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  1. ❨♦✉♥❣ ❘❡s❡❛r❝❤❡rs ❈❊❘▼■❈❙ ❙❡♠✐♥❛r ❙♦❧✈✐♥❣ t❤❡ ❋✐r❡✜❣❤t❡r ♣r♦❜❧❡♠ ♦♥ tr❡❡s ❆❞è❧❡ P❛ss✲▲❛♥♥❡❛✉ ❖❝t♦❜❡r ✶✵✱ ✷✵✶✼ ■♥t❡r♥s❤✐♣ ❉✐r❡❝t♦r✿ ❋ré❞ér✐❝ ▼❡✉♥✐❡r

  2. ■♥tr♦❞✉❝t✐♦♥ t♦ ❋✐r❡✜❣❤t✐♥❣ ❚❤❡ ❋✐r❡✜❣❤t❡r ♣r♦❜❧❡♠ ✭❍❛rt♥❡❧❧✱ ✶✾✾✺✮✿ ♠♦❞❡❧s ♣r♦♣❛❣❛t✐♦♥ ♦❢ ❛ ✜r❡ ✐♥ ❛ ♥❡t✇♦r❦ ✭♦r ❞✐s❡❛s❡ ✐♥ ❛ ♣♦♣✉❧❛t✐♦♥✱ ✈✐r✉s ✐♥ ❝♦♠♣✉t❡r ♥❡t✇♦r❦✱ ❡t❝✮ r • ❆ ❣r❛♣❤✱ ❢♦r ✉s✿ ❛ tr❡❡ T ✇✐t❤ r♦♦t r • ❆ ✜r❡ st❛rts ❛t r • ❆t ❡❛❝❤ t✐♠❡ st❡♣ ✕ ✶ ♥❡✇ ✈❡rt❡① ❝❛♥ ❜❡ ♣r♦t❡❝t❡❞ ✕ ❚❤❡ ✜r❡ s♣r❡❛❞s t = ✵ • ❖❜❥❡❝t✐✈❡✿ ♠❛①✐♠✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ s❛✈❡❞ ✈❡rt✐❝❡s ✶✴✷✹

  3. ■♥tr♦❞✉❝t✐♦♥ t♦ ❋✐r❡✜❣❤t✐♥❣ ❚❤❡ ❋✐r❡✜❣❤t❡r ♣r♦❜❧❡♠ ✭❍❛rt♥❡❧❧✱ ✶✾✾✺✮✿ ♠♦❞❡❧s ♣r♦♣❛❣❛t✐♦♥ ♦❢ ❛ ✜r❡ ✐♥ ❛ ♥❡t✇♦r❦ ✭♦r ❞✐s❡❛s❡ ✐♥ ❛ ♣♦♣✉❧❛t✐♦♥✱ ✈✐r✉s ✐♥ ❝♦♠♣✉t❡r ♥❡t✇♦r❦✱ ❡t❝✮ r • ❆ ❣r❛♣❤✱ ❢♦r ✉s✿ ❛ tr❡❡ T ✇✐t❤ r♦♦t r • ❆ ✜r❡ st❛rts ❛t r • ❆t ❡❛❝❤ t✐♠❡ st❡♣ ✕ ✶ ♥❡✇ ✈❡rt❡① ❝❛♥ ❜❡ ♣r♦t❡❝t❡❞ ✕ ❚❤❡ ✜r❡ s♣r❡❛❞s t = ✶ • ❖❜❥❡❝t✐✈❡✿ ♠❛①✐♠✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ ♣r♦t❡❝t✐♦♥ s❛✈❡❞ ✈❡rt✐❝❡s ✶✴✷✹

