❧❛çã♦ ❞❡ Pr♦❣r❛♠❛çã♦ ■♥t❡✐r❛ ❝♦♠ ✉♠ ♥ú♠❡r♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ r❡str✐çõ❡s ❡ ✉♠❛ ❤❡✉ríst✐❝❛✳ ❖ P❘❱❙▼▼ ❞✐❢❡r❡ ❞♦ ❱❘P❙❉❍❉ ♣♦r ❞♦✐s ❛s♣❡❝t♦s ♣r✐♥✲ ❝✐♣❛✐s✿ ♣♦ss✉✐ r❡str✐çõ❡s ❞❡ ❝♦❜❡t✉r❛ ❡ ♥❛t✉r❡③❛ s❡❧❡t✐✈❛✳
P♦r ✜♠✱ ❇❛❧❞❛❝❝✐ ❡t ❛❧✳ ❬✷✵✵✼❪ ♣r♦♣✉s❡r❛♠ ✉♠ ♣r♦❜❧❡♠❛ q✉❡✱ ❛ss✐♠ ❝♦♠♦ ♦ P❘❱❙▼▼✱ s✉r❣✐✉ ❝♦♠♦ ♠♦❞❡❧♦ ♣❛r❛ ♣r♦❥❡t❛r r❡❞❡s ❞❡ ❝♦♠✉♥✐❝❛çã♦✳ ❖ ❈❛♣❛❝✐t❛t❡❞ ♠✲❘✐♥❣✲❙t❛r Pr♦❜❧❡♠ ✭❈m❘❙P✮ ❝♦♥s✐st❡ ❡♠ ❞❡✜♥✐r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ m ❝✐❝❧♦s ❞✐s❥✉♥t♦s ❝♦♠ ♦r✐❣❡♠ ♥♦ ❞❡♣ós✐t♦ ❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❛tr✐❜✉✐çõ❡s ❡♥tr❡ ❝❛❞❛ ✈ért✐❝❡ ♥ã♦ ✈✐s✐t❛❞♦ ❡ ❛q✉❡❧❡ ✈✐s✐t❛❞♦ ♣♦r ❛❧❣✉♠ ❝✐❝❧♦✳ ❖ ♥ú♠❡r♦ ❞❡ ♥ós ✈✐s✐t❛❞♦s ♣♦r ✉♠ ❝✐❝❧♦ ♦✉ ❛ ❡❧❡ ❛tr✐❜✉í❞♦s ♥ã♦ ❞❡✈❡ ❡①❝❡❞❡r ✉♠ ✈❛❧♦r ♣ré✲❡st❛❜❡❧❡❝✐❞♦✳ ◆❛q✉❡❧❡ ♣r♦❜❧❡♠❛✱ ❞❡s❡❥❛✲s❡ ♠✐♥✐♠✐③❛r ♦ ❝✉st♦ ❞❡ ❝♦♥str✉çã♦ ❞♦s ❝✐❝❧♦s ❡ ♦ ❝✉st♦ ❞❡ ❝♦♥❡①ã♦ ❞♦s ✈ért✐❝❡s ♥ã♦ ✈✐s✲ ✐t❛❞♦s✳ ❖ P❘❱❙▼▼ ❞✐❢❡r❡ ❞♦ ❈m❘❙P ♣♦r três ❛s♣❡❝t♦s✿ ♣♦ss✉✐ ♥❛t✉r❡③❛ ♠✐♥✲♠❛①✱ ♥ã♦ ❤á ❝✉st♦ ♣❛r❛ ❛tr✐❜✉✐r ✭❝♦❜r✐r✮ ✈ért✐❝❡s ♥ã♦ ✈✐s✐t❛❞♦s ❛ ❛❧❣✉♠❛ r♦t❛ ❡✱ ♣♦r ✜♠✱ ♥ã♦ ❤á r❡str✐çõ❡s ❞❡ ❝❛♣❛❝✐❞❛❞❡✳
✷✳✸ ❋♦r♠✉❧❛çõ❡s ▼❛t❡♠át✐❝❛s ♣❛r❛ ♦ P❘❱❙▼▼
✷✳✸✳ ❋♦r♠✉❧❛çõ❡s ▼❛t❡♠át✐❝❛s ♣❛r❛ ♦ P❘❱❙▼▼ ✶✶
