◆♦♥❛♠❡ ♠❛♥✉s❝r✐♣t ◆♦✳ ✭✇✐❧❧ ❜❡ ✐♥s❡rt❡❞ ❜② t❤❡ ❡❞✐t♦r✮
❈♦♠♣❧❡①✐t② ❛♥❞ ✐♥❛♣♣r♦①✐♠❛❜✐❧✐t② r❡s✉❧ts ❢♦r t❤❡
P♦✇❡r ❊❞❣❡ ❙❡t ♣r♦❜❧❡♠
❙♦♥✐❛ ❚♦✉❜❛❧✐♥❡ · ❈❧❛✉❞✐❛ ❉✬❆♠❜r♦s✐♦ · ▲❡♦ ▲✐❜❡rt✐ · P✐❡rr❡✲▲♦✉✐s P♦✐r✐♦♥ · ❇❛r✉❝❤ ❙❝❤✐❡❜❡r · ❍❛❞❛s ❙❤❛❝❤♥❛✐ ❘❡❝❡✐✈❡❞✿ ❞❛t❡ ✴ ❆❝❝❡♣t❡❞✿ ❞❛t❡ ✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖ ❆❜str❛❝t ❲❡ ❝♦♥s✐❞❡r t❤❡ s✐♥❣❧❡ ❝❤❛♥♥❡❧ P▼❯ ♣❧❛❝❡♠❡♥t ♣r♦❜❧❡♠ ❝❛❧❧❡❞ t❤❡ P♦✇❡r ❊❞❣❡ ❙❡t ✭P❊❙✮ ♣r♦❜❧❡♠✳ ■♥ t❤✐s ✈❛r✐❛♥t ♦❢ t❤❡ P▼❯ ♣❧❛❝❡✲ ♠❡♥t ♣r♦❜❧❡♠✱ ✭s✐♥❣❧❡ ❝❤❛♥♥❡❧✮ P▼❯s ❛r❡ ♣❧❛❝❡❞ ♦♥ t❤❡ ❡❞❣❡s ♦❢ ❛♥ ❡❧❡❝tr✐❝❛❧ ♥❡t✇♦r❦✳ ❙✉❝❤ ❛ P▼❯ ♠❡❛s✉r❡s t❤❡ ❝✉rr❡♥t ❛❧♦♥❣ t❤❡ ❡❞❣❡ ♦♥ ✇❤✐❝❤ ✐t ✐s ♣❧❛❝❡❞ ❛♥❞ t❤❡ ✈♦❧t❛❣❡ ❛t ✐ts t✇♦ ❡♥❞♣♦✐♥ts✳ ❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ t❤❡ ♠✐♥✐♠✉♠ ♣❧❛❝❡♠❡♥t ♦❢ P▼❯s ✐♥ t❤❡ ♥❡t✇♦r❦ t❤❛t ❡♥s✉r❡s ✐ts ❢✉❧❧ ♦❜s❡r✈❛❜✐❧✐t②✱ ♥❛♠❡❧② ♠❡❛s✉r❡♠❡♥t ♦❢ ❛❧❧ t❤❡ ✈♦❧t❛❣❡s ❛♥❞ ❝✉rr❡♥ts✳ ❲❡ ♣r♦✈❡ t❤❛t P❊❙ ✐s ◆P✲❤❛r❞ t♦ ❛♣♣r♦①✐♠❛t❡ ✇✐t❤✐♥ ❛ ❢❛❝t♦r ✭✶✳✶✷✮✲ǫ✱ ❢♦r ❛♥② ǫ > 0✳ ❖♥ t❤❡ ♣♦s✐t✐✈❡ s✐❞❡ ✇❡ ♣r♦✈❡ t❤❛t P❊❙ ♣r♦❜❧❡♠ ✐s s♦❧✈❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❢♦r tr❡❡s ❛♥❞ ❣r✐❞s✳ ❑❡②✇♦r❞s P▼❯ ♣❧❛❝❡♠❡♥t ♣r♦❜❧❡♠ · P♦✇❡r ❊❞❣❡ ❙❡t · ◆P✲❤❛r❞♥❡ss · ✐♥❛♣♣r♦①✐♠❛❜✐❧✐t② ❚❤✐s ✇♦r❦ ✇❛s ❝❛rr✐❡❞ ♦✉t ❛s ♣❛rt ♦❢ t❤❡ ❙❖●❘■❉ ♣r♦❥❡❝t ✭✇✇✇✳s♦✲❣r✐❞✳❝♦♠✮✱ ❝♦✲❢✉♥❞❡❞ ❜② t❤❡ ❋r❡♥❝❤ ❛❣❡♥❝② ❢♦r ❊♥✈✐r♦♥♠❡♥t ❛♥❞ ❊♥❡r❣② ▼❛♥❛❣❡♠❡♥t ✭❆❉❊▼❊✮ ❛♥❞ ❞❡✈❡❧♦♣❡❞ ✐♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ❜❡t✇❡❡♥ ♣❛rt✐❝✐♣❛t✐♥❣ ❛❝❛❞❡♠✐❝ ❛♥❞ ✐♥❞✉str✐❛❧ ♣❛rt♥❡rs✳ ❙✳ ❚♦✉❜❛❧✐♥❡ ❯♥✐✈❡rs✐té P❛r✐s✲❉❛✉♣❤✐♥❡✱ P❙▲ ❘❡s❡❛r❝❤ ❯♥✐✈❡rs✐t②✱ ❈◆❘❙✱ ▲❆▼❙❆❉❊✱ ✼✺✵✶✻ P❛r✐s✱ ❋r❛♥❝❡ ❈◆❘❙ ▲■❳✱ ❊❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡✱ ✾✶✶✷✽ P❛❧❛✐s❡❛✉✱ ❋r❛♥❝❡ ❊✲♠❛✐❧✿ s♦♥✐❛✳t♦✉❜❛❧✐♥❡❅❞❛✉♣❤✐♥❡✳❢r ❈✳ ❉✬❆♠❜r♦s✐♦✱ ▲✳ ▲✐❜❡rt✐✱ P✳▲✳ P♦✐r✐♦♥ ❈◆❘❙ ▲■❳✱ ❊❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡✱ ✾✶✶✷✽ P❛❧❛✐s❡❛✉✱ ❋r❛♥❝❡ ❊✲♠❛✐❧✿ ♥❛♠❡❅❧✐①✳♣♦❧②t❡❝❤♥✐q✉❡✳❢r ❇✳ ❙❝❤✐❡❜❡r ■❇▼ ❚✳❏✳ ❲❛ts♦♥ ❘❡s❡❛r❝❤ ❈❡♥t❡r✱ ❨♦r❦t♦✇♥ ❍❡✐❣❤ts✱ ◆❨ ✶✵✺✾✽✳ ❊✲♠❛✐❧✿ s❜❛r❅✉s✳✐❜♠✳❝♦♠ ❍✳ ❙❤❛❝❤♥❛✐ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ❉❡♣❛rt♠❡♥t✱ ❚❡❝❤♥✐♦♥✱ ❍❛✐❢❛ ✸✷✵✵✵✱ ■sr❛❡❧✳ ❊✲♠❛✐❧✿ ❤❛❞❛s❅❝s✳t❡❝❤♥✐♦♥✳❛❝✳✐❧✷ ❙♦♥✐❛ ❚♦✉❜❛❧✐♥❡ ❡t ❛❧✳ ✶ ■♥tr♦❞✉❝t✐♦♥ ▼♦♥✐t♦r✐♥❣ ❛♥ ❡❧❡❝tr✐❝❛❧ ♥❡t✇♦r❦ ✐s ❛♥ ✐♠♣♦rt❛♥t ❛♥❞ ❝❤❛❧❧❡♥❣✐♥❣ t❛s❦✳ ❚♦ ❡♥s✉r❡ ♦♥❣♦✐♥❣ r❡❧✐❛❜✐❧✐t② ❛♥❞ q✉❛❧✐t② ♦❢ ❡❧❡❝tr✐❝✐t② s✉♣♣❧② t♦ ❝✉st♦♠❡rs✱ t❤❡ st❛t❡ ♦❢ t❤❡ ❡❧❡❝tr✐❝❛❧ ♥❡t✇♦r❦ ♠✉st ❜❡ ♠♦♥✐t♦r❡❞ ❝♦♥t✐♥✉♦✉s❧②✳ ❚❤❡ st❛t❡ ♦❢ s✉❝❤ ❛ ♥❡t✇♦r❦ ✐s ✉s✉❛❧❧② ❞❡✜♥❡❞ ❛s t❤❡ ✈❛❧✉❡s ♦❢ ❛❧❧ ✈♦❧t❛❣❡s ♦♥ ✐ts ♥♦❞❡s ❛♥❞ t❤❡ ❜r❛♥❝❤ ❝✉rr❡♥ts✳ P❤❛s♦r ♠❡❛s✉r❡♠❡♥t ✉♥✐ts ✭P▼❯s✮ ❛r❡ ♠♦♥✐t♦r✐♥❣ ❞❡✈✐❝❡s t❤❛t ❝❛♥ ❜❡ ✉s❡❞ ❢♦r t❤✐s ♣✉r♣♦s❡✳ P▼❯s ❛r❡ ❞❡s✐❣♥❡❞ t♦ ❜❡ ♣❧❛❝❡❞ ❛t ✭s✉❜✮st❛t✐♦♥s ❛♥❞ ❝❛♥ ♠❡❛s✉r❡ t❤❡✐r ✈♦❧t❛❣❡ ❛♥❞ t❤❡ ❝✉rr❡♥t ♦♥ ❛❧❧ t❤❡✐r ♦✉t❣♦✐♥❣ tr❛♥s♠✐ss✐♦♥ ❧✐♥❡s ✭▼❛♥♦✉s❛❦✐s ❡t ❛❧✳✱ ✷✵✶✷✮✳ ◆♦t❡ t❤❛t ✐t ✐s ♥♦t ♥❡❝✲ ❡ss❛r② t♦ ♣❧❛❝❡ P▼❯s ❛t ❛❧❧ st❛t✐♦♥s✱ ❛s s♦♠❡ ♦❢ t❤❡ ❝✉rr❡♥ts ❛♥❞ ✈♦❧t❛❣❡s ❝❛♥ ❜❡ ❞❡❞✉❝❡❞ ✉s✐♥❣ ❖❤♠ ❛♥❞ ❑✐r❝❤❤♦✛ ▲❛✇s✳ ❉✉❡ t♦ t❤❡✐r ❤✐❣❤ ❝♦st✱ ✜♥❞✲ ✐♥❣ ❛ ♣❧❛❝❡♠❡♥t ♦❢ P▼❯s t❤❛t ♠✐♥✐♠✐③❡s t❤❡✐r ♥✉♠❜❡r ✇❤✐❧❡ st✐❧❧ ❡♥s✉r✐♥❣ ♠♦♥✐t♦r✐♥❣ ♦❢ t❤❡ ✇❤♦❧❡ ♥❡t✇♦r❦ ✐s ❛♥ ✐♠♣♦rt❛♥t ♣r♦❜❧❡♠✳ ▲❡t t❤❡ ❡❧❡❝tr✐❝❛❧ ♥❡t✇♦r❦ ❜❡ ♠♦❞❡❧❧❡❞ ❜② ❛ ❣r❛♣❤ ✇❤❡r❡ t❤❡ ✈❡rt✐❝❡s r❡♣✲ r❡s❡♥t ❡❧❡❝tr✐❝❛❧ ♥♦❞❡s ❛♥❞ t❤❡ ❡❞❣❡s ❝♦rr❡s♣♦♥❞ t♦ tr❛♥s♠✐ss✐♦♥ ❧✐♥❡s ❥♦✐♥✐♥❣ t✇♦ ♥♦❞❡s✳ ❚❤❡ ✭♠✉❧t✐ ❝❤❛♥♥❡❧✮ P▼❯ ♣❧❛❝❡♠❡♥t ♣r♦❜❧❡♠✱ ❛❧s♦ ❝❛❧❧❡❞ t❤❡ ♣♦✇❡r ❞♦♠✐♥❛t✐♥❣ s❡t ✭P❉❙✮ ♣r♦❜❧❡♠✱ ❝♦♥s✐sts ♦❢ ✜♥❞✐♥❣ ❛ ♠✐♥✐♠✉♠ ♥✉♠✲ ❜❡r ♦❢ P▼❯s t♦ ✐♥st❛❧❧ ♦♥ t❤❡ ✈❡rt✐❝❡s s✉❝❤ t❤❛t ❛❧❧ t❤❡ ❣r❛♣❤ ✐s ♦❜s❡r✈❡❞✱ t❤❛t ✐s✱ ❛❧❧ ✈♦❧t❛❣❡s ❛♥❞ ❝✉rr❡♥ts ❛r❡ ♠❡❛s✉r❡❞✳ ❇r✉❡♥✐ ❛♥❞ ❍❡❛t❤ ✭❇r✉❡♥✐ ❛♥❞ ❍❡❛t❤✱ ✷✵✵✺✮ s❤♦✇❡❞ t❤❛t t❤❡ ♦❜s❡r✈❛❜✐❧✐t② ♦❢ ❛ ❣r❛♣❤ ❜② ✭♠✉❧t✐ ❝❤❛♥♥❡❧✮ P▼❯s ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② t✇♦ r✉❧❡s✿ ✭R1✮ ✐❢ ❛ P▼❯ ✐s ♣❧❛❝❡❞ ❛t ❛ ✈❡rt❡① t❤❡♥ t❤✐s ✈❡rt❡① ❛♥❞ ❛❧❧ ✐ts ♥❡✐❣❤❜♦✉rs ❛r❡ ♦❜s❡r✈❡❞❀ ✭R2✮ ✐❢ ❛❧❧ t❤❡ ♥❡✐❣❤❜♦✉rs ♦❢ ❛♥ ♦❜s❡r✈❡❞ ✈❡rt❡① ❡①❝❡♣t ♦♥❡ ❛r❡ ♦❜s❡r✈❡❞✱ t❤❡♥ t❤✐s ❧❛tt❡r ✐s ♦❜s❡r✈❡❞✳ ❙❡✈❡r❛❧ ❝♦♠♣❧❡①✐t② r❡s✉❧ts ❤❛✈❡ ❜❡❡♥ s❤♦✇♥ ❢♦r P❉❙✿ ◆P✲❝♦♠♣❧❡t❡♥❡ss ♣r♦♦❢s ❢♦r ❜✐♣❛rt✐t❡✱ ❝♦❣r❛♣❤s ✭❍❛②♥❡s ❡t ❛❧✳✱ ✷✵✵✷✮✱ ❛♥❞ ♣❧❛♥❛r ❜✐♣❛rt✐t❡ ❣r❛♣❤s ✭❇r✉❡♥✐ ❛♥❞ ❍❡❛t❤✱ ✷✵✵✺✮✱ ❛♥❞ ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ❛❧❣♦r✐t❤♠s ❢♦r tr❡❡s✱ ❣r✐❞s ✭❉♦r✲ ✢✐♥❣ ❛♥❞ ❍❡♥♥✐♥❣✱ ✷✵✵✻✮✱ ❜❧♦❝❦ ❣r❛♣❤s ✭❳✉ ❡t ❛❧✳✱ ✷✵✵✻✮✱ ❛♥❞ ❜♦✉♥❞❡❞ tr❡❡✇✐❞t❤ ✭●✉♦ ❡t ❛❧✳✱ ✷✵✵✺✮✳ ❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❛♥❞ ❤❛r❞♥❡ss r❡s✉❧ts ❛r❡ ♣r❡✲ s❡♥t❡❞ ✐♥ ✭❆❛③❛♠✐ ❛♥❞ ❙t✐❧♣✱ ✷✵✵✼✮✿ O(√n)✲❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r ♣❧❛♥❛r ❣r❛♣❤s ❛♥❞ ◆P✲❤❛r❞♥❡ss ♦❢ ❛♣♣r♦①✐♠❛❜✐❧✐t② ✇✐t❤✐♥ ❛ ❢❛❝t♦r 2log1−ǫn ✳ ❆❧s♦✱ ✈❛r✐♦✉s s♦✲ ❧✉t✐♦♥ ♠❡t❤♦❞s ❤❛✈❡ ❜❡❡♥ ♣r♦♣♦s❡❞ t♦ s♦❧✈❡ t❤❡ P▼❯ ♣❧❛❝❡♠❡♥t ♣r♦❜❧❡♠ ✭▼❛♥♦✉s❛❦✐s ❡t ❛❧✳✱ ✷✵✶✷✮✳ ❙♦♠❡ ♦❢ t❤❡ P▼❯s ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ ♠❛r❦❡t ❤❛✈❡ ❛ ❧✐♠✐t❡❞ ♥✉♠❜❡r ♦❢ ❝❤❛♥✲ ♥❡❧s✳ ❆ P▼❯ ✇✐t❤ k ❝❤❛♥♥❡❧s ✐♥st❛❧❧❡❞ ❛t ❛ ✈❡rt❡① v ❝❛♥ ♦❜s❡r✈❡ ♦♥❧② v ❛♥❞ k ♦❢ ✐ts ♥❡✐❣❤❜♦rs ✭❛♥❞ t❤❡ ❡❞❣❡s ❝♦♥♥❡❝t✐♥❣ v t♦ t❤❡s❡ k ♥❡✐❣❤❜♦rs✮✳ ❚❤❡ ✐❞❡♥✲ t✐t② ♦❢ t❤❡s❡ k ♥❡✐❣❤❜♦rs ✐s ❞❡t❡r♠✐♥❡❞ ❛t t❤❡ t✐♠❡ ♦❢ ✐♥st❛❧❧❛t✐♦♥✳ ❲❡ ❝♦♥s✐❞❡r P▼❯s ✇✐t❤ ❛ s✐♥❣❧❡ ❝❤❛♥♥❡❧ t❤❛t ✇❤❡♥ ♣❧❛❝❡❞ ❛t ✈❡rt❡① v ♦❜s❡r✈❡ ♦♥❧② v ❛♥❞ ♦♥❡ ♦❢ ✐ts ♥❡✐❣❤❜♦rs✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ s✐♥❣❧❡ ♥❡✐❣❤❜♦r ♦❢ v ♦❜s❡r✈❡❞ ❜② t❤❡ P▼❯ ✐s u✳ ❙✐♥❝❡ ❜♦t❤ v ❛♥❞ u ❛r❡ ♦❜s❡r✈❡❞✱ ✇❡ ❝❛♥ ❡q✉✐✈❛❧❡♥t❧② ✈✐❡✇ t❤✐s ❛s ♣❧❛❝✐♥❣ t❤❡ P▼❯ ♦♥ t❤❡ ❡❞❣❡ (v, u)✱ ✇❤❡r❡ ❛ P▼❯ ♣❧❛❝❡❞ ♦♥ ❛♥ ❡❞❣❡ ✐s ❛ss✉♠❡❞ t♦ ♦❜s❡r✈❡ ❜♦t❤ ✐ts ❡♥❞♣♦✐♥ts✳ ❚❤❡ ♦❜s❡r✈❛❜✐❧✐t② ♦❢ ❛ ❣r❛♣❤ ✉s✐♥❣ s✐♥❣❧❡ ❝❤❛♥♥❡❧ P▼❯s ❝❛♥ ❛❧s♦ ❜❡ ❞❡t❡r✲ ♠✐♥❡❞ ❜② t✇♦ r✉❧❡s s✐♠✐❧❛r t♦ t❤❡ ♠✉❧t✐ ❝❤❛♥♥❡❧ ❝❛s❡✳ ❚❤❡ s❡❝♦♥❞ r✉❧❡ ✭R2✮ ✐s t❤❡ s❛♠❡ ❛s ❛❜♦✈❡ ✇❤✐❧❡ ✭R1✮ ♥❡❡❞s t♦ ❜❡ ♠♦❞✐✜❡❞ t♦ ❛❝❝♦✉♥t ❢♦r t❤❡ s✐♥❣❧❡ ❝❤❛♥♥❡❧ t♦✿ ✭R1E✮ ✐❢ ❛ P▼❯ ✐s ✐♥st❛❧❧❡❞ ♦♥ ❛♥ ❡❞❣❡ t❤❡♥ ✐ts t✇♦ ❡♥❞♣♦✐♥t
❈♦♠♣❧❡①✐t② ❛♥❞ ✐♥❛♣♣r♦①✐♠❛❜✐❧✐t② r❡s✉❧ts ❢♦r t❤❡ P♦✇❡r ❊❞❣❡ ❙❡t ♣r♦❜❧❡♠ ✸ ✈❡rt✐❝❡s ❛r❡ ♦❜s❡r✈❡❞✳ ❚❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ✜♥❞ ❛ ♠✐♥✐♠✉♠ ♣❧❛❝❡♠❡♥t ♦❢ s✐♥❣❧❡ ❝❤❛♥♥❡❧ P▼❯s t❤❛t ❡♥s✉r❡s t❤❡ ♦❜s❡r✈❛❜✐❧✐t② ♦❢ t❤❡ ✇❤♦❧❡ ❣r❛♣❤✳ ❲❡ ❝❛❧❧ t❤✐s ♣r♦❜❧❡♠ t❤❡ P♦✇❡r ❊❞❣❡ ❙❡t ✭P❊❙✮ ♣r♦❜❧❡♠✳ ❊♠❛♠✐ ❡t ❛❧✳ ♣r♦♣♦s❡❞ ❛ ❜✐♥❛r② ❧✐♥❡❛r ♣r♦❣r❛♠ ❢♦r t❤✐s P▼❯ ♣❧❛❝❡♠❡♥t ♣r♦❜❧❡♠ ✭❊♠❛♠✐ ❛♥❞ ❆❜✉r✱ ✷✵✶✵✮✱ ✇❤✐❝❤ ❝♦♥s✐❞❡rs ♦♥❧② r✉❧❡ R1E✱ ❛♥❞ ❤❡♥❝❡ t✉r♥s ♦✉t t♦ ❜❡ ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♠✐♥✐♠✉♠ ❡❞❣❡ ❝♦✈❡r ♣r♦❜❧❡♠✱ ✇❤✐❝❤ ❤❛s ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ s♦❧✉t✐♦♥✳ ❚❤❡ ❛✉t❤♦rs ❞✐s❝✉ss❡❞ t❤❡ ❝♦♥s✐❞❡r❛t✐♦♥ ♦❢ ❛ r❡✲ str✐❝t❡❞ ✈❡rs✐♦♥ ♦❢ R2 ✐♥ ✭❊♠❛♠✐ ❡t ❛❧✳✱ ✷✵✵✽✮✳ ■♥ ✭P♦✐r✐♦♥ ❡t ❛❧✳✱ ✷✵✶✻✮✱ ✇❡ st✉❞✐❡❞ t❤✐s ♣r♦❜❧❡♠ ❢r♦♠ ❛ ♣r❛❝t✐❝❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✳ ❲❡ ✜rst ♣r♦♣♦s❡❞ ❛ ♥❛t✲ ✉r❛❧❧② ✐t❡r❛t✐✈❡ ✐♥❞❡① ❜✐♥❛r② ❧✐♥❡❛r ♠♦❞❡❧ t❤❛t t✉r♥s ♦✉t t♦ ❜❡ t♦♦ ❧❛r❣❡ ❢♦r ♣r❛❝t✐❝❛❧ ♣✉r♣♦s❡s✳ ❯s✐♥❣ ❛ ✜①❡❞ ♣♦✐♥t ❛r❣✉♠❡♥t✱ ✇❡ r❡♠♦✈❡❞ t❤❡ ✐t❡r❛t✐♦♥ ✐♥❞✐❝❡s ❛♥❞ ♦❜t❛✐♥❡❞ ❛ ❜✐❧❡✈❡❧ ❢♦r♠✉❧❛t✐♦♥✳ ❲❡ t❤❡♥ r❡❢♦r♠✉❧❛t❡❞ t❤❡ ❧❛tt❡r t♦ ❛ s✐♥❣❧❡✲❧❡✈❡❧ ♠✐①❡❞✲✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠✱ ✇❤✐❝❤ ♣❡r❢♦r♠s ❜❡tt❡r t❤❛♥ t❤❡ ♥❛t✉r❛❧ ❢♦r♠✉❧❛t✐♦♥✳ ❲❡ t❤❡♥ ♣r♦✈✐❞❡❞ ❛ ❝✉tt✐♥❣ ♣❧❛♥❡ ❛❧❣♦r✐t❤♠ t❤❛t s♦❧✈❡s t❤❡ ❜✐❧❡✈❡❧ ♣r♦❣r❛♠ ♠✉❝❤ ❢❛st❡r t❤❛♥ ❛♥ ♦✛✲t❤❡✲s❤❡❧❢ s♦❧✈❡r ❝❛♥ s♦❧✈❡ t❤❡ ♣r❡✲ ✈✐♦✉s ♠♦❞❡❧s✳ ■♥ t❤✐s ♣❛♣❡r✱ ✇❡ st✉❞② t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤✐s ♣r♦❜❧❡♠ ✇❤❡♥ ❜♦t❤ r✉❧❡s ♦❢ ♦❜s❡r✈❛❜✐❧✐t② ❛r❡ ❝♦♥s✐❞❡r❡❞✳ ❲❡ s❤♦✇ t❤❛t t❤❡ P❊❙ ♣r♦❜❧❡♠ ✐s ◆P✲❤❛r❞ t♦ ❛♣♣r♦①✐♠❛t❡ ✇✐t❤✐♥ ❛ ❢❛❝t♦r ✭✶✳✶✷✮✲ǫ✱ ❢♦r ❛♥② ǫ > 0✳ ❲❡ ❛❧s♦ ♣r♦✈❡ t❤❛t ✐t ✐s ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ s♦❧✈❛❜❧❡ ❢♦r tr❡❡s ❛♥❞ ❣r✐❞s✳ ✷ Pr♦❜❧❡♠ st❛t❡♠❡♥ts ❛♥❞ ♣r❡❧✐♠✐♥❛r✐❡s ▲❡t G = (V, E) ❜❡ ❛♥ ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤ ✇✐t❤ |V | = n✳ ❉❡♥♦t❡ ❜② N(v) t❤❡ s❡t ♦❢ ♥❡✐❣❤❜♦rs ♦❢ ❛ ✈❡rt❡① v ∈ V ✳ G ✐s k✲r❡❣✉❧❛r ✐❢ ❛❧❧ ✈❡rt✐❝❡s ❤❛✈❡ ❛ ❞❡❣r❡❡ k✳ ●✐✈❡♥ ❛ s✉❜s❡t S ⊆ V ✱ ❞❡✜♥❡ t❤❡ s❡t B(S) ⊆ V ♦❢ ✧♦❜s❡r✈❡❞✧ ✈❡rt✐❝❡s st❛rt✐♥❣ ❛t S✳ ■♥✐t✐❛❧❧②✱ B(S) ✐s s❡t t♦ ❜❡ S✳ ❚❤❡♥✱ ❛s ❧♦♥❣ ❛s t❤❡r❡ ❡①✐sts ❛ ✈❡rt❡① v /∈ B(S) s✉❝❤ t❤❛t v ❤❛s ❛ ♥❡✐❣❤❜♦r u ∈ B(S) ❛♥❞ N(u) \ {v} ⊆ B(S)✱ ✐✳❡✳✱ v ✐s t❤❡ ♦♥❧② ♥❡✐❣❤❜♦r ♦❢ u t❤❛t ✐s ♥♦t ✐♥ B(S)✱ t❤❡♥ v ✐s ❛❞❞❡❞ t♦ B(S) ✭r✉❧❡ R2✮✳ ■♥ t❤✐s ❝❛s❡✱ ✇❡ s❛② t❤❛t u ✐s t❤❡ ♣❛r❡♥t ♦❢ v ❛♥❞ v ✐s t❤❡ ❝❤✐❧❞ ♦❢ u✳ ◆♦t❡ t❤❛t ❡❛❝❤ ♣❛r❡♥t ❤❛s ❛t ♠♦st ♦♥❡ ❝❤✐❧❞ ❛♥❞ ❡❛❝❤ ❝❤✐❧❞ ❤❛s ❡①❛❝t❧② ♦♥❡ ♣❛r❡♥t✳ ❲❡ ❡①t❡♥❞ t❤❡ ♣❛r❡♥t✴❝❤✐❧❞ r❡❧❛t✐♦♥s t♦ ❛♥❝❡st♦r✴❞❡s❝❡♥❞❛♥t r❡❧❛t✐♦♥s ✐♥ ❛ ♥❛t✉r❛❧ ✇❛②✳ ■♥ t❤❡ P♦✇❡r ❊❞❣❡ ❙❡t ✭P❊❙✮ ♣r♦❜❧❡♠✱ ✇❡ ❛r❡ ❣✐✈❡♥ ❛ s✉❜s❡t ♦❢ ❡❞❣❡s F ⊆ E ❛♥❞ t❤❡ ✐♥✐t✐❛❧ s❡t ♦❢ ♦❜s❡r✈❡❞ ✈❡rt✐❝❡s S(F ) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ❡♥❞♣♦✐♥ts ♦❢ t❤❡ ❡❞❣❡s ✐♥ F ✭r✉❧❡ R1E✮✳ B(S(F )) ✐s ❝♦♥str✉❝t❡❞ ❛s ❞❡s❝r✐❜❡ ❛❜♦✈❡ ✭r✉❧❡ R2✮✳ ❲❡ s❛② t❤❛t F ✐s ❛ P♦✇❡r ❊❞❣❡ ❙❡t ✭P❊❙✮ ✐❢ B(S(F )) = V ✳ ❚❤❡ P♦✇❡r ❊❞❣❡ ❙❡t ♣r♦❜❧❡♠ ✐s t♦ s❡❧❡❝t ❛ P❊❙ F ♦❢ ♠✐♥✐♠✉♠ ❝❛r❞✐♥❛❧✐t②✳
