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❖♣t✐♠✐s✐♥❣ r❡❛❝❤❛❜✐❧✐t② ❜② r❡♦r❞❡r✐♥❣

❑✐tt② ▼❡❡❦s

❙❝❤♦♦❧ ♦❢ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡✱ ❯♥✐✈❡rs✐t② ♦❢ ●❧❛s❣♦✇

❆❧❣♦r✐t❤♠✐❝ ❆s♣❡❝ts ♦❢ ❚❡♠♣♦r❛❧ ●r❛♣❤s ■■✱ P❛tr❛s✱ ●r❡❡❝❡ ✽t❤ ❏✉❧② ✷✵✶✾

❏♦✐♥t ✇♦r❦ ✇✐t❤ ❏❡ss✐❝❛ ❊♥r✐❣❤t ✭❯♥✐✈❡rs✐t② ♦❢ ❊❞✐♥❜✉r❣❤✮ ❛♥❞ ❋✐♦♥❛ ❙❦❡r♠❛♥ ✭▼❛s❛r②❦ ❯♥✐✈❡rs✐t②✮

❘❡❛❝❤❛❜✐❧✐t② s❡ts ✐♥ t❡♠♣♦r❛❧ ❣r❛♣❤s

1,2,3 1 2 3 4,1 2 3,5 1 2 4,3

❆ss✐❣♥✐♥❣ t✐♠❡s t♦ r❡❞✉❝❡ r❡❛❝❤❛❜✐❧✐t②

◮ ❙✉♣♣♦s❡ ✇❡ ❝❛♥ ❝❤❛♥❣❡ t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ t❤❡ ❡❞❣❡s ❛r❡ ❛❝t✐✈❡✱

✇✐t❤♦✉t ❝❤❛♥❣✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ❛❝t✐✈❡ t✐♠❡s ❢♦r ❡❛❝❤ ❡❞❣❡✳ ❖✉r ❣♦❛❧ ✐s t♦ ♠✐♥✐♠✐s❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❧❛r❣❡st r❡❛❝❤❛❜✐❧✐t② s❡t✳

❆ss✐❣♥✐♥❣ t✐♠❡s t♦ r❡❞✉❝❡ r❡❛❝❤❛❜✐❧✐t②

◮ ❙✉♣♣♦s❡ ✇❡ ❝❛♥ ❝❤❛♥❣❡ t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ t❤❡ ❡❞❣❡s ❛r❡ ❛❝t✐✈❡✱

✇✐t❤♦✉t ❝❤❛♥❣✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ❛❝t✐✈❡ t✐♠❡s ❢♦r ❡❛❝❤ ❡❞❣❡✳

◮ ❖✉r ❣♦❛❧ ✐s t♦ ♠✐♥✐♠✐s❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❧❛r❣❡st r❡❛❝❤❛❜✐❧✐t② s❡t✳

❆ss✐❣♥✐♥❣ t✐♠❡s t♦ r❡❞✉❝❡ r❡❛❝❤❛❜✐❧✐t②

◮ ❙✉♣♣♦s❡ ✇❡ ❝❛♥ ❝❤❛♥❣❡ t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ t❤❡ ❡❞❣❡s ❛r❡ ❛❝t✐✈❡✱

✇✐t❤♦✉t ❝❤❛♥❣✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ❛❝t✐✈❡ t✐♠❡s ❢♦r ❡❛❝❤ ❡❞❣❡✳

◮ ❖✉r ❣♦❛❧ ✐s t♦ ♠✐♥✐♠✐s❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❧❛r❣❡st r❡❛❝❤❛❜✐❧✐t② s❡t✳

❖✉r ❜❛s✐❝ ♣r♦❜❧❡♠

❚❤❡ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ ❛ ✈❡rt❡① v ✐♥ ❛ t❡♠♣♦r❛❧ ❣r❛♣❤ (G, T ) ✐s t❤❡ s❡t ♦❢ ✈❡rt✐❝❡s u s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ❛ str✐❝t t❡♠♣♦r❛❧ ♣❛t❤ ❢r♦♠ v t♦ u❀ ✇❡ ✐♥❝❧✉❞❡ v ✐♥ t❤✐s s❡t✳ ❚❤❡ ♠❛①✐♠✉♠ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② ♦❢ ❛ t❡♠♣♦r❛❧ ❣r❛♣❤ ✐s t❤❡ ♠❛①✐♠✉♠ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ ❛♥② ✈❡rt❡① v ✐♥ t❤❡ ❣r❛♣❤✳

