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  1. ❊①❛❝t ▼❡t❤♦❞s ❢♦r ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣✿ ✏♣r✐❝❡✲❛♥❞✲✈❡r✐❢②✑ ✇✐t❤ ♥♦♥❧✐♥❡❛r s✉❜♣r♦❜❧❡♠s ❆♠❜r♦s ●❧❡✐①♥❡r✱ ❇❡♥❥❛♠✐♥ ▼ü❧❧❡r✱ ❙t❡♣❤❡♥ ❏✳ ▼❛❤❡r✱ ❏♦ã♦ P✳ P❡❞r♦s♦ ❩✉s❡ ■♥st✐t✉t❡ ❇❡r❧✐♥✱ ❣❧❡✐①♥❡r❅③✐❜✳❞❡ ❙❈■P ❲♦r❦s❤♦♣ ✷✵✶✽ · ❘❲❚❍ ❆❛❝❤❡♥ · ▼❛r❝❤ ✽ ●❧❡✐①♥❡r✱ ▼ü❧❧❡r✱ ▼❛❤❡r✱ P❡❞r♦s♦ ✕ ❊①❛❝t ▼❡t❤♦❞s ❢♦r ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ✶ ✴ ✸✷

  2. ▼♦t✐✈❛t✐♦♥ ◮ r❡❞✉❝❡ ❧❛r❣❡ s❤✐♣♣✐♥❣ ❝♦sts ✐♥ t✉❜❡ ♣r♦❞✉❝t✐♦♥ ✐♥❞✉str② ◮ t✉❜❡s ❛r❡ ❝✉t t♦ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝♦♥t❛✐♥❡r → ✷✲❞✐♠ ♣r♦❜❧❡♠ ◮ r❡❝✉rs✐✈❡ ♣❛❝❦✐♥❣ ♦❢ t✉❜❡s ✐♥t♦ ❝♦♥t❛✐♥❡rs ●❧❡✐①♥❡r✱ ▼ü❧❧❡r✱ ▼❛❤❡r✱ P❡❞r♦s♦ ✕ ❊①❛❝t ▼❡t❤♦❞s ❢♦r ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ✷ ✴ ✸✷

  3. ♠✐♥✐♠✐③❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝♦♥t❛✐♥❡rs ♠❛①✐♠✐③❡ ❧♦❛❞ ♦❢ ❛ s✐♥❣❧❡ ❝♦♥t❛✐♥❡r ❱❛r✐❛♥ts ♦❢ ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ❡①t✳ ✴ ✐♥t✳ r❛❞✐✐ ✶✸ ✺ ✷ ❞❡♠❛♥❞s ●❧❡✐①♥❡r✱ ▼ü❧❧❡r✱ ▼❛❤❡r✱ P❡❞r♦s♦ ✕ ❊①❛❝t ▼❡t❤♦❞s ❢♦r ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ✸ ✴ ✸✷

  4. ♠✐♥✐♠✐③❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝♦♥t❛✐♥❡rs ❱❛r✐❛♥ts ♦❢ ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ❡①t✳ ✴ ✐♥t✳ r❛❞✐✐ ✶✸ ✺ ✷ ❞❡♠❛♥❞s ♠❛①✐♠✐③❡ ❧♦❛❞ ♦❢ ❛ s✐♥❣❧❡ ❝♦♥t❛✐♥❡r ●❧❡✐①♥❡r✱ ▼ü❧❧❡r✱ ▼❛❤❡r✱ P❡❞r♦s♦ ✕ ❊①❛❝t ▼❡t❤♦❞s ❢♦r ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ✸ ✴ ✸✷

  5. ♠✐♥✐♠✐③❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝♦♥t❛✐♥❡rs ❱❛r✐❛♥ts ♦❢ ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ❡①t✳ ✴ ✐♥t✳ r❛❞✐✐ ✶✸ ✺ ✷ ❞❡♠❛♥❞s ♠❛①✐♠✐③❡ ❧♦❛❞ ♦❢ ❛ s✐♥❣❧❡ ❝♦♥t❛✐♥❡r ♠✐♥✐♠✐③❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝♦♥t❛✐♥❡rs ●❧❡✐①♥❡r✱ ▼ü❧❧❡r✱ ▼❛❤❡r✱ P❡❞r♦s♦ ✕ ❊①❛❝t ▼❡t❤♦❞s ❢♦r ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ✸ ✴ ✸✷