  4. ■♥tr♦❞✉❝t✐♦♥ t♦ ❋✐r❡✜❣❤t✐♥❣ ❚❤❡ ❋✐r❡✜❣❤t❡r ♣r♦❜❧❡♠ ✭❍❛rt♥❡❧❧✱ ✶✾✾✺✮✿ ♠♦❞❡❧s ♣r♦♣❛❣❛t✐♦♥ ♦❢ ❛ ✜r❡ ✐♥ ❛ ♥❡t✇♦r❦ ✭♦r ❞✐s❡❛s❡ ✐♥ ❛ ♣♦♣✉❧❛t✐♦♥✱ ✈✐r✉s ✐♥ ❝♦♠♣✉t❡r ♥❡t✇♦r❦✱ ❡t❝✮ r • ❆ ❣r❛♣❤✱ ❢♦r ✉s✿ ❛ tr❡❡ T ✇✐t❤ r♦♦t r • ❆ ✜r❡ st❛rts ❛t r • ❆t ❡❛❝❤ t✐♠❡ st❡♣ ✕ ✶ ♥❡✇ ✈❡rt❡① ❝❛♥ ❜❡ ♣r♦t❡❝t❡❞ ✕ ❚❤❡ ✜r❡ s♣r❡❛❞s t = ✶ • ❖❜❥❡❝t✐✈❡✿ ♠❛①✐♠✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ ♣r♦♣❛❣❛t✐♦♥ s❛✈❡❞ ✈❡rt✐❝❡s ✶✴✷✹

  5. ■♥tr♦❞✉❝t✐♦♥ t♦ ❋✐r❡✜❣❤t✐♥❣ ❚❤❡ ❋✐r❡✜❣❤t❡r ♣r♦❜❧❡♠ ✭❍❛rt♥❡❧❧✱ ✶✾✾✺✮✿ ♠♦❞❡❧s ♣r♦♣❛❣❛t✐♦♥ ♦❢ ❛ ✜r❡ ✐♥ ❛ ♥❡t✇♦r❦ ✭♦r ❞✐s❡❛s❡ ✐♥ ❛ ♣♦♣✉❧❛t✐♦♥✱ ✈✐r✉s ✐♥ ❝♦♠♣✉t❡r ♥❡t✇♦r❦✱ ❡t❝✮ r • ❆ ❣r❛♣❤✱ ❢♦r ✉s✿ ❛ tr❡❡ T ✇✐t❤ r♦♦t r • ❆ ✜r❡ st❛rts ❛t r • ❆t ❡❛❝❤ t✐♠❡ st❡♣ ✕ ✶ ♥❡✇ ✈❡rt❡① ❝❛♥ ❜❡ ♣r♦t❡❝t❡❞ ✕ ❚❤❡ ✜r❡ s♣r❡❛❞s t = ✷ • ❖❜❥❡❝t✐✈❡✿ ♠❛①✐♠✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ ♣r♦t❡❝t✐♦♥ s❛✈❡❞ ✈❡rt✐❝❡s ✶✴✷✹

  6. ■♥tr♦❞✉❝t✐♦♥ t♦ ❋✐r❡✜❣❤t✐♥❣ ❚❤❡ ❋✐r❡✜❣❤t❡r ♣r♦❜❧❡♠ ✭❍❛rt♥❡❧❧✱ ✶✾✾✺✮✿ ♠♦❞❡❧s ♣r♦♣❛❣❛t✐♦♥ ♦❢ ❛ ✜r❡ ✐♥ ❛ ♥❡t✇♦r❦ ✭♦r ❞✐s❡❛s❡ ✐♥ ❛ ♣♦♣✉❧❛t✐♦♥✱ ✈✐r✉s ✐♥ ❝♦♠♣✉t❡r ♥❡t✇♦r❦✱ ❡t❝✮ r • ❆ ❣r❛♣❤✱ ❢♦r ✉s✿ ❛ tr❡❡ T ✇✐t❤ r♦♦t r • ❆ ✜r❡ st❛rts ❛t r • ❆t ❡❛❝❤ t✐♠❡ st❡♣ ✕ ✶ ♥❡✇ ✈❡rt❡① ❝❛♥ ❜❡ ♣r♦t❡❝t❡❞ ✕ ❚❤❡ ✜r❡ s♣r❡❛❞s t = ✷ • ❖❜❥❡❝t✐✈❡✿ ♠❛①✐♠✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ ♣r♦♣❛❣❛t✐♦♥ s❛✈❡❞ ✈❡rt✐❝❡s ✶✴✷✹