✹ ❙♦♥✐❛ ❚♦✉❜❛❧✐♥❡ ❡t ❛❧✳ P♦✇❡r ❊❞❣❡ ❙❡t Pr♦❜❧❡♠ ■♥♣✉t✿ ❆ ❣r❛♣❤ G = (V, E)✳ ❖✉t♣✉t✿ ❆ ♠✐♥✐♠✉♠ ❝❛r❞✐♥❛❧✐t② s❡t F ⊂ E s✉❝❤ t❤❛t B(S(F )) = V ✳ ❚♦ ❡st❛❜❧✐s❤ t❤❡ ◆P✲❤❛r❞♥❡ss ❛♣♣r♦①✐♠❛t✐♦♥ r❡s✉❧t✱ ✇❡ ✉s❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❛♥ E✲r❡❞✉❝t✐♦♥ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✳ E✲r❡❞✉❝t✐♦♥ ❈♦♥s✐❞❡r ❛♥ ◆P ❖♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❛♥❞ ❛♥ ✐♥st❛♥❝❡ I ♦❢ t❤✐s ♣r♦❜❧❡♠✳ ❲❡ ❞❡♥♦t❡ |I| t❤❡ s✐③❡ ♦❢ I✱ opt(I) t❤❡ ♦♣t✐♠✉♠ ✈❛❧✉❡ ♦❢ I✱ ❛♥❞ val(I, S) t❤❡ ✈❛❧✉❡ ♦❢ ❛ ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥ S ♦❢ I✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❢❛❝t♦r r(I, S)♦❢ S ✐s ❣✐✈❡♥ ❜② maxnval(I,S)opt(I) ,val(I,S)opt(I) o✳ ❚❤❡ ❡rr♦r ♦❢ S✱ ♥♦t❡❞ ε(I, S)✱ ✐s ❞❡✜♥❡❞ ❜② ε(I, S) = r(I, S) − 1. ❋♦r ❛ ❢✉♥❝t✐♦♥ f✱ ❛♥ ❛❧❣♦r✐t❤♠ ✐s ❛♥ f(n)✲❛♣♣r♦①✐♠❛t✐♦♥✱ ✐❢ ❢♦r ❡✈❡r② ✐♥✲ st❛♥❝❡ I ♦❢ t❤❡ ♣r♦❜❧❡♠✱ ✐t r❡t✉r♥s ❛ s♦❧✉t✐♦♥ S s✉❝❤ t❤❛t r(I, S) ≤ f(|I|). ❑❤❛♥♥❛ ❡t ❛❧✳ ✐♥tr♦❞✉❝❡❞ t❤❡ ♥♦t✐♦♥ ♦❢ ❛♥ E✲r❡❞✉❝t✐♦♥ ✭❡rr♦r✲♣r❡s❡r✈✐♥❣ r❡❞✉❝t✐♦♥✮ ✐♥ ✭❑❤❛♥♥❛ ❡t ❛❧✳✱ ✶✾✾✾✮✳ ❆ ♣r♦❜❧❡♠ Π ✐s ❝❛❧❧❡❞ E✲r❡❞✉❝✐❜❧❡ t♦ ❛ ♣r♦❜❧❡♠ Π′✱ ✐❢ t❤❡r❡ ❡①✐st ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❝♦♠♣✉t❛❜❧❡ ❢✉♥❝t✐♦♥s f✱ g ❛♥❞ ❛ ❝♦♥st❛♥t β s✉❝❤ t❤❛t ✕ f ♠❛♣s ❛♥ ✐♥st❛♥❝❡ I ♦❢ Π t♦ ❛♥ ✐♥st❛♥❝❡ I′ ♦❢ Π′ s✉❝❤ t❤❛t opt(I) ❛♥❞ opt(I′)❛r❡ r❡❧❛t❡❞ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ ❢❛❝t♦r✱ ✐✳❡✳ t❤❡r❡ ❡①✐sts ❛ ♣♦❧②♥♦♠✐❛❧ p s✉❝❤ t❤❛t opt(I′) ≤ p(|I|)opt(I)✱ ✕ g ♠❛♣s ❛♥② s♦❧✉t✐♦♥ S′ ♦❢ I′ t♦ ♦♥❡ s♦❧✉t✐♦♥ S ♦❢ I s✉❝❤ t❤❛t ε(I, S) ≤ βε(I′, S′)✳ ❆♥ ✐♠♣♦rt❛♥t ♣r♦♣❡rt② ♦❢ ❛♥ E✲r❡❞✉❝t✐♦♥ ✐s t❤❛t ✐t ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ ✉♥✐✲ ❢♦r♠❧② t♦ ❛❧❧ ❧❡✈❡❧s ♦❢ ❛♣♣r♦①✐♠❛❜✐❧✐t②❀ t❤❛t ✐s✱ ✐❢ Π ✐s E✲r❡❞✉❝✐❜❧❡ t♦ Π′ ❛♥❞ Π′❜❡❧♦♥❣s t♦ C t❤❡♥ Π ❜❡❧♦♥❣s t♦ C ❛s ✇❡❧❧✱ ✇❤❡r❡ C ✐s ❛ ❝❧❛ss ♦❢ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ✇✐t❤ ❛♥② ❦✐♥❞ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❣✉❛r❛♥t❡❡ ✭s❡❡ ❛❧s♦ ✭❑❤❛♥♥❛ ❡t ❛❧✳✱ ✶✾✾✾✮✮✳ ✸ ◆P✲❍❛r❞♥❡ss ♦❢ ❆♣♣r♦①✐♠❛t✐♦♥ ❢♦r P❊❙ ❲❡ ♣r♦✈❡ t❤❛t P❊❙ ✐s ❤❛r❞ t♦ ❛♣♣r♦①✐♠❛t❡ ✇✐t❤✐♥ s♦♠❡ ❝♦♥st❛♥t✱ ✉♥❧❡ss P❂◆P✳ ❚♦ t❤✐s ❡♥❞✱ ✇❡ ❞❡✜♥❡ ❛♥ E✲r❡❞✉❝t✐♦♥ ❢r♦♠ ▼✐♥ ❱❡rt❡① ❈♦✈❡r r❡str✐❝t❡❞ t♦ ✸✲r❡❣✉❧❛r ❣r❛♣❤s✳ ❚❤✐s ♣r♦❜❧❡♠ ✐s ◆P✲❤❛r❞ t♦ ❛♣♣r♦①✐♠❛t❡ ✇✐t❤✐♥ ❛ ❢❛❝t♦r ♦❢ ✶✳✸✻ ✭❉✐♥✉r ❛♥❞ ❙❛❢r❛✱ ✷✵✵✺❀ ❋❡✐❣❡✱ ✷✵✵✸✮✳ ❲❡ ✜rst ❞❡s❝r✐❜❡ ♦✉r r❡❞✉❝t✐♦♥✳ ▲❡t I ❜❡ ❛♥ ✐♥st❛♥❝❡ ♦❢ ▼✐♥ ❱❡rt❡① ❈♦✈❡r ❢♦r♠❡❞ ❜② ❛ ✸✲r❡❣✉❧❛r ❣r❛♣❤ G = (V, E)✳ ❲❡ ❝♦♥str✉❝t ❛♥ ✐♥st❛♥❝❡ I′ ♦❢ P❊❙ ❝♦♥s✐st✐♥❣ ♦❢ ❛ ❣r❛♣❤ G′ = (V′, E′)❛s ❢♦❧❧♦✇s ✭s❡❡ ❋✐❣✳ ✷✮✳ ❲❡ ❛ss♦❝✐❛t❡ t♦ ❡❛❝❤ ✈❡rt❡① v ∈ V ❛ ❣❛❞❣❡t G v ✐♥ G′ ✭s❡❡ ❋✐❣✳ ✶✮✳ ❋♦r v ∈ V ✱ G v ✐s ❝♦♠♣♦s❡❞ ♦❢ ✶✵ ✈❡rt✐❝❡s {v0, . . . , v9}✳ ❋♦r ❡❛❝❤ ❡❞❣❡ (v, u) ∈ E✱ ✇❡ ❛❞❞ t❤❡ ❡❞❣❡ (vi, uj)t♦ E′ ❢♦r ♦♥❡ i ∈ {1, 2, 3}, ❛♥❞ uj∈ Gu ❢♦r ♦♥❡ j ∈ {1, 2, 3}✳ ❚❤❡ ✈❡rt❡① s✉❜s❡ts {v1, . . . , v5} ❛♥❞ {v6, . . . , v9} ❢♦r♠ t✇♦ ❝❧✐q✉❡s ❧✐♥❦❡❞ t♦ ❡❛❝❤ ♦t❤❡r ❜② t❤❡ ❡❞❣❡s {vi, vi+5}, i = 1, . . . , 4✳ ❲❡
❈♦♠♣❧❡①✐t② ❛♥❞ ✐♥❛♣♣r♦①✐♠❛❜✐❧✐t② r❡s✉❧ts ❢♦r t❤❡ P♦✇❡r ❊❞❣❡ ❙❡t ♣r♦❜❧❡♠ ✺ ❛❧s♦ ❛❞❞ t❤❡ ❡❞❣❡ (v0, v5)t♦ E′✳ ▲❡t u, w, t ❜❡ t❤❡ t❤r❡❡ ♥❡✐❣❤❜♦rs ♦❢ v ∈ V ✳ ❲✐t❤♦✉t ❧♦s❡ ♦❢ ❣❡♥❡r❛❧✐t②✱ ✇❡ ❛ss✉♠❡ t❤❛t (v1, u1), (v2, w2)✱ ❛♥❞ (v3, t3)❛r❡ ✐♥ E′✱ ❢♦r v i∈ Gv, i = 1, 2, 3✱ u1∈ Gu✱ w2∈ Gw❛♥❞ t3∈ Gt✳ ❚❤❡ ✈❡rt✐❝❡s u1, w2✱ ❛♥❞ t3 ❛r❡ ❝❛❧❧❡❞ ✧♥❡✐❣❤❜♦r ✈❡rt✐❝❡s✧ ♦❢ Gv ❛♥❞ v1, v2, v3 ✐ts ❥✉♥❝t✐♦♥ ✈❡rt✐❝❡s✳ v1 v2 v3 v4 v6 v7 v8 v9 u1 w2 t3 v5 v0 ❋✐❣✳ ✶ ●❛❞❣❡t Gv ❛ss♦❝✐❛t❡❞ t♦ ❛ ✈❡rt❡① v ∈ V ✳ u1, w2 ❛♥❞ t3 ❛r❡ ✈❡rt✐❝❡s ♦❢ Gu✱ Gw ❛♥❞ Gtr❡s♣❡❝t✐✈❡❧② ✇❤❡r❡ u, w ❛♥❞ t ❛r❡ t❤❡ ❛❞❥❛❝❡♥t ✈❡rt✐❝❡s ♦❢ v ✐♥ G✳ v u t w
❀
v1 v2 v3 u1 u3 u2 w2 w3 w1 t3 t2 t1G
vG
uG
wG
t ❋✐❣✳ ✷ ●❛❞❣❡t ❛ss♦❝✐❛t❡❞ t♦ ❛ ✈❡rt❡①✱ u ∈ V ✇✐t❤ t❤r❡❡ ♥❡✐❣❤❜♦rs v, w ❛♥❞ t ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✇❡ ♣r❡s❡♥t ❢♦✉r r❡s✉❧ts ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ P▼❯s t♦ ✐♥st❛❧❧ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞❡❣r❡❡ ♦❢ ✈❡rt✐❝❡s ❛♥❞ t❤❡ ♦❜s❡r✈❛❜✐❧✐t② st❛t✉s ♦❢ t❤❡ ❥✉♥❝t✐♦♥ ❛♥❞✴♦r ♥❡✐❣❤❜♦r ✈❡rt✐❝❡s✳✻ ❙♦♥✐❛ ❚♦✉❜❛❧✐♥❡ ❡t ❛❧✳ ▲❡♠♠❛ ✶ ▲❡t G = (V, E) ❜❡ ❛ ❝♦♥♥❡❝t❡❞ ❣r❛♣❤ ✇✐t❤ ❛t ❧❡❛st ❢♦✉r ✈❡rt✐❝❡s✱ s✉❝❤ t❤❛t ♦♥❧② ♦♥❡ P▼❯ ✐s ♣❧❛❝❡❞ ❛t ❛ ❣✐✈❡♥ ❡❞❣❡ {u, v} ♦❢ t❤❡ ❣r❛♣❤✳ ■❢ ♥♦ ✈❡rt❡① ♦❢ G ❤❛s ❞❡❣r❡❡ ✷✱ t❤❡♥ ✐t ✐s ✐♠♣♦ss✐❜❧❡ t❤❛t ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ G ❛r❡ ♦❜s❡r✈❡❞✳ Pr♦♦❢ ❇② t❤❡ P▼❯ ♣❧❛❝❡❞ ❛t {u, v}✱ t❤❡ ✈❡rt✐❝❡s u ❛♥❞ v ❛r❡ ♦❜s❡r✈❡❞ ✉s✐♥❣ R1E✳ ❙✐♥❝❡ ♥♦ ✈❡rt❡① ✐♥ G ❤❛s ❞❡❣r❡❡ ✷✱ d(u) ♦r d(v) ✐s ❛t ❧❡❛st ✸ ✭❡✐t❤❡r ❜♦t❤ ❤❛✈❡ ❞❡❣r❡❡ ❛t ❧❡❛st ✸ ♦r ♦♥❡ ❤❛s ❞❡❣r❡❡ ✶ ❛♥❞ t❤❡ ♦t❤❡r ❞❡❣r❡❡ ❛t ❧❡❛st ✸✮✳ ▲❡t ✉s ❛ss✉♠❡ t❤❛t d(u) ≥ 3✳ ❚❤❡ ✈❡rt❡① u ❤❛s t❤❡♥ ❛t ❧❡❛st ✷ ❛❞❥❛❝❡♥t ✈❡rt✐❝❡s t❤❛t ❛r❡ ♥♦t ♦❜s❡r✈❡❞✳ ❚❤❡ s❛♠❡ ❤♦❧❞s ✐❢ d(v) ≥ 3✳ ❚❤❡♥ R2 ❝❛♥♥♦t ❜❡ ✉s❡❞ ❛t ❡✐t❤❡r u ♦r v✳ ❚❤❡♥ G ✐s ♥♦t ♦❜s❡r✈❡❞✳ ❉❡✜♥✐t✐♦♥ ✶ ●✐✈❡♥ ❛ ❣❛❞❣❡t Gv✐♥ G′✱ ❛ ❥✉♥❝t✐♦♥ ✈❡rt❡① vi♦❢ Gv✱ i ∈ {1, 2, 3}✱ ✐s ✧♦❜s❡r✈❡❞ ❡①t❡r♥❛❧❧②✧ ✐❢ vi❝❛♥ ❜❡ ♦❜s❡r✈❡❞ ❜② ❛♣♣❧②✐♥❣ R2 ♦♥ t❤❡ ♥❡✐❣❤❜♦r ✈❡rt❡① ♦❢ Gv ❛❞❥❛❝❡♥t t♦ vi✳ ▲❡♠♠❛ ✷ ●✐✈❡♥ ❛ ❣❛❞❣❡t Gv ♦❢ G′✱ t❤❡ ♣❧❛❝❡♠❡♥t ♦❢ t✇♦ P▼❯s ✐s s✉✣❝✐❡♥t t♦ ♦❜s❡r✈❡ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ Gv ❛♥❞ ✐ts ♥❡✐❣❤❜♦r ✈❡rt✐❝❡s✳ ❋✉rt❤❡r♠♦r❡✱ ✉♥❧❡ss t❤❡ ♥❡✐❣❤❜♦r ✈❡rt✐❝❡s ❛r❡ ♦❜s❡r✈❡❞ ❡①t❡r♥❛❧❧②✱ t✇♦ P▼❯s ❛r❡ ❛❧s♦ ♥❡❝❡ss❛r② t♦ ♦❜s❡r✈❡ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ ❣❛❞❣❡t✳ Pr♦♦❢ ❲❡ ♣❧❛❝❡ ♦♥❡ P▼❯ ♦♥ {v6, v7} ❛♥❞ ♦♥❡ ♦♥ {v8, v9}✳ ❚❤❡ ✈❡rt✐❝❡s v6, . . . , v9 ❛r❡ ♦❜s❡r✈❡❞ ❜② R1E✳ ❯s✐♥❣ R2✱ t❤❡ ✈❡rt✐❝❡s vi−5✱ ❢♦r i = 6, . . . , 9✱ ❛r❡ t❤❡♥ ♦❜s❡r✈❡❞✳ ❇② ❛♣♣❧②✐♥❣ R2 t♦ v4 ✭✇❤✐❝❤ ✐s ♥♦✇ ♦❜s❡r✈❡❞✮✱ t❤❡ ✈❡rt✐❝❡s v5 ❛♥❞ t❤❡♥ v0 ❛r❡ ♦❜s❡r✈❡❞✳ ❋✐♥❛❧❧② ❛♣♣❧②✐♥❣ R2 t♦ t❤❡ ❥✉♥❝t✐♦♥ ✈❡rt✐❝❡s ❛❧❧♦✇ ✉s t♦ ♦❜s❡r✈❡ t❤❡ ♥❡✐❣❤❜♦r ✈❡rt✐❝❡s✳ ◆♦✇ ❛ss✉♠❡ t❤❛t ❛❧❧ ♥❡✐❣❤❜♦r ✈❡rt✐❝❡s ♦❢ Gv ❛r❡ ♥♦t ♦❜s❡r✈❡❞ ❛♥❞ t❤❛t ♦♥❧② ♦♥❡ P▼❯ ✐♥st❛❧❧❡❞ ✐♥ Gv ✐s s✉✣❝✐❡♥t t♦ ♦❜s❡r✈❡ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ Gv✳ ❇② ▲❡♠♠❛ ✶✱ Gv ❝❛♥♥♦t ❜❡ ♦❜s❡r✈❡❞ ❜② ♦♥❧② ♦♥❡ P▼❯ s✐♥❝❡ ♥♦ ✈❡rt❡① ✐♥ Gv ❤❛s ❛ ❞❡❣r❡❡ ✷✱ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❚❤❡♥✱ ❛t ❧❡❛st t✇♦ P▼❯s ♥❡❡❞ t♦ ❜❡ ✐♥st❛❧❧❡❞✳ ▲❡♠♠❛ ✸ ●✐✈❡♥ ❛ ❣❛❞❣❡t Gv ♦❢ G′ ✇❤❡r❡ v1, v2 ❛♥❞ v3 ❛r❡ ♦❜s❡r✈❡❞ ❡①t❡r✲ ♥❛❧❧②✱ ♦♥❡ P▼❯ ✐s ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t t♦ ♦❜s❡r✈❡ ❛❧❧ t❤❡ r❡♠❛✐♥✐♥❣ ✈❡rt✐❝❡s ♦❢ Gv✳ Pr♦♦❢ ❋♦r ❡❛❝❤ i ∈ {1, 2, 3}, vi ❤❛s t❤r❡❡ ♥♦♥✲♦❜s❡r✈❡❞ ❛❞❥❛❝❡♥t ✈❡rt✐❝❡s✱ ✇✐t❤ v4❛♥❞ v5✐♥ ❝♦♠♠♦♥✳ ■t ✐s t❤❡♥ ✐♠♣♦ss✐❜❧❡ t♦ ♦❜s❡r✈❡ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ Gv✇✐t❤ ♥♦ P▼❯✳ ▲❡t ✉s ♣❧❛❝❡ ♦♥❡ P▼❯ ♦♥ {v4, v5}✳ ❯s✐♥❣ R2 ♦♥ v5✱ v0 ✐s ♦❜s❡r✈❡❞✳ ❋✉rt❤❡r♠♦r❡✱ s✐♥❝❡ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ ❝❧✐q✉❡ {v1, ..., v5} ❛r❡ ♥♦✇ ♦❜s❡r✈❡❞✱ ❢♦r i ∈ {6, . . . , 9}✱ t❤❡ ✈❡rt✐❝❡s vi ❛r❡ ♦❜s❡r✈❡❞ ❜② ❛♣♣❧②✐♥❣ R2 ♦♥ vi−5✳ ▲❡♠♠❛ ✹ ●✐✈❡♥ ❛ ❣❛❞❣❡t Gv ♦❢ G′✱ s✉❝❤ t❤❛t ❛t ♠♦st t✇♦ ✈❡rt✐❝❡s ♦❢ Gv ❛r❡ ♦❜s❡r✈❡❞ ❡①t❡r♥❛❧❧②✱ t❤❡♥ t❤❡ r❡♠❛✐♥✐♥❣ ✈❡rt✐❝❡s ♦❢ Gv ❝❛♥♥♦t ❜❡ ♦❜s❡r✈❡❞ ✐❢ ♦♥❧② ♦♥❡ P▼❯ ✐s ♣❧❛❝❡❞ ✐♥ Gv✳ Pr♦♦❢ ❆ss✉♠❡ t❤❛t v1 ❛♥❞ v2 ❛r❡ ♦❜s❡r✈❡❞ ❡①t❡r♥❛❧❧② ✭❜② s②♠♠❡tr② ✇❡ ❝❛♥ ♣r♦✈❡ t❤❡ t✇♦ ♦t❤❡r ❝❛s❡s✮ ❛♥❞ t❤❛t ♦♥❧② ♦♥❡ P▼❯ ✐s ✐♥st❛❧❧❡❞ ✐♥ Gv✳ ❚❤❡ ✈❡rt✐❝❡s v1❛♥❞ v2❤❛✈❡ ❡❛❝❤ ❢♦✉r ♥♦♥ ♦❜s❡r✈❡❞ ❛❞❥❛❝❡♥t ✈❡rt✐❝❡s✱ ✇✐t❤ ✈❡rt✐❝❡s
❈♦♠♣❧❡①✐t② ❛♥❞ ✐♥❛♣♣r♦①✐♠❛❜✐❧✐t② r❡s✉❧ts ❢♦r t❤❡ P♦✇❡r ❊❞❣❡ ❙❡t ♣r♦❜❧❡♠ ✼ v3, v4 ❛♥❞ v5 ✐♥ ❝♦♠♠♦♥✳ P❧❛❝✐♥❣ t❤❡ P▼❯ ♦♥ ♦♥❡ ♦❢ t❤❡ ❡❞❣❡s ❧✐♥❦✐♥❣ v1 ❛♥❞ v2t♦ t❤❡s❡ ♥♦♥✲♦❜s❡r✈❡❞ ✈❡rt✐❝❡s ✇✐❧❧ ♥♦t ❛❧❧♦✇ ✉s t♦ ❛♣♣❧② R2✳ ❙✐♠✐❧❛r❧②✱ v3 ❛♥❞ v4 ❤❛✈✐♥❣ t❤r❡❡ ♥♦♥✲♦❜s❡r✈❡❞ ❛❞❥❛❝❡♥t ✈❡rt✐❝❡s✱ v5 ❤❛✈✐♥❣ t❤r❡❡ ♥♦♥✲ ♦❜s❡r✈❡❞ ❛❞❥❛❝❡♥t ✈❡rt✐❝❡s✱ ❛♥❞ v6, . . . , v9 ❤❛✈✐♥❣ ❛t ❧❡❛st t❤r❡❡ ♥♦♥✲♦❜s❡r✈❡❞ ❛❞❥❛❝❡♥t ✈❡rt✐❝❡s✱ R2 ❝❛♥♥♦t ❜❡ ✉s❡❞✳ ❚❤❡r❡❢♦r❡✱ ♦♥❡ P▼❯ ✐s ♥♦t s✉✣❝✐❡♥t t♦ ♦❜s❡r✈❡ Gv✳ ❲❡ ♥♦✇ ♣r❡s❡♥t ♦✉r ✐♥❛♣♣r♦①✐♠❛❜✐❧✐t② r❡s✉❧t✳ ❚❤❡♦r❡♠ ✶ P❊❙ ✐s ◆P✲❤❛r❞ t♦ ❛♣♣r♦①✐♠❛t❡ ✇✐t❤✐♥ ❛ ❢❛❝t♦r ✭✶✳✶✷✮✲ǫ✱ ❢♦r ❛♥② ǫ > 0✳ Pr♦♦❢ ❲❡ ♣r♦✈❡ ✜rst t❤❛t opt(I′)❛♥❞ opt(I) ❛r❡ ♣♦❧②♥♦♠✐❛❧❧② r❡❧❛t❡❞✳ ❈♦♥s✐❞❡r ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ C∗♦❢ I✳ ▲❡t Π = {{v 6, v7}, {v8, v9} : v ∈ C∗} ∪ {{v4, v5} : v 6∈ C∗} ❜❡ t❤❡ ♣❧❛❝❡♠❡♥t ❝♦♥s✐st✐♥❣ ♦❢ ✐♥st❛❧❧✐♥❣ t✇♦ P▼❯s ✐♥ t❤❡ ❣❛❞❣❡ts ❛ss♦❝✐❛t❡❞ t♦ ✈❡rt✐❝❡s ✐♥ C∗ ❛♥❞ ♦♥❡ P▼❯ ✐♥ t❤❡ ♦t❤❡r ❣❛❞❣❡ts✳ ❆❝❝♦r❞✐♥❣ t♦ ▲❡♠♠❛ ✷✱ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ Gv, v ∈ C∗ ❛♥❞ t❤❡✐r ♥❡✐❣❤❜♦r ✈❡rt✐❝❡s ❛r❡ ♦❜s❡r✈❡❞✳ ❋♦r v 6∈ C∗✱ ✐ts ♥❡✐❣❤❜♦r ✈❡rt✐❝❡s u, w, t ∈ C∗ ❛r❡ ♦❜s❡r✈❡❞ s✐♥❝❡ ♦t❤❡r✇✐s❡ t❤❡ ❡❞❣❡s (v, u)✱ (v, w) ❛♥❞ (v, t) ❛r❡ ♥♦t ❝♦✈❡r❡❞✳ ❚❤❡♥ t✇♦ P▼❯s ❛r❡ ✐♥st❛❧❧❡❞ ✐♥ Gu✱ Gw ❛♥❞ Gt✱ ❛♥❞ ❜② ▲❡♠♠❛ ✷✱ ❛❧❧ t❤❡✐r ♥❡✐❣❤❜♦r ✈❡rt✐❝❡s t❤❛t ✐♥❝❧✉❞❡ v1✱ v2 ❛♥❞ v3✱ ❛r❡ ♦❜s❡r✈❡❞✳ ❍❡♥❝❡✱ ❜② ▲❡♠♠❛ ✸✱ t❤❡ r❡♠❛✐♥✐♥❣ ✈❡rt✐❝❡s ♦❢ Gv ❝❛♥ ❜❡ ♦❜s❡r✈❡❞ ❜② ♣❧❛❝✐♥❣ ♦♥❧② ♦♥❡ P▼❯ ✐♥ Gv✳ ❚❤❡r❡❢♦r❡✱ opt(I′) ≤ |Π| = opt(I) + n✳ ❙✐♥❝❡ |C∗| ≥ 2 27(n−1)✱ ✇❡ ❤❛✈❡ t❤❛t n ≤ 27 2opt(I)+1✳ ❚❤❡♥ opt(I ′) ≤ 31 2opt(I)✳ ❚❤❡r❡❢♦r❡✱ ❢♦r n ❧❛r❣❡ ❡♥♦✉❣❤✱ opt(I′) ≤ n opt(I)✳ ❈♦♥s✐❞❡r ♥♦✇ ❛ s♦❧✉t✐♦♥ F ⊂ E′♦❢ I′✳ ■♥ F ✱ P▼❯s ❝❛♥ ❜❡ ✐♥st❛❧❧❡❞ ♦♥ ❡❞❣❡s ❜❡t✇❡❡♥ ❣❛❞❣❡ts ❛♥❞ s♦♠❡ ❣❛❞❣❡ts ❝❛♥ ❤❛✈❡ ♠♦r❡ t❤❛♥ t✇♦ P▼❯s ♣❧❛❝❡❞ ♦♥ t❤❡♠✳ ❲❡ s❤♦✇ t❤❛t t❤❡r❡ ❡①✐sts ❛♥♦t❤❡r s♦❧✉t✐♦♥ F′ ♦❢ I′ t❤❛t ✐s ❛t ❧❡❛st ❛s ❣♦♦❞ ❛s F ❛♥❞ ❝♦♥t❛✐♥✐♥❣ ♥♦ P▼❯s ♦♥ t❤❡ ❡❞❣❡s ❧✐♥❦✐♥❣ ❣❛❞❣❡ts ❛♥❞ ♦♥❧② ♦♥❡ ♦r t✇♦ P▼❯s ❜② ❣❛❞❣❡t✳ ✶✳ ■❢ ❛ P▼❯ ✐s ♣❧❛❝❡❞ ♦♥ {u1, v1} t❤❡ ❡❞❣❡ ❧✐♥❦✐♥❣ Gu ❛♥❞ Gv ✭t❤❡ ♦t❤❡r ❝❛s❡s ❛r❡ s②♠♠❡tr✐❝✮✳ ■❢ vi✱ ❢♦r i ∈ {2, . . . , 6}✱ ✐s t❤❡ ❝❤✐❧❞ ♦❢ v1 t❤❡♥ F′= F \ {{u1, v1}} ∪ {{v1, vi}}✳ ❙✐♥❝❡ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ Gv ❛r❡ ♦❜s❡r✈❡❞ ✭F ✐s ❛ P❊❙✮✱ u1 ❝❛♥ ❜❡ ♦❜s❡r✈❡❞ ✉s✐♥❣ R2 ♦♥ v1✳ ■❢ v1 ❤❛s ♥♦ ❝❤✐❧❞ t❤❡♥ F′ = F \ {{u 1, v1}}✳ ❙✐♥❝❡ ❛❧❧ t❤❡ ♦t❤❡r ✈❡rt✐❝❡s ♦❢ Gv ❛r❡ ♦❜s❡r✈❡❞ ✭F ✐s ❛ P❊❙✮✱ v1 ✐s ♦❜s❡r✈❡❞ ❜② ♦♥❡ ♦❢ vi✱ ❢♦r i ∈ {2, . . . , 6}✳ u1 ❝❛♥ ❜❡ ♦❜s❡r✈❡❞ ✉s✐♥❣ R2 ♦♥ v1✳ ✷✳ ❚❤❡r❡ ❛r❡ ❛t ❧❡❛st ✸ P▼❯s ✐♥st❛❧❧❡❞ ✐♥ ❛ ❣✐✈❡♥ ❣❛❞❣❡t Gv✳ ▲❡t Fv ❜❡ s✉❜s❡t ♦❢ ❡❞❣❡s ♦❢ Gv✇❤❡r❡ ❛ P▼❯ ✐s ✐♥st❛❧❧❡❞✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ F′ ❞❡♣❡♥❞s ♦♥ t❤❡ ♦❜s❡r✈❛❜✐❧✐t② ♦❢ ♥❡✐❣❤❜♦r ✈❡rt✐❝❡s u1✱ w2 ❛♥❞ t3 ♦❢ Gv✳ ❲❡ ❞✐st✐♥❣✉✐s❤ ✸ ❝❛s❡s✿ ❛✳ ❆❧❧ t❤❡ ♥❡✐❣❤❜♦r ✈❡rt✐❝❡s ❛r❡ ♦❜s❡r✈❡❞✿ ❜② ▲❡♠♠❛ ✸✱ ♦♥❧② ♦♥❡ P▼❯ ✐s ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t t♦ ♦❜s❡r✈❡ Gv✳ ❚❤❡♥ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ♣❧❛❝❡♠❡♥t ♣r♦♣♦s❡❞ ✐♥ ▲❡♠♠❛ ✸✱ F′ = F \ F v∪ {{v4, v5}}✳ ❜✳ ❆❧❧ t❤❡ ♥❡✐❣❤❜♦r ✈❡rt✐❝❡s ❛r❡ ♥♦t ♦❜s❡r✈❡❞✿ ❇② ▲❡♠♠❛ ✷✱ ♦♥❧② t✇♦ P▼❯s ❛r❡ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t t♦ ♦❜s❡r✈❡ Gv✳ ❚❤❡♥ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ♣❧❛❝❡✲ ♠❡♥t ♣r♦♣♦s❡❞ ✐♥ ▲❡♠♠❛ ✷✱ F′ = F \ F v∪ {{v6, v7}, {v8, v9}}✳
✽ ❙♦♥✐❛ ❚♦✉❜❛❧✐♥❡ ❡t ❛❧✳ ❝✳ ❆t ♠♦st ✷ ❛r❡ ♦❜s❡r✈❡❞✿ ❇② ▲❡♠♠❛ ✹✱ ♦♥❡ P▼❯ ✐s ♥♦t ❡♥♦✉❣❤ t♦ ♦❜s❡r✈❡ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ Gv✳ ❇② ▲❡♠♠❛ ✷✱ t✇♦ P▼❯s ❛r❡ s✉✣❝✐❡♥t t♦ ♦❜s❡r✈❡ ❛❧❧ t❤❡ ✈❡rt✐❝❡s ♦❢ Gv✳ ❚❤❡♥ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ♣❧❛❝❡♠❡♥t ♣r♦♣♦s❡❞ ✐♥ ▲❡♠♠❛ ✷✱ F′= F \ F v∪ {{v6, v7}, {v8, v9}}✳ ❚❤❡r❡❢♦r❡✱ F′✐s s✉❝❤ t❤❛t |F′| ≤ |F | ≤ k✱ ❛❧❧ t❤❡ P▼❯s ❛r❡ ♦♥❧② ♣❧❛❝❡❞ ♦♥ ❡❞❣❡s ♦❢ ❣❛❞❣❡ts ❛♥❞ ❡❛❝❤ ❣❛❞❣❡t ❤❛s ❡✐t❤❡r ♦♥❡ ♦r t✇♦ P▼❯s ✐♥st❛❧❧❡❞ ♦♥ ✐t✳ ❈♦♥s✐❞❡r C = {u : Gu ❤❛s t✇♦ P▼❯s ✐♥st❛❧❧❡❞ ♦♥ ✐t} ❛ s✉❜s❡t ♦❢ ✈❡rt✐❝❡s ✐♥ G✳ ❲❡ ♣r♦✈❡ t❤❛t C ✐s ❛ ❝♦✈❡r ❜② ❝♦♥tr❛❞✐❝t✐♦♥✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡r❡ ❡①✐sts ❛♥ ❡❞❣❡ {u, v} t❤❛t ✐s ♥♦t ❝♦✈❡r❡❞ ❜② C✱ ✐✳❡✳ u, v 6∈ C✳ ❚❤✉s✱ Gu ❛♥❞ Gv ❤❛s ♦♥❧② ♦♥❡ P▼❯ ✐♥st❛❧❧❡❞ ♦♥ t❤❡♠ ♦♥ G′✳ ❇② ▲❡♠♠❛s ✸ ❛♥❞ ✹✱ u1, u2✱ ❛♥❞ u3 ❛r❡ ❡①t❡r♥❛❧❧② ♦❜s❡r✈❡❞ ❛♥❞ s♦ ❛r❡ v1, v2 ✱❛♥❞ v3✳ ■❢ ✇❡ ❛ss✉♠❡ t❤❛t {u1, v1} ✐s t❤❡ ❡❞❣❡ ❧✐♥❦✐♥❣ Gu ❛♥❞ Gv✱ t❤❡♥ u1 ✐s ♦❜s❡r✈❡❞ ❡①t❡r♥❛❧❧② ❜② v1 ✉s✐♥❣ R2 ❛♥❞ v1 ✐s ♦❜s❡r✈❡❞ ❡①t❡r♥❛❧❧② ❜② u1 ✉s✐♥❣ R2✱ ✇❤✐❝❤ ✐s ✐♠♣♦ss✐❜❧❡✱ ❝♦♥tr❛❞✐❝t✐♥❣ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t F′ ✐s ❛ P❊❙ ♦❢ I′✳ ❚❤❡♥ C ✐s ❛ ❝♦✈❡r ❛♥❞ val(I, C) = val(I′, F′) − n✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❤❡♥ F′ ✐s ❛♥ ♦♣t✐♠✉♠ s♦❧✉t✐♦♥✱ ✇❡
❤❛✈❡ opt(I′) = val(I, C) + n ≥ opt(I) + n✳ ■t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ♣r❡✈✐♦✉s r❡s✉❧t
t❤❛t opt(I′) = opt(I) + n✳
❚❤❡r❡❢♦r❡✱ ✇❡ ❤❛✈❡ opt(I′) ≤ n opt(I) ❛♥❞
ε(I, C) = val(I,C)opt(I) − 1 =
val(I′,F′)−n opt(I′)−n − 1 = val(I′,F′)−opt(I′) opt(I′)−n ❂ val(I′,F′)−opt(I′) opt(I′) × opt(I′) opt(I′)−n✳ ❙✐♥❝❡ I ✐s ❛♥ ✐♥st❛♥❝❡ ♦❢ ❛ ▼✐♥✐♠✉♠ ❱❡rt❡① ❈♦✈❡r ❞❡✜♥❡❞ ♦♥ ❛ ✸✲r❡❣✉❧❛r ❣r❛♣❤s ✇❡ ❤❛✈❡ t❤❛t n 2 ≤ opt(I) ≤ 3n 4 ✭❋❡✐❣❡✱ ✷✵✵✸✮✳ ❚❤❡♥ 3n 2 ≤ opt(I′) ≤ 7n 4✳ ❲❡ ♦❜t❛✐♥ t❤❛t n ≤ 2opt(I′) 3 ✱ ❛♥❞ ✇❡ ❣❡t opt(I′) opt(I′)−n ≤ 3✳ ❚❤❡r❡❢♦r❡ ε(I, C) ≤
3val(I′opt(I,F′)−opt(I′) ′) = 3 ε(I ′, F′)✳
❚❤✉s✱ r(I, C) − 1 ≤ 3(r(I′, F′) − 1) ❛♥❞ t❤❡♥ r(I′, F′) ≥ r(I,C)+2 3 ✳ ❙✐♥❝❡ r(I, C) = ρ = 1.36✱ ✇❡ ❤❛✈❡ r(I′, F′) ≥ ρ+2 3 = 1.12✳ ✹ P♦❧②♥♦♠✐❛❧✲t✐♠❡ ❝❛s❡s ❢♦r P❊❙ ❲❡ ♣r♦✈❡ ✐♥ t❤✐s s❡❝t✐♦♥ t❤❛t t❤❡ P❊❙ ♣r♦❜❧❡♠ ❤❛s ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ s♦❧✉✲ t✐♦♥s ❢♦r tr❡❡s ❛♥❞ ❣r✐❞s✳ ✹✳✶ ❚r❡❡s ❲❡ ♣r♦✈❡ t❤❛t✱ ✐♥ tr❡❡s✱ P❊❙ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ P❛t❤ ❈♦✈❡r Pr♦❜❧❡♠✱ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✳ P❛t❤ ❈♦✈❡r Pr♦❜❧❡♠ ■♥♣✉t✿ ❆ ❣r❛♣❤ G = (V, E)✳ ❖✉t♣✉t✿ ❆ ♠✐♥✐♠✉♠ ❝❛r❞✐♥❛❧✐t② s❡t ♦❢ ✈❡rt❡① ❞✐s❥♦✐♥t ♣❛t❤s✱ s✉❝❤ t❤❛t ❡❛❝❤ ✈❡rt❡① ❜❡❧♦♥❣s t♦ ❛ ♣❛t❤✳ ✭❆ s✐♥❣❧❡t♦♥ ✈❡rt❡① ✐s ❛❧s♦ ❝♦♥s✐❞❡r❡❞ ❛ ♣❛t❤✳✮