❡t ❛❧✳✱ ✷✵✶✶❪✳ P♦r s✉❛ ✈❡③✱ ❛ s❡❣✉♥❞❛ ❢♦r♠✉❧❛çã♦✱ ♣r♦♣♦st❛ ♥❡st❡ tr❛❜❛❧❤♦✱ é ✉♠❛ r❡❢♦r♠✉❧❛çã♦ ❞♦ P❘❱❙▼▼ ♣♦r ❘❡❝♦❜r✐♠❡♥t♦ ❞❡ ❈♦♥❥✉♥t♦s q✉❡ ♣♦ss✉✐ ✉♠ ♥ú♠❡r♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ✈❛r✐á✈❡✐s✳ ◆♦ r❡st❛♥t❡ ❞❡st❛ s❡çã♦✱ ❞❛❞❛ ✉♠❛ ❢♦r♠✉❧❛çã♦ P ♣❛r❛ ♦ P❘❱❙▼▼✱ ❛ss✉♠❛ q✉❡ z(P)❞❡♥♦t❡ ♦ ❧✐♠✐t❡ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ❢♦r♥❡❝✐❞♦ ♣♦r P✳ ❈♦♥s✐❞❡r❡ q✉❡ B = {0,1}✱ Z ❞❡♥♦t❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡✱ ♣♦r ✜♠✱ R ❞❡♥♦t❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳
✷✳✸✳✶ ❋♦r♠✉❧❛çã♦ ●❙❊❈
❉❛❞♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ W ⊆ V ❞❡ ✈ért✐❝❡s✱ E[W, V \W] = {{i, j} ∈ E : i ∈ W, j ∈ V \W} r❡♣r❡s❡♥t❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s ♥♦ ❝♦rt❡[W, V \W]✱ ❡ E[W] ={{i, j} ∈E :
i, j ∈ W} r❡♣r❡s❡♥t❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s ❝♦♠ ❛♠❜❛s ❛s ❡①tr❡♠✐❞❛❞❡s ❡♠ W✳ ❆❧é♠ ❞✐ss♦✱δ(i) =E[{i}, V\{i}]r❡♣r❡s❡♥t❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❛r❡st❛s ✐♥❝✐❞❡♥t❡s ❛♦ ✈ért✐❝❡i∈V✳ ❆s ✈❛r✐á✈❡✐s ❞❡ ❞❡❝✐sã♦ ❛ s❡❣✉✐r sã♦ ✉t✐❧✐③❛❞❛s ♥❛ ❢♦r♠✉❧❛çã♦✿ ✭✐✮ xk
ij ∈ B,∀{i, j} ∈
E,∀k ∈ K✱ ❛ss✉♠✐♥❞♦ ✈❛❧♦r1s❡ ❛ ❛r❡st❛{i, j} ♣❡rt❡♥❝❡r à r♦t❛ ♣❡r❝♦rr✐❞❛ ♣❡❧♦ ✈❡í❝✉❧♦ k ♦✉ ✈❛❧♦r 0 ❝❛s♦ ❝♦♥trár✐♦✱ ✭✐✐✮ yk
i ∈ B,∀i ∈ V,∀k ∈ K✱ ❛ss✉♠✐♥❞♦ ✈❛❧♦r 1 s❡ ♦
✈ért✐❝❡ i ❢♦r ✈✐s✐t❛❞♦ ♣❡❧❛ r♦t❛ ♣❡r❝♦rr✐❞❛ ♣❡❧♦ ✈❡í❝✉❧♦ k ♦✉ ✈❛❧♦r ✵ ❝❛s♦ ❝♦♥trár✐♦✱ ❡ ✭✐✐✐✮ w ∈ R✱ ❞❡♥♦t❛♥❞♦ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❛ ♠❛✐♦r r♦t❛✳ ❈♦♠ ❜❛s❡ ♥✐ss♦✱ ❛ ♣r✐♠❡✐r❛