  7. ■♥tr♦❞✉❝t✐♦♥ t♦ ❋✐r❡✜❣❤t✐♥❣ ❚❤❡ ❋✐r❡✜❣❤t❡r ♣r♦❜❧❡♠ ✭❍❛rt♥❡❧❧✱ ✶✾✾✺✮✿ ♠♦❞❡❧s ♣r♦♣❛❣❛t✐♦♥ ♦❢ ❛ ✜r❡ ✐♥ ❛ ♥❡t✇♦r❦ ✭♦r ❞✐s❡❛s❡ ✐♥ ❛ ♣♦♣✉❧❛t✐♦♥✱ ✈✐r✉s ✐♥ ❝♦♠♣✉t❡r ♥❡t✇♦r❦✱ ❡t❝✮ r • ❆ ❣r❛♣❤✱ ❢♦r ✉s✿ ❛ tr❡❡ T ✇✐t❤ r♦♦t r • ❆ ✜r❡ st❛rts ❛t r • ❆t ❡❛❝❤ t✐♠❡ st❡♣ ✕ ✶ ♥❡✇ ✈❡rt❡① ❝❛♥ ❜❡ ♣r♦t❡❝t❡❞ ✕ ❚❤❡ ✜r❡ s♣r❡❛❞s − → ✻ ✈❡rt✐❝❡s s❛✈❡❞ • ❖❜❥❡❝t✐✈❡✿ ♠❛①✐♠✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ s❛✈❡❞ ✈❡rt✐❝❡s ✶✴✷✹

  8. ❆♥ ■♥t❡❣❡r Pr♦❣r❛♠♠✐♥❣ ❢♦r♠✉❧❛t✐♦♥ • ▲❡✈❡❧ ✿ s❡t ♦❢ ✈❡rt✐❝❡s ✇✐t❤ s❛♠❡ ❞✐st❛♥❝❡ t♦ t❤❡ r♦♦t • ❲❡✐❣❤t ✿ ✐❢ v ♣r♦t❡❝t❡❞✱ w v ✈❡rt✐❝❡s s❛✈❡❞ r ✭✺✮ t❤✐r❞ ❧❡✈❡❧ ✭✶✮ ✭✸✮ s❡❝♦♥❞ ❧❡✈❡❧ ✭✶✮ ✜rst ❧❡✈❡❧ ✷✴✷✹

  9. ❆♥ ■♥t❡❣❡r Pr♦❣r❛♠♠✐♥❣ ❢♦r♠✉❧❛t✐♦♥ ❉❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡s ✿ x v = ✶ ✐✛ v ✐s ♣r♦t❡❝t❡❞ � max x v w v v ∈ V \{ r } s✳ t✳ x u + x v ≤ ✶ ∀ u , v ♦♥ s❛♠❡ ❧❡✈❡❧ ✭✶✮ x u + x v ≤ ✶ ∀ u ❛♥❝❡st♦r ♦❢ v ✭✷✮ x v ∈ { ✵ , ✶ } ✭✶✮✿ ✶ ✈❡rt❡① ♣r♦t❡❝t❡❞ ♣❡r ❧❡✈❡❧ ✭✷✮✿ ❛ ✈❡rt❡① ❛❧r❡❛❞② s❛✈❡❞ ♠✉st ♥♦t ❜❡ ♣r♦t❡❝t❡❞ ✷✴✷✹

  10. ❙♦❧✈✐♥❣ t❤❡ ❋✐r❡✜❣❤t❡r ♣r♦❜❧❡♠ • ❋✐r❡✜❣❤t❡r ♣r♦❜❧❡♠ ✐s ◆P✲❤❛r❞ ♦♥ tr❡❡s ✭❋✐♥❜♦✇ ❡t ❛❧✳✱ ✷✵✵✾✮ • ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ♠❡t❤♦❞s ❢♦r s♦❧✈✐♥❣ ✐t ❉✐✛❡r❡♥t ❛♣♣r♦❛❝❤❡s✿ • P♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠s ✐♥ s♣❡❝✐❛❧ ❝❛s❡s • ▼❡t❤♦❞s ❜❛s❡❞ ♦♥ ✐♥t❡❣❡r ♣r♦❣r❛♠♠✐♥❣ • ▲❛❣r❛♥❣✐❛♥ r❡❧❛①❛t✐♦♥ ✸✴✷✹