  6. ❱❛r✐❛♥ts ♦❢ ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ❡①t✳ ✴ ✐♥t✳ r❛❞✐✐ ✶✸ ✺ ✷ ❞❡♠❛♥❞s ♠❛①✐♠✐③❡ ❧♦❛❞ ♦❢ ❛ s✐♥❣❧❡ ❝♦♥t❛✐♥❡r ♠✐♥✐♠✐③❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝♦♥t❛✐♥❡rs ●❧❡✐①♥❡r✱ ▼ü❧❧❡r✱ ▼❛❤❡r✱ P❡❞r♦s♦ ✕ ❊①❛❝t ▼❡t❤♦❞s ❢♦r ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ✸ ✴ ✸✷

  7. ♦❜❥❡❝t✐✈❡ ✳ ✳ ✳ ♠✐♥✐♠✐③❡ ♥✉♠❜❡r ♦❢ r❡❝t❛♥❣❧❡s s✳t✳ ♠❛♥② r✐♥❣s ♦❢ t②♣❡ ❛r❡ ♣❛❝❦❛❜❧❡ ✐♥t♦ r❡❝t❛♥❣❧❡s ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ Pr♦❜❧❡♠ ✭❘❈PP✮ ✐♥♣✉t ✳ ✳ ✳ ◮ s❡t ♦❢ r✐♥❣ t②♣❡s T := { ✶ , . . . , T } ◮ ❡①t❡r♥❛❧ r❛❞✐✉s r ext ❛♥❞ ✐♥t❡r♥❛❧ r❛❞✐✉s r int ❢♦r ❡❛❝❤ t ∈ T t t ◮ ❞❡♠❛♥❞ D t ∈ N ❢♦r ❡❛❝❤ t ∈ T ◮ ✐♥✜♥✐t❡❧② ♠❛♥② r❡❝t❛♥❣❧❡s ✇✐t❤ ✇✐❞t❤ W ❛♥❞ ❤❡✐❣❤t H ●❧❡✐①♥❡r✱ ▼ü❧❧❡r✱ ▼❛❤❡r✱ P❡❞r♦s♦ ✕ ❊①❛❝t ▼❡t❤♦❞s ❢♦r ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ✹ ✴ ✸✷

  8. ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ Pr♦❜❧❡♠ ✭❘❈PP✮ ✐♥♣✉t ✳ ✳ ✳ ◮ s❡t ♦❢ r✐♥❣ t②♣❡s T := { ✶ , . . . , T } ◮ ❡①t❡r♥❛❧ r❛❞✐✉s r ext ❛♥❞ ✐♥t❡r♥❛❧ r❛❞✐✉s r int ❢♦r ❡❛❝❤ t ∈ T t t ◮ ❞❡♠❛♥❞ D t ∈ N ❢♦r ❡❛❝❤ t ∈ T ◮ ✐♥✜♥✐t❡❧② ♠❛♥② r❡❝t❛♥❣❧❡s ✇✐t❤ ✇✐❞t❤ W ❛♥❞ ❤❡✐❣❤t H ♦❜❥❡❝t✐✈❡ ✳ ✳ ✳ ◮ ♠✐♥✐♠✐③❡ z ∗ = ♥✉♠❜❡r ♦❢ r❡❝t❛♥❣❧❡s t { D t ♠❛♥② r✐♥❣s ♦❢ t②♣❡ t } ❛r❡ ♣❛❝❦❛❜❧❡ ✐♥t♦ z ∗ r❡❝t❛♥❣❧❡s ◮ s✳t✳ � ●❧❡✐①♥❡r✱ ▼ü❧❧❡r✱ ▼❛❤❡r✱ P❡❞r♦s♦ ✕ ❊①❛❝t ▼❡t❤♦❞s ❢♦r ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ✹ ✴ ✸✷

  9. ❘❡❧❛t❡❞ ❲♦r❦ ❈PP ❡①t❡♥s✐✈❡❧② st✉❞✐❡❞ ✳ ✳ ✳ ◮ s♣❤❡r❡ ♣❛❝❦✐♥❣ ❬❑❡♣❧❡r ✶✻✶✶❪ ◮ ♣❛❝❦✐♥❣ ✳ ✳ ✳ ◮ ✐❞❡♥t✐❝❛❧ ❝✐r❝❧❡s ❬●♦❧❞❜❡r❣ ✶✾✼✵✱ ▲♦❝❛t❡❧❧✐✱ ❘❛❜❡r ✷✵✵✷✱ ✳ ✳ ✳ ❪ ◮ ❞✐✛❡r❡♥t ❝✐r❝❧❡s ❬●❡♦r❣❡✱ ●❡♦r❣❡✱ ▲❛♠❛❡r ✶✾✾✺✱ ❍✉❛♥❣ ❡t ❛❧✳ ✷✵✵✹✱ ❑❛❧❧r❛t❤ ✷✵✵✼✱ ✳ ✳ ✳ ❪ ◮ ❡❧❧✐♣s♦✐❞s ❬❑❛❧❧r❛t❤ ✷✵✶✺❪ ◮ r❡❝❡♥t s✉r✈❡②s ❬❈❛st✐❧❧♦✱ ❑❛♠♣❛s✱ ❛♥❞ P✐♥tér ✷✵✵✽✱ ❍✐✜ ❛♥❞ ▼❵❍❛❧❧❛❤ ✷✵✵✾❪ ❘❈PP ❤❛s ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ ❜② ❬P❡❞r♦s♦✱ ❈✉♥❤❛✱ ❚❛✈❛r❡s ✷✵✶✸❪ ◮ ❝♦♠♣❛❝t ♥♦♥❝♦♥✈❡① ▼■◆▲P ❢♦r♠✉❧❛t✐♦♥ ◮ r❛♥❞♦♠✐③❡❞ ❣r❡❡❞② ❤❡✉r✐st✐❝ ✭●❘❆❙P✮ ●❧❡✐①♥❡r✱ ▼ü❧❧❡r✱ ▼❛❤❡r✱ P❡❞r♦s♦ ✕ ❊①❛❝t ▼❡t❤♦❞s ❢♦r ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ✺ ✴ ✸✷