❙■◆●▲❊❚❖◆ ▼■◆✲▼❆❳ ❘❊❆❈❍❆❇■▲■❚❨ ❚❊▼P❖❘❆▲ ❖❘❉❊❘■◆●

■♥♣✉t✿ ❆ ❣r❛♣❤ ✱ ❛♥❞ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛ ❜✐❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ s✉❝❤ t❤❛t ♠❛①✐♠✉♠ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② ♦❢ ✐s ❛t ♠♦st ❄

❖✉r ❜❛s✐❝ ♣r♦❜❧❡♠

❚❤❡ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ ❛ ✈❡rt❡① v ✐♥ ❛ t❡♠♣♦r❛❧ ❣r❛♣❤ (G, T ) ✐s t❤❡ s❡t ♦❢ ✈❡rt✐❝❡s u s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ❛ str✐❝t t❡♠♣♦r❛❧ ♣❛t❤ ❢r♦♠ v t♦ u❀ ✇❡ ✐♥❝❧✉❞❡ v ✐♥ t❤✐s s❡t✳ ❚❤❡ ♠❛①✐♠✉♠ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② ♦❢ ❛ t❡♠♣♦r❛❧ ❣r❛♣❤ ✐s t❤❡ ♠❛①✐♠✉♠ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ ❛♥② ✈❡rt❡① v ✐♥ t❤❡ ❣r❛♣❤✳

❙■◆●▲❊❚❖◆ ▼■◆✲▼❆❳ ❘❊❆❈❍❆❇■▲■❚❨ ❚❊▼P❖❘❆▲ ❖❘❉❊❘■◆●

■♥♣✉t✿ ❆ ❣r❛♣❤ G = (V , E)✱ ❛♥❞ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r k ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛ ❜✐❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ T : E → [|E|] s✉❝❤ t❤❛t ♠❛①✐♠✉♠ t❡♠♣♦r❛❧ r❡❛❝❤❛❜✐❧✐t② ♦❢ (G, T ) ✐s ❛t ♠♦st k❄

❚❤❡ ♣r♦❜❧❡♠ ✐s ❤❛r❞

❚❤❡♦r❡♠

❙■◆●▲❊❚❖◆ ▼■◆✲▼❆❳ ❘❊❆❈❍❆❇■▲■❚❨ ❚❊▼P❖❘❆▲ ❖❘❉❊❘■◆● ✐s ◆P✲❝♦♠♣❧❡t❡✳ ❲❡ ❣✐✈❡ ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ ▼✐♥✐♠✉♠ ❇✐s❡❝t✐♦♥ ❢♦r ❈✉✲ ❜✐❝ ●r❛♣❤s✳

❚❤❡ ♣r♦❜❧❡♠ ✐s ❤❛r❞

❚❤❡♦r❡♠

❙■◆●▲❊❚❖◆ ▼■◆✲▼❆❳ ❘❊❆❈❍❆❇■▲■❚❨ ❚❊▼P❖❘❆▲ ❖❘❉❊❘■◆● ✐s ◆P✲❝♦♠♣❧❡t❡✳ ❲❡ ❣✐✈❡ ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ ▼✐♥✐♠✉♠ ❇✐s❡❝t✐♦♥ ❢♦r ❈✉✲ ❜✐❝ ●r❛♣❤s✳

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• ❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s

◮ ❍♦✇❡✈❡r ✇❡ ♦r❞❡r t❤❡ ❡❞❣❡s✱ t❤❡ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ ❡✈❡r②

✈❡rt❡① ❝♦♥t❛✐♥s ❛❧❧ ♦❢ ✐ts ✭♦✉t✲✮♥❡✐❣❤❜♦✉rs✳

❚❤❡♦r❡♠

❚❤❡ s♠❛❧❧❡st ❛❝❤✐❡✈❛❜❧❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ❢♦r ❛ ❉❆● ✐s ❡①❛❝t❧② t❤❡ ♠❛①✐♠✉♠ ♦✉t✲❞❡❣r❡❡ ♣❧✉s ♦♥❡✳

• ❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s

◮ ❍♦✇❡✈❡r ✇❡ ♦r❞❡r t❤❡ ❡❞❣❡s✱ t❤❡ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ ❡✈❡r②

✈❡rt❡① ❝♦♥t❛✐♥s ❛❧❧ ♦❢ ✐ts ✭♦✉t✲✮♥❡✐❣❤❜♦✉rs✳

❚❤❡♦r❡♠

❚❤❡ s♠❛❧❧❡st ❛❝❤✐❡✈❛❜❧❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ❢♦r ❛ ❉❆● ✐s ❡①❛❝t❧② t❤❡ ♠❛①✐♠✉♠ ♦✉t✲❞❡❣r❡❡ ♣❧✉s ♦♥❡✳

• ❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s

◮ ❍♦✇❡✈❡r ✇❡ ♦r❞❡r t❤❡ ❡❞❣❡s✱ t❤❡r❡ ✐s ❛ r❡❛❝❤❛❜✐❧✐t② s❡t ✇❤✐❝❤

❝♦♥t❛✐♥s ❛❧❧ ♥❡✐❣❤❜♦✉rs ♦❢ t✇♦ ❛❞❥❛❝❡♥t ✈❡rt✐❝❡s✳

• ❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s

◮ ❍♦✇❡✈❡r ✇❡ ♦r❞❡r t❤❡ ❡❞❣❡s✱ t❤❡r❡ ✐s ❛ r❡❛❝❤❛❜✐❧✐t② s❡t ✇❤✐❝❤

❝♦♥t❛✐♥s ❛❧❧ ♥❡✐❣❤❜♦✉rs ♦❢ t✇♦ ❛❞❥❛❝❡♥t ✈❡rt✐❝❡s✳

❚❤❡♦r❡♠

❋♦r ❛ ♣❛t❤ ♦♥ ❛t ❧❡❛st ✜✈❡ ✈❡rt✐❝❡s✱ t❤❡ s♠❛❧❧❡st ❛❝❤✐❡✈❛❜❧❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ✐s ❡①❛❝t❧② ❢♦✉r✳

• ❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s

◮ ❚❤❡r❡ ✐s ❛♥ ♦r❞❡r ♦❢ ❡❞❣❡s t❤❛t ♠✐♥✐♠✐s❡s t❤❡ ♠❛①✐♠✉♠

r❡❛❝❤❛❜✐❧✐t②✱ s♦ t❤❛t ❛❧❧ ✏❧❡❛❢✑ ❡❞❣❡s ❛r❡ ❛❝t✐✈❡ ❜❡❢♦r❡ ❛❧❧ ♦t❤❡rs✳ ▼♦r❡♦✈❡r✱ t❤❡ r❡❧❛t✐✈❡ ♦r❞❡r ♦❢ ❧❡❛❢ ❡❞❣❡s ❞♦❡s ♥♦t ♠❛tt❡r✳

• ❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s

◮ ❚❤❡r❡ ✐s ❛♥ ♦r❞❡r ♦❢ ❡❞❣❡s t❤❛t ♠✐♥✐♠✐s❡s t❤❡ ♠❛①✐♠✉♠

r❡❛❝❤❛❜✐❧✐t②✱ s♦ t❤❛t ❛❧❧ ✏❧❡❛❢✑ ❡❞❣❡s ❛r❡ ❛❝t✐✈❡ ❜❡❢♦r❡ ❛❧❧ ♦t❤❡rs✳ ▼♦r❡♦✈❡r✱ t❤❡ r❡❧❛t✐✈❡ ♦r❞❡r ♦❢ ❧❡❛❢ ❡❞❣❡s ❞♦❡s ♥♦t ♠❛tt❡r✳

❚❤❡♦r❡♠

❚❤❡r❡ ✐s ❛♥ ❋P❚ ❛❧❣♦r✐t❤♠ t♦ ❝♦♠♣✉t❡ t❤❡ ♦♣t✐♠✉♠ ♦r❞❡r✐♥❣ ♦❢ ❡❞❣❡s✱ ♣❛r❛♠❡t❡r✐s❡❞ ❜② t❤❡ ♥✉♠❜❡r ♦❢ ✈❡rt✐❝❡s ♦❢ ❞❡❣r❡❡ ❛t ❧❡❛st t✇♦✳

• ❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s

◮ ❚❤❡r❡ ✐s ❛♥ ♦r❞❡r ♦❢ ❡❞❣❡s t❤❛t ♠✐♥✐♠✐s❡s t❤❡ ♠❛①✐♠✉♠

r❡❛❝❤❛❜✐❧✐t②✱ s♦ t❤❛t ❛❧❧ ✏❧❡❛❢✑ ❡❞❣❡s ❛r❡ ❛❝t✐✈❡ ❜❡❢♦r❡ ❛❧❧ ♦t❤❡rs✳ ▼♦r❡♦✈❡r✱ t❤❡ r❡❧❛t✐✈❡ ♦r❞❡r ♦❢ ❧❡❛❢ ❡❞❣❡s ❞♦❡s ♥♦t ♠❛tt❡r✳

❚❤❡♦r❡♠

❚❤❡r❡ ✐s ❛♥ ❋P❚ ❛❧❣♦r✐t❤♠ t♦ ❝♦♠♣✉t❡ t❤❡ ♦♣t✐♠✉♠ ♦r❞❡r✐♥❣ ♦❢ ❡❞❣❡s✱ ♣❛r❛♠❡t❡r✐s❡❞ ❜② t❤❡ ♥✉♠❜❡r ♦❢ ✈❡rt✐❝❡s ♦❢ ❞❡❣r❡❡ ❛t ❧❡❛st t✇♦✳

❈♦r♦❧❧❛r②

❚❤❡r❡ ✐s ❛♥ ❋P❚ ❛❧❣♦r✐t❤♠ t♦ ❝♦♠♣✉t❡ t❤❡ ♦♣t✐♠✉♠ ♦r❞❡r✐♥❣ ♦❢ ❡❞❣❡s ❢♦r ❛ tr❡❡✱ ♣❛r❛♠❡t❡r✐s❡❞ ❜② t❤❡ ✈❡rt❡① ❝♦✈❡r ♥✉♠❜❡r✳

❆♥ ❛❧❣♦r✐t❤♠ ❢♦r tr❡❡s

❚❤❡ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ ❧✐♥❡❛r t✐♠❡ ♦♥ tr❡❡s ✇❤❡♥ k ✭♠❛① ♣❡r♠✐tt❡❞ r❡❛❝❤❛❜✐❧✐t②✮ ✐s ❜♦✉♥❞❡❞ ❜② ❛ ✜①❡❞ ❝♦♥st❛♥t✿ ❘❡❝❛❧❧✿ ✐❢ ♠❛① ❞❡❣r❡❡ t❤❡♥ ✇❡ ❤❛✈❡ ❛ ♥♦✲✐♥st❛♥❝❡ ❙t❛rt✐♥❣ ❛t ❧❡❛✈❡s✱ ❛♥❞ ✇♦r❦✐♥❣ t♦✇❛r❞s t❤❡ r♦♦t✿

❈♦♥s✐❞❡r ❛❧❧ r❡❧❛t✐✈❡ ♦r❞❡r✐♥❣s ♦❢ t❤❡ ❡❞❣❡s ✐♥❝✐❞❡♥t ✇✐t❤ ❛ s✐♥❣❧❡ ✈❡rt❡① ✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t t❤❡ ❡❞❣❡ ❢r♦♠ t♦ ✐ts ♣❛r❡♥t ✐s ❛❝t✐✈❡ ❛t t✐♠❡ ✳ ❋♦r ❡❛❝❤ ♣♦ss✐❜❧❡ ♦r❞❡r✐♥❣✱ ❛♥❞ ❡✈❡r② ♣❛✐r ✇✐t❤ ✶ ✱ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r t❤❡r❡ ✐s ❛♥ ♦r❞❡r✐♥❣ ♦❢ ❛❧❧ ❡❞❣❡s ✐♥❝✐❞❡♥t ✇✐t❤ ❛♥❞ ✐ts ❞❡s❝❡♥❞❛♥ts s✉❝❤ t❤❛t✿