❈♦♠♣❧❡①✐t② ❛♥❞ ✐♥❛♣♣r♦①✐♠❛❜✐❧✐t② r❡s✉❧ts ❢♦r t❤❡ P♦✇❡r ❊❞❣❡ ❙❡t ♣r♦❜❧❡♠ ✾ ❚❤❡ P❛t❤ ❈♦✈❡r Pr♦❜❧❡♠ ✐s ◆P✲❤❛r❞ ❢♦r ❣❡♥❡r❛❧ ❣r❛♣❤s ✭❛s t❤❡ ❍❛♠✐❧✲ t♦♥✐❛♥ P❛t❤ ♣r♦❜❧❡♠ ✐s ❡❛s✐❧② r❡❞✉❝❡❞ t♦ ✐t✮✱ ❜✉t ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ s♦❧✈❛❜❧❡ ♦♥ tr❡❡s ✭▼♦r❛♥ ❛♥❞ ❲♦❧❢st❛❤❧✱ ✶✾✾✶✮✳ ❚❤❡♦r❡♠ ✷ ❚❤❡ P♦✇❡r ❊❞❣❡ ❙❡t ♣r♦❜❧❡♠ ✐s ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ s♦❧✈❛❜❧❡ ♦♥ tr❡❡s✳ ❖♥ tr❡❡s ✇✐t❤ n ✈❡rt✐❝❡s✱ t❤❡ ❛❧❣♦r✐t❤♠ r✉♥s ✐♥ O(n) t✐♠❡✳ Pr♦♦❢ ❲❡ ♣r♦✈❡ t❤❛t P❊❙ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ P❛t❤ ❝♦✈❡r ♣r♦❜❧❡♠✳ ❈♦♥s✐❞❡r ❛♥② s♦❧✉t✐♦♥ ♦❢ t❤❡ P❊❙ ♣r♦❜❧❡♠ ♦❢ s✐③❡ k✳ ❲❡ ❞❡t❡r♠✐♥❡ t❤❡ ♣❛r❡♥t✲ ❝❤✐❧❞ ♣❛t❤s st❛rt✐♥❣ ❢r♦♠ ❛❧❧ t❤❡ ❡♥❞♣♦✐♥ts ♦❢ t❤❡s❡ k ❡❞❣❡s✳ ❚❤✐s r❡s✉❧ts ✐♥ ❛ ♣❛t❤ ❝♦✈❡r ♦❢ s✐③❡ ❛t ♠♦st 2k✳ ✭■t ✇✐❧❧ r❡s✉❧t ✐♥ ❧❡ss t❤❛♥ 2k ♣❛t❤s ✐❢ s♦♠❡ ❡❞❣❡s ✐♥ t❤❡ P❊❙ s❤❛r❡ ❡♥❞♣♦✐♥ts✳✮ ❲❡ ♥♦✇ s❤♦✇ ❤♦✇ t♦ r❡❞✉❝❡ t❤❡ s✐③❡ ♦❢ t❤❡ ♣❛t❤ ❝♦✈❡r t♦ k✳ ▲❡t e1, . . . , ek ❜❡ t❤❡ ❡❞❣❡s ✐♥ t❤❡ P❊❙✳ ❋♦r ❛♥ ❡❞❣❡ ei✱ ❧❡t Xi ❛♥❞ Yi❜❡ t❤❡ ♣❛t❤s st❛rt✐♥❣ ❢r♦♠ t❤❡ ❡♥❞♣♦✐♥ts ♦❢ ei✳ ❲❡ ❝♦♥str✉❝t t❤❡ ♣❛t❤ ❝♦✈❡r P1, . . . , Pk ✐t❡r❛t✐✈❡❧②✳ ■♥ t❤❡ i✲t❤ ✐t❡r❛t✐♦♥✱ ✇❡ ❝♦♥s✐❞❡r Xi❛♥❞ Yi✿ ✐❢ ❜♦t❤ ❛r❡ ♥♦t ✐♥ t❤❡ ❝✉rr❡♥t ♣❛rt✐❛❧ ❝♦✈❡r✱ ✇❡ ❛❞❞ t❤❡ ♣❛t❤ Pi = (Xi, ei, Yi) t♦ t❤❡ ♣❛rt✐❛❧ ❝♦✈❡r✳ ■❢ ♦♥❧② ♦♥❡ ♦❢ Xi♦r Yi✱ s❛② Xi✱ ✐s ♥♦t ✐♥ t❤❡ ♣❛rt✐❛❧ ❝♦✈❡r ✭✐♠♣❧②✐♥❣ t❤❛t ei s❤❛r❡s ❛♥ ❡♥❞♣♦✐♥t ✇✐t❤ ♦♥❡ ♦❢ t❤❡ ❡❞❣❡s e1, . . . , ei−1✮✱ ❛❞❞ Xit♦ t❤❡ ♣❛rt✐❛❧ ❝♦✈❡r✳ ◆♦t❡ t❤❛t ❞✉❡ t♦ t❤❡ ♠✐♥✐♠❛❧✐t② ♦❢ t❤❡ P❊❙✱ ✇❡ ❝❛♥♥♦t ❤❛✈❡ ❛ s✐t✉❛t✐♦♥ ✇❤❡r❡ ❜♦t❤ Xi ❛♥❞ Yi ❛r❡ ♥♦t ✐♥ t❤❡ ♣❛rt✐❛❧ ❝♦✈❡r✳ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t✱ ❛❢t❡r k ✐t❡r❛t✐♦♥s✱ ✇❡ ❡♥❞ ✉♣ ✇✐t❤ ❛ ♣❛t❤ ❝♦✈❡r ♦❢ s✐③❡ k✳ ❚❤✉s✱ ❛♥② ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ❢♦r P❊❙ ✐♥❞✉❝❡s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ s❛♠❡ ❝❛r❞✐♥❛❧✐t② ❢♦r t❤❡ P❛t❤ ❈♦✈❡r Pr♦❜❧❡♠✳ ❈♦♥s✐❞❡r ♥♦✇ ❛ ♣❛t❤ ❝♦✈❡r P1, . . . , Pk✳ ❲❡ ❝♦♥str✉❝t ❛ P❊❙ ❜② ♣✉tt✐♥❣ ❛ P▼❯ ♦♥ t❤❡ ❡①tr❡♠✐t② ❡❞❣❡ ♦❢ ❡❛❝❤ ♣❛t❤✳ ■❢ ❢♦r s♦♠❡ i = 1, . . . , k✱ Pi ✐s ❛ s✐♥❣❧❡t♦♥ ✈❡rt❡①✱ ✇❡ ♣✉t ❛ P▼❯ ♦♥ ♦♥❡ ♦❢ ✐ts ✐♥❝✐❞❡♥t ❡❞❣❡✳ ❲❡ ❝❧❛✐♠ t❤❛t R2 ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ♦❜s❡r✈❡ t❤❡ r❡st ♦❢ t❤❡ ✈❡rt✐❝❡s ❛❧♦♥❣ t❤❡ ♣❛t❤s✳ ❲❡ s❤♦✇ t❤✐s ❜② ❝♦♥tr❛❞✐❝t✐♦♥✿ s✉♣♣♦s❡ t❤❛t ❛❢t❡r t❤❡ ✈❡rt✐❝❡s ✐♥ s♦♠❡ ♣r❡✜①❡s ♦❢ P1, . . . , Pk ❛r❡ ♦❜s❡r✈❡❞✱ ✇❡ r❡❛❝❤❡❞ ❛ ♣♦✐♥t ✇❤❡r❡ R2 ❝❛♥♥♦t ❜❡ ❛♣♣❧✐❡❞ ❛♥②♠♦r❡✳ ▲❡t i1, . . . , iℓ ❜❡ t❤❡ ✐♥❞✐❝❡s ♦❢ t❤❡ ℓ ♣❛t❤s t❤❛t st✐❧❧ ❤❛✈❡ ✉♥♦❜s❡r✈❡❞ ✈❡rt✐❝❡s✱ ❛♥❞ ❧❡t xij ❜❡ t❤❡ ❧❛st ♦❜s❡r✈❡❞ ✈❡rt❡① ✐♥ ❡❛❝❤ s✉❝❤ ♣❛t❤✳ ❈♦♥s✐❞❡r t❤❡ ✈❡rt❡① yi1 t❤❛t ❢♦❧❧♦✇s xi1✐♥ ♣❛t❤ Pi1✳ ❙✐♥❝❡ xi1❝❛♥♥♦t ♦❜s❡r✈❡ yi1✱ ✐t ♠✉st ❤❛✈❡ ❛♥♦t❤❡r ✉♥♦❜s❡r✈❡❞ ♥❡✐❣❤❜♦r✳ ❚❤✐s ♥❡✐❣❤❜♦r ❝❛♥♥♦t ❜❡ ♦♥ Pi1✱ s✐♥❝❡ t❤❡♥ ✇❡ ❤❛✈❡ ❛ ❝②❝❧❡✳ ❆ss✉♠❡ t❤❛t t❤✐s ♥❡✐❣❤❜♦r ✐s ♦♥ Pi2❛♥❞ ❝♦♥s✐❞❡r xi2✳ ❱❡rt❡① xi2 ❛❧s♦ ❤❛s ❛♥ ✉♥♦❜s❡r✈❡❞ ♥❡✐❣❤❜♦r ✐♥ ❛❞❞✐t✐♦♥ t♦ yi2✱ ❜✉t t❤✐s ♥❡✐❣❤❜♦r ❝❛♥♥♦t ❜❡ ♦♥ Pi2 ❛♥❞ Pi1✱ ❜❡❝❛✉s❡ ✐♥ ❜♦t❤ ❝❛s❡s ✇❡ ❝❧♦s❡ ❛ ❝②❝❧❡✳ ❙♦ ❛ss✉♠❡ ✐t ✐s ♦♥ Pi3✳ ❲❡ ❝❛♥ r❡♣❡❛t t❤❡ ♣r♦❝❡ss ❛t ♠♦st k t✐♠❡s ✉♥t✐❧ ✇❡ ♠✉st ❤❛✈❡ ❛ ❝②❝❧❡✱ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❚❤❡♥ ❛♥② s♦❧✉t✐♦♥ ♦❢ P❛t❤ ❈♦✈❡r Pr♦❜❧❡♠ ✐♥❞✉❝❡s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ s❛♠❡ ❝❛r❞✐♥❛❧✐t② ❢♦r t❤❡ P❊❙ ♣r♦❜❧❡♠✳ ❚❤❡r❡❢♦r❡✱ s✐♥❝❡ t❤❡ P❛t❤ ❝♦✈❡r ♣r♦❜❧❡♠ ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ O(n) t✐♠❡ ❢♦r tr❡❡s ✭▼♦r❛♥ ❛♥❞ ❲♦❧❢st❛❤❧✱ ✶✾✾✶✮✱ ✇❡ ❞❡❞✉❝❡ t❤❛t t❤❡ P❊❙ ♣r♦❜❧❡♠ ✐s ❛❧s♦ s♦❧✈❛❜❧❡ ✐♥ O(n) t✐♠❡ ❢♦r tr❡❡s✳
✶✵ ❙♦♥✐❛ ❚♦✉❜❛❧✐♥❡ ❡t ❛❧✳
✹✳✷ ●r✐❞s
▲❡t Gm×n = (V, E) ❜❡ ❛ ❣r✐❞ ❣r❛♣❤ t❤❛t ✐s t❤❡ ❣r❛♣❤ ❈❛rt❡s✐❛♥ ♣r♦❞✉❝t
♦❢ Pm× Pn ♦❢ ♣❛t❤ ❣r❛♣❤s ♦♥ m ❛♥❞ n ♥♦❞❡s r❡s♣❡❝t✐✈❡❧②✳ ❲❡ ♣r♦✈❡ ✐♥ t❤❡
❢♦❧❧♦✇✐♥❣ t❤❛t t❤❡ P❊❙ ♣r♦❜❧❡♠ ✐s ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ s♦❧✈❛❜❧❡ ❢♦r ❣r✐❞s ✇✐t❤ opt(Gm×n) = ⌈12min{m, n}⌉✳ ▲❡t ℓ = min{m, n}✳