  10. ♠✐♥ ✷ ✷ ✶ ✷ ✷ ✶ ✳ ✳ ✳ ❍♦♣❡❧❡ss t♦ s♦❧✈❡ ✇✐t❤ ❛ ❣❡♥❡r❛❧ ▼■◆▲P s♦❧✈❡r✦ ❈♦♠♣❛❝t ▼■◆▲P ❋♦r♠✉❧❛t✐♦♥ ◮ ❝♦♥s✐❞❡r ❡❛❝❤ r✐♥❣ ✐♥❞✐✈✐❞✉❛❧❧② ◮ ( x i , y i ) ❝❡♥t❡r ♦❢ r✐♥❣ i ◮ u i , j ∈ { ✵ , ✶ } ✐❢ r✐♥❣ i ✐s ❞✐r❡❝t❧② ♣❧❛❝❡❞ ✐♥s✐❞❡ r✐♥❣ j ◮ w i , r ∈ { ✵ , ✶ } ✐❢ r✐♥❣ i ✐s ❞✐r❡❝t❧② ♣❧❛❝❡❞ ✐♥s✐❞❡ r❡❝t❛♥❣❧❡ r ◮ φ r ∈ { ✵ , ✶ } ✐❢ r❡❝t❛♥❣❧❡ r ✐s ✉s❡❞ ●❧❡✐①♥❡r✱ ▼ü❧❧❡r✱ ▼❛❤❡r✱ P❡❞r♦s♦ ✕ ❊①❛❝t ▼❡t❤♦❞s ❢♦r ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ✻ ✴ ✸✷

  11. ❍♦♣❡❧❡ss t♦ s♦❧✈❡ ✇✐t❤ ❛ ❣❡♥❡r❛❧ ▼■◆▲P s♦❧✈❡r✦ ❈♦♠♣❛❝t ▼■◆▲P ❋♦r♠✉❧❛t✐♦♥ ◮ ❝♦♥s✐❞❡r ❡❛❝❤ r✐♥❣ ✐♥❞✐✈✐❞✉❛❧❧② ◮ ( x i , y i ) ❝❡♥t❡r ♦❢ r✐♥❣ i ◮ u i , j ∈ { ✵ , ✶ } ✐❢ r✐♥❣ i ✐s ❞✐r❡❝t❧② ♣❧❛❝❡❞ ✐♥s✐❞❡ r✐♥❣ j ◮ w i , r ∈ { ✵ , ✶ } ✐❢ r✐♥❣ i ✐s ❞✐r❡❝t❧② ♣❧❛❝❡❞ ✐♥s✐❞❡ r❡❝t❛♥❣❧❡ r ◮ φ r ∈ { ✵ , ✶ } ✐❢ r❡❝t❛♥❣❧❡ r ✐s ✉s❡❞ � ♠✐♥ φ r r ✷ � � x i � � x j �� ) ✷ ( w i , r + w j , r − ✶ ) � � ≥ ( r ext + r ext − ∀ i � = j ∀ r � � i j y i y j � � ✷ � � � � �� x i x j ) ✷ ( u i , k + u j , k − ✶ ) � � ≥ ( r ext + r ext − ∀ i � = j � = k � � i j y i y j � � ✳ ✳ ✳ ●❧❡✐①♥❡r✱ ▼ü❧❧❡r✱ ▼❛❤❡r✱ P❡❞r♦s♦ ✕ ❊①❛❝t ▼❡t❤♦❞s ❢♦r ❘❡❝✉rs✐✈❡ ❈✐r❝❧❡ P❛❝❦✐♥❣ ✻ ✴ ✸✷

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