✶✳ ♥♦ ❞❡s❝❡♥❞❛♥t ♦❢ ❤❛s ❛ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ s✐③❡ ✱ ✷✳ ♥♦ ❞❡s❝❡♥❞❛♥t ♦❢ ✇❤✐❝❤ r❡❛❝❤❡s ❜❡❢♦r❡ t✐♠❡ ❤❛s s✐③❡ ♠♦r❡ t❤❛♥ ✱ ❛♥❞ ✸✳ ❛t ♠♦st ❞❡s❝❡♥❞❛♥ts ♦❢ ❛r❡ r❡❛❝❤❛❜❧❡ ❢r♦♠ ❛❢t❡r t✐♠❡ ✳

❚❤✐s ❣❡♥❡r❛❧ ♠❡t❤♦❞ ❝❛♥ ❛❧s♦ ❜❡ ❣❡♥❡r❛❧✐s❡❞ t♦ ❣r❛♣❤s ♦❢ ❜♦✉♥❞❡❞ tr❡❡✇✐❞t❤✳

❆♥ ❛❧❣♦r✐t❤♠ ❢♦r tr❡❡s

❚❤❡ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ ❧✐♥❡❛r t✐♠❡ ♦♥ tr❡❡s ✇❤❡♥ k ✭♠❛① ♣❡r♠✐tt❡❞ r❡❛❝❤❛❜✐❧✐t②✮ ✐s ❜♦✉♥❞❡❞ ❜② ❛ ✜①❡❞ ❝♦♥st❛♥t✿

◮ ❘❡❝❛❧❧✿ ✐❢ ♠❛① ❞❡❣r❡❡ ≥ k t❤❡♥ ✇❡ ❤❛✈❡ ❛ ♥♦✲✐♥st❛♥❝❡ ◮ ❙t❛rt✐♥❣ ❛t ❧❡❛✈❡s✱ ❛♥❞ ✇♦r❦✐♥❣ t♦✇❛r❞s t❤❡ r♦♦t✿

◮ ❈♦♥s✐❞❡r ❛❧❧ r❡❧❛t✐✈❡ ♦r❞❡r✐♥❣s ♦❢ t❤❡ ❡❞❣❡s ✐♥❝✐❞❡♥t ✇✐t❤ ❛

s✐♥❣❧❡ ✈❡rt❡① v✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t t❤❡ ❡❞❣❡ ❢r♦♠ v t♦ ✐ts ♣❛r❡♥t ✐s ❛❝t✐✈❡ ❛t t✐♠❡ tv✳ ❋♦r ❡❛❝❤ ♣♦ss✐❜❧❡ ♦r❞❡r✐♥❣✱ ❛♥❞ ❡✈❡r② ♣❛✐r ✇✐t❤ ✶ ✱ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r t❤❡r❡ ✐s ❛♥ ♦r❞❡r✐♥❣ ♦❢ ❛❧❧ ❡❞❣❡s ✐♥❝✐❞❡♥t ✇✐t❤ ❛♥❞ ✐ts ❞❡s❝❡♥❞❛♥ts s✉❝❤ t❤❛t✿

✶✳ ♥♦ ❞❡s❝❡♥❞❛♥t ♦❢ ❤❛s ❛ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ s✐③❡ ✱ ✷✳ ♥♦ ❞❡s❝❡♥❞❛♥t ♦❢ ✇❤✐❝❤ r❡❛❝❤❡s ❜❡❢♦r❡ t✐♠❡ ❤❛s s✐③❡ ♠♦r❡ t❤❛♥ ✱ ❛♥❞ ✸✳ ❛t ♠♦st ❞❡s❝❡♥❞❛♥ts ♦❢ ❛r❡ r❡❛❝❤❛❜❧❡ ❢r♦♠ ❛❢t❡r t✐♠❡ ✳

❚❤✐s ❣❡♥❡r❛❧ ♠❡t❤♦❞ ❝❛♥ ❛❧s♦ ❜❡ ❣❡♥❡r❛❧✐s❡❞ t♦ ❣r❛♣❤s ♦❢ ❜♦✉♥❞❡❞ tr❡❡✇✐❞t❤✳

❆♥ ❛❧❣♦r✐t❤♠ ❢♦r tr❡❡s

❚❤❡ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ ❧✐♥❡❛r t✐♠❡ ♦♥ tr❡❡s ✇❤❡♥ k ✭♠❛① ♣❡r♠✐tt❡❞ r❡❛❝❤❛❜✐❧✐t②✮ ✐s ❜♦✉♥❞❡❞ ❜② ❛ ✜①❡❞ ❝♦♥st❛♥t✿