  11. ❖✉t❧✐♥❡ ✶✳ ❋✐r❡✜❣❤t✐♥❣ ❛♥❞ ❙t❛❜❧❡ ❙❡t ♣r♦❜❧❡♠s ✷✳ P❡r❢❡❝t ❣r❛♣❤s✿ ❛ ♣♦❧②♥♦♠✐❛❧ ❝❛s❡ ❢♦r t❤❡ ❋✐r❡✜❣❤t❡r ♣r♦❜❧❡♠ ✸✳ ❋❛❝❡ts ♦❢ ♣♦❧②❤❡❞r❛✿ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❋✐r❡✜❣❤t❡r ♣♦❧②t♦♣❡ ✹✳ ◆✉♠❡r✐❝❛❧ r❡s♦❧✉t✐♦♥ t❤r♦✉❣❤ ▲❛❣r❛♥❣✐❛♥ r❡❧❛①❛t✐♦♥ ✹✴✷✹

  12. ❋✐r❡✜❣❤t✐♥❣ ❛♥❞ ❙t❛❜❧❡ ❙❡t ♣r♦❜❧❡♠s

  13. ❙t❛❜❧❡ ❙❡ts ❙t❛❜❧❡ s❡t ●✐✈❡♥ ❛ ❣r❛♣❤ G = ( V , E ) ✱ ❛ s✉❜s❡t ♦❢ ✈❡rt✐❝❡s S ⊂ V ✐s st❛❜❧❡ ✐❢ ∀ u , v ∈ S ✱ u ❛♥❞ v ❛r❡ ♥♦t ❛❞❥❛❝❡♥t✱ ✐✳❡✳✱ ( u , v ) / ∈ E ✳ • ❈❧❛ss✐❝❛❧ ♣r♦❜❧❡♠s✿ st❛❜❧❡ s❡t ♦❢ ♠❛①✐♠✉♠ s✐③❡ ✱ ♦❢ ♠❛①✐♠✉♠ ✇❡✐❣❤t � v ∈ S w v • ▼❛① ❙t❛❜❧❡ ❙❡t ✐s ◆P✲❝♦♠♣❧❡t❡ ❀ ❡✈❡♥ ❤❛r❞ ✐♥ ♣r❛❝t✐❝❡ ✇✐t❤ ❛ t❤♦✉s❛♥❞ ✈❡rt✐❝❡s✳ ✺✴✷✹

  14. ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❙t❛❜❧❡ ❙❡t ♣r♦❜❧❡♠s • ❚❛s❦ s❝❤❡❞✉❧✐♥❣ ✿ ❣✐✈❡♥ ❛ s❡t ♦❢ ❥♦❜s ✇✐t❤ ❜❡❣✐♥✴❡♥❞ ❞❛t❡✱ ✜♥❞ t❤❡ ♠❛① ♥✉♠❜❡r ♦❢ t❛s❦s t❤❛t ❝❛♥ ❜❡ s❝❤❡❞✉❧❡❞ ♦♥ ❛ s✐♥❣❧❡ ♠❛❝❤✐♥❡✳ ✻✴✷✹

  15. ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❙t❛❜❧❡ ❙❡t ♣r♦❜❧❡♠s • P✐❧♦t✲❈♦♣✐❧♦t ❛❧❧♦❝❛t✐♦♥ ✿ ❣✐✈❡♥ ❛ s❡t ♦❢ ♣❡rs♦♥s s♣❡❛❦✐♥❣ ❞✐✛❡r❡♥t ❧❛♥❣✉❛❣❡s✱ ✜♥❞ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ♣❛✐rs s♣❡❛❦✐♥❣ t❤❡ s❛♠❡ ❧❛♥❣✉❛❣❡✳ ✻✴✷✹

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