▲❡♠♠❛ ✺ ❋♦r ❛♥② ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ S ⊆ E ♦❢ t❤❡ P❊❙ ♣r♦❜❧❡♠ ❛♥❞ ❢♦r ❡❛❝❤ Pi✱ ❢♦r i = 1, . . . , ℓ✱ ❛t ❧❡❛st ♦♥❡ ♥♦❞❡ ♦❢ Pi ✐s ♦❜s❡r✈❡❞ ❜② ❛♥ ❡❞❣❡ ✐♥ S ✉s✐♥❣ r✉❧❡ R1E✳ Pr♦♦❢ ❆ss✉♠❡ t❤❛t ℓ = m✳ ❋♦r ❝♦♥tr❛❞✐❝t✐♦♥✱ ❛ss✉♠❡ t❤❛t ▲❡♠♠❛ ✺ ❞♦❡s ♥♦t ❤♦❧❞✱ ✐✳❡✳✱ t❤❡r❡ ❡①✐sts ❛ s♦❧✉t✐♦♥ S s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ❛ r♦✇ ♣❛t❤ Pi✱ ❢♦r i = 1, . . . , m✱ ✇✐t❤ ♥♦♥❡ ♦❢ ✐ts ♥♦❞❡s ♦❜s❡r✈❡❞ ✉s✐♥❣ R1E✳ ❈♦♥s✐❞❡r Pk s✉❝❤ ❛ ♣❛t❤✱ k ∈ {1, . . . , m}✳ ❚❤✉s ✐ts n ♥♦❞❡s ❛r❡ ♦❜s❡r✈❡❞ ❢r♦♠ n ❞✐st✐♥❝t ♥♦❞❡s ♦❢ Pk−1 ♦r Pk+1 ✉s✐♥❣ r✉❧❡ R2✳ ❇② t❤❡ s❛♠❡ r❡❛s♦♥✐♥❣✱ t❤❡s❡ n ♥♦❞❡s ♦❢ Pk−1 ♦r Pk+1t❤❛t ❛r❡ ♥♦t ♦❜s❡r✈❡❞ ❜② R1E ❛r❡ ♦❜s❡r✈❡❞ ❢r♦♠ t❤❡ n ❞✐st✐♥❝t ♥♦❞❡s ♦❢ Pk−2 ♦r Pk+2✱ ❛♥❞ s♦ ♦♥ ✉♥t✐❧ ✇❡ ❣❡t t♦ t❤❡ ❜♦r❞❡r ♦❢ t❤❡ ❣r✐❞ P1 ♦r Pm✳ ❲❡ ❞❡❞✉❝❡ t❤❛t ❛t ❧❡❛st n ♥♦❞❡s ♦❢ Gm×n ❛r❡ ♦❜s❡r✈❡❞ ❜② r✉❧❡ R1E ✉s✐♥❣ ❛t ❧❡❛st ⌈n 2⌉ ❡❞❣❡s ✐♥ S✳ ❍♦✇❡✈❡r✱ Gm×n ❝❛♥ ❜❡ ♦❜s❡r✈❡❞ ❜② ❡①❛❝t❧② ⌈ m 2⌉ P▼❯s✳ ▲❡t v1, v2, ..., v⌈m/2⌉ ❜❡ t❤❡ ♥♦❞❡s ♦❢ P1✳ ❇② ♣❧❛❝✐♥❣ t❤❡ P▼❯s ♦♥ t❤❡ ❡❞❣❡s {v2i, v2i−1}✱ ❢♦r i = 1, . . . , ⌊m/2⌋ ❛♥❞ ♦♥❡ ♠♦r❡ P▼❯ ♦♥ {vm−1, vm} ✐❢ m ✐s ♦❞❞✱ ✇❡ ❝❛♥ ♦❜s❡r✈❡ t❤❡ ♥♦❞❡s ♦❢ P1 ✉s✐♥❣ R1E ❛♥❞ t❤❡♥ ❛❧❧ t❤❡ ♥♦❞❡s ♦❢ Pk ❢r♦♠ t❤❡ ♥♦❞❡s ♦❢ Pk−1 ✉s✐♥❣ R2✱ ❢♦r k = 2, . . . , n✳ ❍❡♥❝❡✱ S ✐s ♥♦t ♦♣t✐♠❛❧✱ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❚❤❡ ❝❛s❡ ℓ = n ✐s ♣r♦✈❡❞ ✐♥ ❛ s✐♠✐❧❛r ✇❛②✳ ❚❤❡♦r❡♠ ✸ ❚❤❡ P♦✇❡r ❊❞❣❡ ❙❡t ♣r♦❜❧❡♠ ✐s ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ s♦❧✈❛❜❧❡ ♦♥ ❣r✐❞s✳ ❖♥ ❣r✐❞s ✇✐t❤ s✐③❡ m × n✱ ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ✐s ♦❜t❛✐♥❡❞ ✐♥ O(ℓ) t✐♠❡ ❛♥❞ ✐ts ♦♣t✐♠❛❧ ✈❛❧✉❡ ✐♥ O(1) t✐♠❡✱ ✇✐t❤ ℓ = min{m, n}✳ Pr♦♦❢ ▲❡t S ❜❡ ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ❢♦r t❤❡ P❊❙ ♣r♦❜❧❡♠✳ ❇② ▲❡♠♠❛ ✺✱ s✐♥❝❡ ❛t ❧❡❛st ♦♥❡ ♥♦❞❡ ♦❢ ❡❛❝❤ Pi✱ ❢♦r i = 1, . . . , ℓ✱ ✐s ♦❜s❡r✈❡❞ ❜② ❛♥ ❡❞❣❡ ✐♥ S ✉s✐♥❣ r✉❧❡ R1E✱ t❤❡♥ |S| ≥ ⌈2ℓ⌉✳ ❲❡ ♣r♦✈❡ t❤❛t t❤❡r❡ ❡①✐sts ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ S∗ ✇✐t❤ s✐③❡ ❡①❛❝t❧② ⌈ℓ 2⌉✳ ❆ss✉♠❡ t❤❛t ℓ = m✳ ❆s ♣r♦✈❡❞ ✐♥ t❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✺✱ ✐❢ ✇❡ ♥♦t❡ v1, v2, . . . , v⌈m/2⌉ t❤❡ ♥♦❞❡s ♦❢ t❤❡ ❝♦❧✉♠♥ ♣❛t❤ P1✱ t❤❡♥ S∗ ❝♦♥s✐sts ♦❢ t❤❡ s❡t ♦❢ ❡❞❣❡s {v2i, v2i−1}✱ ❢♦r i = 1, . . . , ⌊m/2⌋ ❛♥❞ t❤❡ ❡❞❣❡ {vm−1, vm} ✐❢ m ✐s ♦❞❞✳ ❚❤❡ ♥♦❞❡s ♦❢ P1 ❛r❡ ♦❜s❡r✈❡❞ ✉s✐♥❣ R1E ❛♥❞ t❤❡♥ ❛❧❧ t❤❡ ♥♦❞❡s ♦❢ Pk ❛r❡ ♦❜s❡r✈❡❞ ❢r♦♠ t❤❡ ♥♦❞❡s ♦❢ Pk−1✉s✐♥❣ R2✱ ❢♦r k = 2, . . . , n✳ ❲❡ ❤❛✈❡ t❤❡♥ B(∪m i=1{vi}) = V ✳ ◆♦✇✱ ✐❢ ℓ = n✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ r♦✇ ♣❛t❤ P1 ❛♥❞ ❞❡♥♦t❡ ✐ts ♥♦❞❡s ❜② v1, v2, . . . , v⌈n/2⌉✳ ❆s ❢♦r t❤❡ ♣r❡✈✐♦✉s ❝❛s❡ ✇❡ ♣r♦✈❡ t❤❛t ❢♦r S∗ ❝♦♥s✐st✐♥❣ ♦❢ t❤❡ s❡t ♦❢ ❡❞❣❡s {v2i, v2i−1}✱ ❢♦r i = 1, . . . , ⌊n/2⌋ ❛♥❞ t❤❡ ❡❞❣❡ {vn−1, vn} ✐❢ n ✐s ♦❞❞✱ ✇❡ ❤❛✈❡ B(∪n i=1{vi}) = V ✳ ❚❤❡r❡❢♦r❡✱ S∗ ✐s ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ❢♦r t❤❡ P❊❙ ♣r♦❜❧❡♠ ❛♥❞ ✐s ♦❜t❛✐♥❡❞
✐♥ O(ℓ) t✐♠❡✳ ■ts ♦♣t✐♠❛❧ ✈❛❧✉❡ opt(Gm×n) = ⌈12min{m, n}⌉ ❛♥❞ ✐s ♦❜t❛✐♥❡❞
❈♦♠♣❧❡①✐t② ❛♥❞ ✐♥❛♣♣r♦①✐♠❛❜✐❧✐t② r❡s✉❧ts ❢♦r t❤❡ P♦✇❡r ❊❞❣❡ ❙❡t ♣r♦❜❧❡♠ ✶✶ ❘❡♠❛r❦ ✶ ❉❡✜♥✐♥❣ t❤❡ P❊❙ ♣r♦❜❧❡♠ ✉s✐♥❣ R1E❛♥❞ R2 ❣✐✈❡s r✐s❡ t♦ ❛ ♥❛t✉r❛❧ ✈❛r✐❛♥t ♦❢ t❤✐s ♣r♦❜❧❡♠ t❤❛t ❝♦rr❡s♣♦♥❞s t♦ P▼❯s ✇✐t❤ ✧③❡r♦✧ ❝❤❛♥♥❡❧s✳ ■♥ t❤✐s ✈❛r✐❛♥t P▼❯s ❛r❡ ♣❧❛❝❡❞ ♦♥ ❛ ✈❡rt❡① ❛♥❞ ❝❛♥ ♦❜s❡r✈❡ ♦♥❧② t❤❡ ✈❡rt❡① t❤❡② ❛r❡ ✐♥st❛❧❧❡❞ ♦♥✳ ❚❤❡ ♦❜s❡r✈❛❜✐❧✐t② ♦❢ ❛ ❣r❛♣❤ ✉s✐♥❣ ✧③❡r♦✧ ❝❤❛♥♥❡❧ P▼❯s ❝❛♥ ❛❧s♦ ❜❡ ❞❡t❡r♠✐♥❡❞ ❜② t✇♦ r✉❧❡s✳ ❚❤❡ s❡❝♦♥❞ r✉❧❡ ✭R2✮ ✐s t❤❡ s❛♠❡ ❛s ❛❜♦✈❡ ✇❤✐❧❡ ✭R1✮ ♥❡❡❞s t♦ ❜❡ ♠♦❞✐✜❡❞ t♦✿ ✭R1V✮ ✐❢ ❛ P▼❯ ✐s ✐♥st❛❧❧❡❞ ♦♥ ❛ ✈❡rt❡① t❤❡♥ ♦♥❧② t❤✐s ✈❡rt❡① ✐s ♦❜s❡r✈❡❞✳ ❆❣❛✐♥✱ t❤❡ ♦❜❥❡❝t✐✈❡ ✐s ❛ ♠✐♥✐♠❛❧ ♣❧❛❝❡♠❡♥t ♦❢ P▼❯s t❤❛t ❡♥s✉r❡s t❤❡ ♦❜s❡r✈❛❜✐❧✐t② ♦❢ t❤❡ ✇❤♦❧❡ ❣r❛♣❤✳ ❲❡ ❝❛❧❧ t❤✐s ♣r♦❜❧❡♠ t❤❡ P♦✇❡r ❱❡rt❡① ❙❡t ✭P❱❙✮ ♣r♦❜❧❡♠✳ ■t ✐s ❛❧s♦ ❦♥♦✇♥ ❛s t❤❡ ❩❡r♦ ❋♦r❝✐♥❣ ❙❡t ♣r♦❜❧❡♠ ✭▼✐♥✐♠✉♠ ❘❛♥❦✲❙♣❡❝✐❛❧ ●r❛♣❤s ❲♦r❦ ●r♦✉♣✱ ✷✵✵✽✮✳ ❚❤❡ P❱❙ ♣r♦❜❧❡♠ ✐s ◆P✲❤❛r❞ ❢♦r ❣❡♥❡r❛❧ ❣r❛♣❤s ✭❆❛③❛♠✐✱ ✷✵✵✽✮ ❛♥❞ ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ s♦❧✈❛❜❧❡ ❢♦r tr❡❡s ✭▼✐♥✐♠✉♠ ❘❛♥❦✲❙♣❡❝✐❛❧ ●r❛♣❤s ❲♦r❦ ●r♦✉♣✱ ✷✵✵✽✮✳ ■t ✐s ❡❛s② t♦ ♦❜s❡r✈❡ t❤❛t t❤❡ s✐③❡ ♦❢ t❤❡ ♦♣t✐♠❛❧ P❱❙ ✐s ❧♦✇❡r ❜♦✉♥❞❡❞ ❜② t❤❡ s✐③❡ ♦❢ t❤❡ ♦♣t✐♠❛❧ P❊❙ ✇❤✐❝❤ ✐♥ t✉r♥ ✐s ❧♦✇❡r ❜♦✉♥❞❡❞ ❜② t❤❡ s✐③❡ ♦❢ t❤❡ ♦♣t✐♠❛❧ P❉❙✳ ❲❤✐❧❡ t❤❡ r❛t✐♦ ♦❢ t❤❡ ♦♣t✐♠❛❧ P❱❙ s✐③❡ t♦ t❤❡ ♦♣t✐♠❛❧ P❊❙ s✐③❡ ✐s ❛t ♠♦st ✷✱ t❤❡ r❛t✐♦ ♦❢ t❤❡ ♦♣t✐♠❛❧ P❊❙ s✐③❡ t♦ t❤❡ ♦♣t✐♠❛❧ P❉❙ s✐③❡ ♠❛② ❜❡ ❛s ❧❛r❣❡ ❛s n − 2✳ ❚♦ s❡❡ t❤✐s ❝♦♥s✐❞❡r ❛ ✧st❛r✧ ❣r❛♣❤ ✇✐t❤ n ✈❡rt✐❝❡s ❛♥❞ n − 1 ❡❞❣❡s✳ ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t t❤❡ P❉❙ s✐③❡ ♦❢ t❤✐s ❣r❛♣❤ ✐s 1 ✭♣❧❛❝✐♥❣ t❤❡ P▼❯ ✐♥ t❤❡ ❝❡♥t❡r✮✱ ✇❤✐❧❡ t❤❡ s✐③❡ ♦❢ t❤❡ P❱❙ ❛♥❞ t❤❡ P❊❙ ✐s n − 2✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ❞❡❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❜♦✉♥❞s ❢♦r t❤❡ ♦♣t✐♠✉♠ ✈❛❧✉❡ ♦❢ ❛♥ ✐♥st❛♥❝❡ ♦❢ t❤❡ P❊❙ ♣r♦❜❧❡♠✿