◮ ❘❡❝❛❧❧✿ ✐❢ ♠❛① ❞❡❣r❡❡ ≥ k t❤❡♥ ✇❡ ❤❛✈❡ ❛ ♥♦✲✐♥st❛♥❝❡ ◮ ❙t❛rt✐♥❣ ❛t ❧❡❛✈❡s✱ ❛♥❞ ✇♦r❦✐♥❣ t♦✇❛r❞s t❤❡ r♦♦t✿

◮ ❈♦♥s✐❞❡r ❛❧❧ r❡❧❛t✐✈❡ ♦r❞❡r✐♥❣s ♦❢ t❤❡ ❡❞❣❡s ✐♥❝✐❞❡♥t ✇✐t❤ ❛

s✐♥❣❧❡ ✈❡rt❡① v✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t t❤❡ ❡❞❣❡ ❢r♦♠ v t♦ ✐ts ♣❛r❡♥t ✐s ❛❝t✐✈❡ ❛t t✐♠❡ tv✳

◮ ❋♦r ❡❛❝❤ ♣♦ss✐❜❧❡ ♦r❞❡r✐♥❣✱ ❛♥❞ ❡✈❡r② ♣❛✐r (α, β) ✇✐t❤

✶ ≤ α, β ≤ k✱ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r t❤❡r❡ ✐s ❛♥ ♦r❞❡r✐♥❣ ♦❢ ❛❧❧ ❡❞❣❡s ✐♥❝✐❞❡♥t ✇✐t❤ v ❛♥❞ ✐ts ❞❡s❝❡♥❞❛♥ts s✉❝❤ t❤❛t✿

✶✳ ♥♦ ❞❡s❝❡♥❞❛♥t ♦❢ v ❤❛s ❛ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ s✐③❡ > k✱ ✷✳ ♥♦ ❞❡s❝❡♥❞❛♥t ♦❢ v ✇❤✐❝❤ r❡❛❝❤❡s v ❜❡❢♦r❡ t✐♠❡ tv ❤❛s s✐③❡ ♠♦r❡ t❤❛♥ α✱ ❛♥❞ ✸✳ ❛t ♠♦st β ❞❡s❝❡♥❞❛♥ts ♦❢ v ❛r❡ r❡❛❝❤❛❜❧❡ ❢r♦♠ v ❛❢t❡r t✐♠❡ tv✳

❚❤✐s ❣❡♥❡r❛❧ ♠❡t❤♦❞ ❝❛♥ ❛❧s♦ ❜❡ ❣❡♥❡r❛❧✐s❡❞ t♦ ❣r❛♣❤s ♦❢ ❜♦✉♥❞❡❞ tr❡❡✇✐❞t❤✳

❆♥ ❛❧❣♦r✐t❤♠ ❢♦r tr❡❡s

❚❤❡ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ ❧✐♥❡❛r t✐♠❡ ♦♥ tr❡❡s ✇❤❡♥ k ✭♠❛① ♣❡r♠✐tt❡❞ r❡❛❝❤❛❜✐❧✐t②✮ ✐s ❜♦✉♥❞❡❞ ❜② ❛ ✜①❡❞ ❝♦♥st❛♥t✿

◮ ❘❡❝❛❧❧✿ ✐❢ ♠❛① ❞❡❣r❡❡ ≥ k t❤❡♥ ✇❡ ❤❛✈❡ ❛ ♥♦✲✐♥st❛♥❝❡ ◮ ❙t❛rt✐♥❣ ❛t ❧❡❛✈❡s✱ ❛♥❞ ✇♦r❦✐♥❣ t♦✇❛r❞s t❤❡ r♦♦t✿

◮ ❈♦♥s✐❞❡r ❛❧❧ r❡❧❛t✐✈❡ ♦r❞❡r✐♥❣s ♦❢ t❤❡ ❡❞❣❡s ✐♥❝✐❞❡♥t ✇✐t❤ ❛

s✐♥❣❧❡ ✈❡rt❡① v✱ ❛♥❞ s✉♣♣♦s❡ t❤❛t t❤❡ ❡❞❣❡ ❢r♦♠ v t♦ ✐ts ♣❛r❡♥t ✐s ❛❝t✐✈❡ ❛t t✐♠❡ tv✳