❢♦r♠✉❧❛çã♦ ♠❛t❡♠át✐❝❛ ♣❛r❛ ♦ P❘❱❙▼▼✱ ♣r♦♣♦st❛ ❡♠ ❬❱❛❧❧❡ ❡t ❛❧✳✱ ✷✵✶✶❪✱ é ❞❛❞❛ ♣♦r✿
♠✐♥ {w: (w, x, y)∈PGSEC ∩(R,BK|E|,BK|V|)} ✭✷✳✶✮
s❡♥❞♦ ♦ ♣♦❧✐❡❞r♦ PGSEC ❞❡✜♥✐❞♦ ❝♦♠♦✿
w≥ X
{i,j}∈E
dijxkij,∀k ∈ K ✭✷✳✷✮
X
{i,j}∈δ(i)
xkij = 2y k
i, ∀i∈V, ∀k ∈ K, ✭✷✳✸✮
X
k∈K
X
j∈ω(i)
yk
j ≥1, ∀i∈V, ✭✷✳✹✮
−X
k∈K
yki ≥ −1, ∀i∈V \ {1}, ✭✷✳✺✮
X
{i,j}∈E[W,V\W]
xk ij ≥2y
k
z,∀W ⊂V,1∈W, z 6∈W,∀k∈ K, ✭✷✳✻✮
yk1 = 1, ∀k∈ K. ✭✷✳✼✮
✶✷ ❈❛♣ít✉❧♦ ✷✳ ❆❧❣♦r✐t♠♦ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞❡❘♦t❡❛♠❡♥t♦ ❞❡ ❱❡í❝✉❧♦s ❙❡❧❡t✐✈♦ ▼✐♥✲♠❛①
♣r✐♠❡♥t♦ ❞❡ ❝❛❞❛ r♦t❛✳ ❆s r❡str✐çõ❡s ✭✷✳✸✮ ❣❛r❛♥t❡♠ q✉❡ s❡ ♦ ✈ért✐❝❡ i ❢♦r ✈✐s✐t❛❞♦ ♣❡❧♦ ✈❡í❝✉❧♦ k✱ ❡♥tã♦ ❡①❛t❛♠❡♥t❡ ❞✉❛s ❞❛s ❛r❡st❛s ✉t✐❧✐③❛❞❛s ♣♦r ❡st❡ ✈❡í❝✉❧♦ ❞❡✈❡♠ ✐♥❝✐❞✐r s♦❜r❡ i✳ ❆s r❡str✐çõ❡s ✭✷✳✹✮ r❡❢❡r❡♠✲s❡ ❛♦ r❡q✉✐s✐t♦ ❞❡ ❝♦❜❡rt✉r❛✱ ❣❛r❛♥t✐♥❞♦ q✉❡ t♦❞♦ ✈ért✐❝❡ é ❝♦❜❡rt♦✳ P♦r s✉❛ ✈❡③✱ ❛s r❡str✐çõ❡s ✭✷✳✺✮ ❛ss❡❣✉r❛♠ q✉❡ ✉♠ ✈ért✐❝❡ s❡rá ✈✐s✐t❛❞♦ ♣♦r ♥♦ ♠á①✐♠♦ ✉♠ ✈❡í❝✉❧♦✳ ❆s r❡str✐çõ❡s ✭✷✳✻✮ r❡❢❡r❡♠✲s❡ às r❡str✐çõ❡s ❞❡ ❡❧✐♠✐♥❛çã♦ ❞❡ s✉❜❝✐❝❧♦ ●❙❊❈✱ ♣r❡✈❡♥✐♥❞♦ ♦ ♠♦❞❡❧♦ ❞❡ ❢♦r♠❛r r♦t❛s q✉❡ ♥ã♦ ✈✐s✐t❛♠ ♦ ❞❡♣ós✐t♦✳ P♦r ✜♠✱ ❛s r❡str✐çõ❡s ✭✷✳✼✮ ♦❜r✐❣❛♠ q✉❡ ♦ ❞❡♣ós✐t♦ s❡❥❛ s❡♠♣r❡ ✈✐s✐t❛❞♦ ♣❛r❛ ❝❛❞❛ ✈❡í❝✉❧♦✳
✷✳✸✳✷ ❘❡❢♦r♠✉❧❛çã♦ ♣♦r ❘❡❝♦❜r✐♠❡♥t♦ ❞❡ ❈♦♥❥✉♥t♦s