max{opt(I2V), opt(ID)} ≤ opt(IE) ≤ min{opt(IV), (n − 2)opt(ID)},
✇❤❡r❡ IV, IE ❛♥❞ ID ❛r❡ ✐♥st❛♥❝❡s ♦❢ t❤❡ P❱❙✱ P❊❙ ❛♥❞ P❉❙ ♣r♦❜❧❡♠s✳ ✺ ❈♦♥❝❧✉s✐♦♥s ❲❡ ♣r❡s❡♥t❡❞ ◆P✲❤❛r❞♥❡ss ♦❢ ❛♣♣r♦①✐♠❛❜✐❧✐t② ❢♦r t❤❡ P❊❙ ♣r♦❜❧❡♠ ✐♥ ❣❡♥✲ ❡r❛❧ ❣r❛♣❤s ❛♥❞ ♣♦❧②♥♦♠❛❧✐t② r❡s✉❧ts ✐♥ tr❡❡s ❛♥❞ ❣r✐❞s✳ ❆♥ ✐♥t❡r❡st✐♥❣ ❛✈❡♥✉❡ ❢♦r ❢✉t✉r❡ ✇♦r❦ ✇♦✉❧❞ ❜❡ t♦ st✉❞② t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤✐s ♣r♦❜❧❡♠ ♦♥ ♦t❤❡r ❝❧❛ss❡s ♦❢ ❣r❛♣❤s✱ s✉❝❤ ❛s ❣r❛♣❤s ✇✐t❤ ❜♦✉♥❞❡❞ tr❡❡✇✐❞t❤✱ ❝♦❣r❛♣❤s✱ ❛♥❞ r❡❣✲ ✉❧❛r ❛♥❞ ❜✐♣❛rt✐t❡ ❣r❛♣❤s✳ ❲❡ ❝♦♥❥❡❝t✉r❡ t❤❛t P❱❙ ❛♥❞ P❊❙ ❛r❡ ❡❛s✐❡r t❤❛♥ t❤❡ P❉❙ ♣r♦❜❧❡♠✳ ❚❤✐s ❝❛♥ ❜❡ s✉❜st❛♥t✐❛t❡❞ ❜② ✜♥❞✐♥❣ ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ❢♦r ❛ ❝❧❛ss ♦❢ ❣r❛♣❤s ♦♥ ✇❤✐❝❤ P❉❙ ✐s ◆P✲❤❛r❞✳ ❆♥♦t❤❡r ❞✐r❡❝t✐♦♥ ❢♦r ❢✉rt❤❡r st✉❞② ✐s ✜♥❞✐♥❣ ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r P❊❙✳ ❆❧s♦✱ ❢✉rt❤❡r ✇♦r❦ ✇♦✉❧❞ ❜❡ t♦ ✜♥❞ ✐♥t❡r❡st✐♥❣ ✉♣♣❡r ❛♥❞ ❧♦✇❡r ❜♦✉♥❞s ❢♦r t❤❡ ♦♣t✐♠✉♠ ✈❛❧✉❡ ♦❢ ❛♥ ✐♥st❛♥❝❡ ♦❢ t❤❡ P❊❙ ♣r♦❜❧❡♠✳ ❘❡❢❡r❡♥❝❡s ❆✳ ❆❛③❛♠✐✳ ❍❛r❞♥❡ss r❡s✉❧ts ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r s♦♠❡ ♣r♦❜❧❡♠s ♦♥ ❣r❛♣❤s✳ P❤❉ t❤❡s✐s✱ ❉❡♣❛rt♠❡♥t ♦❢ ❈♦♠❜✐♥❛t♦r✐❝s ❛♥❞ ❖♣t✐♠✐③❛t✐♦♥✱ ❯♥✐✈❡rs✐t② ♦❢ ❲❛t❡r❧♦♦✱ ❖♥t❛r✐♦✱ ❈❛♥❛❞❛✱ ✷✵✵✽✳
✶✷ ❙♦♥✐❛ ❚♦✉❜❛❧✐♥❡ ❡t ❛❧✳ ❆✳ ❆❛③❛♠✐ ❛♥❞ ▼✳ ❙t✐❧♣✳ ❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❛♥❞ ❤❛r❞♥❡ss ❢♦r ❞♦♠✲ ✐♥❛t✐♦♥ ✇✐t❤ ♣r♦♣❛❣❛t✐♦♥✳ ■♥ ▼✳ ❈❤❛r✐❦❛r✱ ❑✳ ❏❛♥s❡♥✱ ❖✳ ❘❡✐♥❣♦❧❞✱ ❛♥❞ ❏✳ ❘♦❧✐♠✱ ❡❞✐t♦rs✱ ❆♣♣r♦①✐♠❛t✐♦♥✱ ❘❛♥❞♦♠✐③❛t✐♦♥✱ ❛♥❞ ❈♦♠❜✐♥❛t♦r✐❛❧ ❖♣✲ t✐♠✐③❛t✐♦♥✿ ❆❧❣♦r✐t❤♠s ❛♥❞ ❚❡❝❤♥✐q✉❡s✱ ✈♦❧✉♠❡ ✹✻✷✼ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✶✕✶✺✱ ❇❡r❧✐♥✱ ✷✵✵✼✳ ❙♣r✐♥❣❡r✳ ❉✳ ❏✳ ❇r✉❡♥✐ ❛♥❞ ▲✳ ❙✳ ❍❡❛t❤✳ ❚❤❡ P▼❯ ♣❧❛❝❡♠❡♥t ♣r♦❜❧❡♠✳ ❙■❆▼ ❏♦✉r♥❛❧ ♦♥ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✱ ✶✾✭✸✮✿✼✹✹✕✼✻✶✱ ✷✵✵✺✳ ■✳ ❉✐♥✉r ❛♥❞ ❙✳ ❙❛❢r❛✳ ❖♥ t❤❡ ❤❛r❞♥❡ss ♦❢ ❛♣♣r♦①✐♠❛t✐♥❣ ♠✐♥✐♠✉♠ ✈❡rt❡① ❝♦✈❡r✳ ❆♥♥❛❧s ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✶✻✷✭✶✮✿✹✸✾✕✹✽✺✱ ✷✵✵✺✳ ▼✳ ❉♦r✢✐♥❣ ❛♥❞ ▼✳ ❆✳ ❍❡♥♥✐♥❣✳ ❆ ♥♦t❡ ♦♥ ♣♦✇❡r ❞♦♠✐♥❛t✐♦♥ ✐♥ ❣r✐❞ ❣r❛♣❤s✳ ❉✐s❝r❡t❡ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ✶✺✹✭✻✮✿✶✵✷✸✕✶✵✷✼✱ ✷✵✵✻✳ ❘✳ ❊♠❛♠✐ ❛♥❞ ❆✳ ❆❜✉r✳ ❘♦❜✉st ♠❡❛s✉r❡♠❡♥t ❞❡s✐❣♥ ❜② ♣❧❛❝✐♥❣ s②♥❝❤r♦♥✐③❡❞ ♣❤❛s♦r ♠❡❛s✉r❡♠❡♥ts ♦♥ ♥❡t✇♦r❦ ❜r❛♥❝❤❡s✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ P♦✇❡r ❙②st❡♠s✱ ✷✺✭✶✮✿✸✽✕✹✸✱ ✷✵✶✵✳ ❘✳ ❊♠❛♠✐✱ ❆✳ ❆❜✉r✱ ❛♥❞ ❋✳ ●❛❧✈❛♥✳ ❖♣t✐♠❛❧ ♣❧❛❝❡♠❡♥t ♦❢ ♣❤❛s♦r ♠❡❛s✉r❡✲ ♠❡♥ts ❢♦r ❡♥❤❛♥❝❡❞ st❛t❡ ❡st✐♠❛t✐♦♥✿ ❆ ❝❛s❡ st✉❞②✳ ■♥ ✶✻t❤ P♦✇❡r ❙②st❡♠s ❈♦♠♣✉t❛t✐♦♥ ❈♦♥❢❡r❡♥❝❡✱ ♣❛❣❡s ✾✷✸✕✾✷✽✱ P✐s❝❛t❛✇❛②✱ ✷✵✵✽✳ ■❊❊❊✳ ❯✳ ❋❡✐❣❡✳ ❱❡rt❡① ❝♦✈❡r ✐s ❤❛r❞❡st t♦ ❛♣♣r♦①✐♠❛t❡ ♦♥ r❡❣✉❧❛r ❣r❛♣❤s✳ ❚❡❝❤♥✐❝❛❧ ❘❡♣♦rt ▼❈❙✵✸✲✶✺✱ ❲❡✐③♠❛♥♥ ■♥st✐t✉t❡✱ ✷✵✵✸✳ ❏✳ ●✉♦✱ ❘✳ ◆✐❡❞❡r♠❡✐❡r✱ ❛♥❞ ❉✳ ❘❛✐❜❧❡✳ ■♠♣r♦✈❡❞ ❛❧❣♦r✐t❤♠s ❛♥❞ ❝♦♠♣❧❡①✐t② r❡s✉❧ts ❢♦r ♣♦✇❡r ❞♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s✳ ■♥ ▼✳ ▲✐➧❦✐❡✇✐❝③ ❛♥❞ ❘✳ ❘❡✐s❝❤✉❦✱ ❡❞✐t♦rs✱ ❋✉♥❞❛♠❡♥t❛❧s ♦❢ ❈♦♠♣✉t❛t✐♦♥ ❚❤❡♦r②✱ ✈♦❧✉♠❡ ✸✻✷✸ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✶✼✷✕✶✽✹✱ ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ✷✵✵✺✳ ❙♣r✐♥❣❡r✳ ❚✳ ❲✳ ❍❛②♥❡s✱ ❙✳ ▼✳ ❍❡❞❡t♥✐❡♠✐✱ ❙✳ ❚✳ ❍❡❞❡t♥✐❡♠✐✱ ❛♥❞ ▼✳ ❆✳ ❍❡♥♥✐♥❣✳ ❉♦♠✐♥❛t✐♦♥ ✐♥ ❣r❛♣❤s ❛♣♣❧✐❡❞ t♦ ❡❧❡❝tr✐❝ ♣♦✇❡r ♥❡t✇♦r❦s✳ ❙■❆▼ ❏♦✉r♥❛❧ ♦♥ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✱ ✶✺✭✹✮✿✺✶✾✕✺✷✾✱ ✷✵✵✷✳ ❙✳ ❑❤❛♥♥❛✱ ❘✳ ▼♦t✇❛♥✐✱ ▼✳ ❙✉❞❛♥✱ ❛♥❞ ❯✳ ❱❛③✐r❛♥✐✳ ❖♥ s②♥t❛❝t✐❝ ✈❡rs✉s ❝♦♠♣✉t❛t✐♦♥❛❧ ✈✐❡✇s ♦❢ ❛♣♣r♦①✐♠❛❜✐❧✐t②✳ ❙■❆▼ ❏♦✉r♥❛❧ ♦♥ ❈♦♠♣✉t✐♥❣✱ ✷✽ ✭✶✮✿✶✻✹✕✶✾✶✱ ✶✾✾✾✳ ◆✳ ▼✳ ▼❛♥♦✉s❛❦✐s✱ ●✳ ◆✳ ❑♦rr❡✱ ❛♥❞ P✳ ❙✳ ●❡♦r❣✐❧❛❦✐s✳ ❚❛①♦♥♦♠② ♦❢ P▼❯ ♣❧❛❝❡♠❡♥t ♠❡t❤♦❞♦❧♦❣✐❡s✳ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ P♦✇❡r ❙②st❡♠s✱ ✷✼✭✷✮✿ ✶✵✼✵✕✶✵✼✼✱ ✷✵✶✷✳ ▼✐♥✐♠✉♠ ❘❛♥❦✲❙♣❡❝✐❛❧ ●r❛♣❤s ❲♦r❦ ●r♦✉♣✳ ❩❡r♦ ❢♦r❝✐♥❣ s❡ts ❛♥❞ t❤❡ ♠✐♥✐✲ ♠✉♠ r❛♥❦ ♦❢ ❣r❛♣❤s✳ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✱ ✹✷✽✭✼✮✿✶✻✷✽✕✶✻✹✽✱ ✷✵✵✽✳ ❙✳ ▼♦r❛♥ ❛♥❞ ❨✳ ❲♦❧❢st❛❤❧✳ ❖♣t✐♠❛❧ ❝♦✈❡r✐♥❣ ♦❢ ❝❛❝t✐ ❜② ✈❡rt❡①✲❞✐s❥♦✐♥t ♣❛t❤s✳ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✽✹✭✷✮✿✶✼✾✕✶✾✼✱ ✶✾✾✶✳ P✳▲✳ P♦✐r✐♦♥✱ ❙✳ ❚♦✉❜❛❧✐♥❡✱ ❈✳ ❉✬❆♠❜r♦s✐♦✱ ❛♥❞ ▲✳ ▲✐❜❡rt✐✳ ❚❤❡ ♣♦✇❡r ❡❞❣❡ s❡t ♣r♦❜❧❡♠✳ ◆❡t✇♦r❦s✱ ✻✽✭✷✮✿✶✵✹✕✶✷✵✱ ✷✵✶✻✳ ●✳ ❳✉✱ ▲✳ ❑❛♥❣✱ ❊✳ ❙❤❛♥✱ ❛♥❞ ▼✳ ❩❤❛♦✳ P♦✇❡r ❞♦♠✐♥❛t✐♦♥ ✐♥ ❜❧♦❝❦ ❣r❛♣❤s✳ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✸✺✾✭✶✲✸✮✿✷✾✾✕✸✵✺✱ ✷✵✵✻✳ Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org)