❆ ❢♦r♠✉❧❛çã♦ ♣r♦♣♦st❛ ♥❡st❡ tr❛❜❛❧❤♦ é ✉♠❛ r❡❢♦r♠✉❧❛çã♦ ❞♦ P❘❱❙▼▼ ♣♦r ❘❡❝♦❜r✐✲ ♠❡♥t♦ ❞❡ ❈♦♥❥✉♥t♦s✳ ❊st❛ r❡❢♦r♠✉❧❛çã♦ ♣♦ss✉✐ ✉♠ ♥ú♠❡r♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ✈❛r✐á✈❡✐s✳ P♦rt❛♥t♦✱ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ♦ ✉s♦ ❞♦ ♠ét♦❞♦ ❞❡ ●❡r❛çã♦ ❞❡ ❈♦❧✉♥❛s ♣❛r❛ ♦❜t❡r ❧✐♠✐t❡s ✐♥❢❡r✐♦r❡s ✈á❧✐❞♦s ♣❛r❛ ❛ r❡❢♦r♠✉❧❛çã♦ ♣r♦♣♦st❛✳ ❆♥t❡s ❞❡ ❛♣r❡s❡♥t❛r ❛ r❡❢♦r♠✉❧❛çã♦✱ ❢❛③❡♠✲s❡ ♥❡❝❡ssár✐❛s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ ♥♦t❛çã♦✳
❙❡❥❛ P ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s r♦t❛s ✈✐á✈❡✐s ♣❛r❛ ♦ P❘❱❙▼▼ s♦❜r❡ ✉♠ ❣r❛❢♦ G= (V, E)✱ ❞❡ ♠♦❞♦ q✉❡Pkr❡♣r❡s❡♥t❛ ♦ ❝♦♥❥✉♥t♦ ❞❛s r♦t❛s ❡♠P ❛ss♦❝✐❛❞❛s ❛♦ ✈❡í❝✉❧♦
k✳ P❛r❛ ❝❛❞❛ r♦t❛ p ∈ P✱ ♦s s❡❣✉✐♥t❡s ♣❛râ♠❡tr♦s sã♦ ❞❡✜♥✐❞♦s✿ ✭✐✮ dp r❡♣r❡s❡♥t❛ ♦
❝♦♠♣r✐♠❡♥t♦ ❞❛ r♦t❛✱ ✭✐✐✮βip ❛ss✉♠❡ ✈❛❧♦r1s❡ ♦ ✈ért✐❝❡i❢♦r ✈✐s✐t❛❞♦ ♣❡❧❛ r♦t❛ ♦✉ ✈❛❧♦r
0 ❝❛s♦ ❝♦♥trár✐♦✱ ✭✐✐✐✮ δpi ❛ss✉♠❡ ✈❛❧♦r 1 s❡ ♦ ✈ért✐❝❡ i ❢♦r ❝♦❜❡rt♦ ♣❡❧❛ r♦t❛ ♦✉ ✈❛❧♦r 0
❝❛s♦ ❝♦♥trár✐♦✱ ❡ ✭✐✈✮ςijp ❛ss✉♠❡ ✈❛❧♦r1s❡ ❛ ❛r❡st❛{i, j} ∈E ❢♦r ✉t✐❧✐③❛❞❛ ♣❡❧❛ r♦t❛ ♦✉
✈❛❧♦r0 ❝❛s♦ ❝♦♥trár✐♦✳ P♦r ✜♠✱ s❡❥❛ λpk ∈B✉♠❛ ✈❛r✐á✈❡❧ q✉❡ ❛ss✉♠❡ ✈❛❧♦r ✶ s❡ ❛ r♦t❛ p∈ Pk ❢♦r ✉t✐❧✐③❛❞❛ ♣❡❧♦ ✈❡í❝✉❧♦ k ♦✉ ✈❛❧♦r ✵ ❝❛s♦ ❝♦♥trár✐♦✳ ❉❡st❡ ♠♦❞♦✱ ♦ Pr♦❜❧❡♠❛
▼❡str❡ ✭P▼✮ é ❞❡✜♥✐❞♦ ❝♦♠♦✿
♠✐♥ {w: (w, λ)∈Pc∩(R+,BK|P|)}, ✭✷✳✽✮
s❡♥❞♦ ♦ ♣♦❧✐❡❞r♦ Pc ❞❡✜♥✐❞♦ ❝♦♠♦✿
w≥ X
p∈Pk
dpλpk,∀k ∈ K, ✭✷✳✾✮
X
p∈Pk
λpk = 1,∀k ∈ K, ✭✷✳✶✵✮
X
k∈K
X
p∈Pk
δipλ p
k ≥1,∀i∈V \ {1}, ✭✷✳✶✶✮
−X
k∈K
X
p∈Pk
βipλ p