◮ ❋♦r ❡❛❝❤ ♣♦ss✐❜❧❡ ♦r❞❡r✐♥❣✱ ❛♥❞ ❡✈❡r② ♣❛✐r (α, β) ✇✐t❤

✶ ≤ α, β ≤ k✱ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r t❤❡r❡ ✐s ❛♥ ♦r❞❡r✐♥❣ ♦❢ ❛❧❧ ❡❞❣❡s ✐♥❝✐❞❡♥t ✇✐t❤ v ❛♥❞ ✐ts ❞❡s❝❡♥❞❛♥ts s✉❝❤ t❤❛t✿

✶✳ ♥♦ ❞❡s❝❡♥❞❛♥t ♦❢ v ❤❛s ❛ r❡❛❝❤❛❜✐❧✐t② s❡t ♦❢ s✐③❡ > k✱ ✷✳ ♥♦ ❞❡s❝❡♥❞❛♥t ♦❢ v ✇❤✐❝❤ r❡❛❝❤❡s v ❜❡❢♦r❡ t✐♠❡ tv ❤❛s s✐③❡ ♠♦r❡ t❤❛♥ α✱ ❛♥❞ ✸✳ ❛t ♠♦st β ❞❡s❝❡♥❞❛♥ts ♦❢ v ❛r❡ r❡❛❝❤❛❜❧❡ ❢r♦♠ v ❛❢t❡r t✐♠❡ tv✳

❚❤✐s ❣❡♥❡r❛❧ ♠❡t❤♦❞ ❝❛♥ ❛❧s♦ ❜❡ ❣❡♥❡r❛❧✐s❡❞ t♦ ❣r❛♣❤s ♦❢ ❜♦✉♥❞❡❞ tr❡❡✇✐❞t❤✳

• ❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ✭❝♦♥t✳✮

◮ ❈❤❛♥❣✐♥❣ t❤❡ r❡❧❛t✐✈❡ ♦r❞❡r ♦❢ t✇♦ ♥♦♥✲✐♥❝✐❞❡♥t ❡❞❣❡s ✭✇❤✐❧❡

❧❡❛✈✐♥❣ ❛❧❧ ♦t❤❡r r❡❧❛t✐✈❡ ♦r❞❡r✐♥❣s ✉♥❝❤❛♥❣❡❞✮ ❞♦❡s♥✬t ❝❤❛♥❣❡ ❛♥② r❡❛❝❤❛❜✐❧✐t② s❡ts ■❢ t❤❡ ❡❞❣❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ t❤❡ ❣r❛♣❤ ✐s ✱ t❤❡r❡ ✐s ❛♥ ♦r❞❡r✐♥❣ ♦❢ t❤❡ ❡❞❣❡s s♦ t❤❛t ♥♦ r❡❛❝❤❛❜✐❧✐t② s❡t ❤❛s s✐③❡ ♠♦r❡ t❤❛♥ ✷ ✿

✇❡ ❝❛♥ ❞❡❝♦♠♣♦s❡ t❤❡ ❡❞❣❡s ✐♥t♦ ❛t ♠♦st s❡ts ♦❢ ❞✐s❥♦✐♥t ❡❞❣❡s ❛❞❞✐♥❣ ❛ s❡t ♦❢ ❞✐s❥♦✐♥t ❡❞❣❡s✱ ✇✐t❤ ❛❝t✐✈❡ t✐♠❡s ❧❛t❡r t❤❛♥ ❛♥② ♦t❤❡r ❡❞❣❡s✱ ❛t ♠♦st ❞♦✉❜❧❡s t❤❡ s✐③❡ ♦❢ ❛♥② r❡❛❝❤❛❜✐❧✐t② s❡t

• ❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ✭❝♦♥t✳✮

◮ ❈❤❛♥❣✐♥❣ t❤❡ r❡❧❛t✐✈❡ ♦r❞❡r ♦❢ t✇♦ ♥♦♥✲✐♥❝✐❞❡♥t ❡❞❣❡s ✭✇❤✐❧❡

❧❡❛✈✐♥❣ ❛❧❧ ♦t❤❡r r❡❧❛t✐✈❡ ♦r❞❡r✐♥❣s ✉♥❝❤❛♥❣❡❞✮ ❞♦❡s♥✬t ❝❤❛♥❣❡ ❛♥② r❡❛❝❤❛❜✐❧✐t② s❡ts

◮ ■❢ t❤❡ ❡❞❣❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ t❤❡ ❣r❛♣❤ ✐s c✱ t❤❡r❡ ✐s ❛♥

♦r❞❡r✐♥❣ ♦❢ t❤❡ ❡❞❣❡s s♦ t❤❛t ♥♦ r❡❛❝❤❛❜✐❧✐t② s❡t ❤❛s s✐③❡ ♠♦r❡ t❤❛♥ ✷c✿

✇❡ ❝❛♥ ❞❡❝♦♠♣♦s❡ t❤❡ ❡❞❣❡s ✐♥t♦ ❛t ♠♦st s❡ts ♦❢ ❞✐s❥♦✐♥t ❡❞❣❡s ❛❞❞✐♥❣ ❛ s❡t ♦❢ ❞✐s❥♦✐♥t ❡❞❣❡s✱ ✇✐t❤ ❛❝t✐✈❡ t✐♠❡s ❧❛t❡r t❤❛♥ ❛♥② ♦t❤❡r ❡❞❣❡s✱ ❛t ♠♦st ❞♦✉❜❧❡s t❤❡ s✐③❡ ♦❢ ❛♥② r❡❛❝❤❛❜✐❧✐t② s❡t

• ❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ✭❝♦♥t✳✮

◮ ❈❤❛♥❣✐♥❣ t❤❡ r❡❧❛t✐✈❡ ♦r❞❡r ♦❢ t✇♦ ♥♦♥✲✐♥❝✐❞❡♥t ❡❞❣❡s ✭✇❤✐❧❡

❧❡❛✈✐♥❣ ❛❧❧ ♦t❤❡r r❡❧❛t✐✈❡ ♦r❞❡r✐♥❣s ✉♥❝❤❛♥❣❡❞✮ ❞♦❡s♥✬t ❝❤❛♥❣❡ ❛♥② r❡❛❝❤❛❜✐❧✐t② s❡ts

◮ ■❢ t❤❡ ❡❞❣❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ t❤❡ ❣r❛♣❤ ✐s c✱ t❤❡r❡ ✐s ❛♥

♦r❞❡r✐♥❣ ♦❢ t❤❡ ❡❞❣❡s s♦ t❤❛t ♥♦ r❡❛❝❤❛❜✐❧✐t② s❡t ❤❛s s✐③❡ ♠♦r❡ t❤❛♥ ✷c✿

◮ ✇❡ ❝❛♥ ❞❡❝♦♠♣♦s❡ t❤❡ ❡❞❣❡s ✐♥t♦ ❛t ♠♦st c s❡ts ♦❢ ❞✐s❥♦✐♥t

❡❞❣❡s ❛❞❞✐♥❣ ❛ s❡t ♦❢ ❞✐s❥♦✐♥t ❡❞❣❡s✱ ✇✐t❤ ❛❝t✐✈❡ t✐♠❡s ❧❛t❡r t❤❛♥ ❛♥② ♦t❤❡r ❡❞❣❡s✱ ❛t ♠♦st ❞♦✉❜❧❡s t❤❡ s✐③❡ ♦❢ ❛♥② r❡❛❝❤❛❜✐❧✐t② s❡t

• ❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ✭❝♦♥t✳✮

◮ ❈❤❛♥❣✐♥❣ t❤❡ r❡❧❛t✐✈❡ ♦r❞❡r ♦❢ t✇♦ ♥♦♥✲✐♥❝✐❞❡♥t ❡❞❣❡s ✭✇❤✐❧❡

❧❡❛✈✐♥❣ ❛❧❧ ♦t❤❡r r❡❧❛t✐✈❡ ♦r❞❡r✐♥❣s ✉♥❝❤❛♥❣❡❞✮ ❞♦❡s♥✬t ❝❤❛♥❣❡ ❛♥② r❡❛❝❤❛❜✐❧✐t② s❡ts

◮ ■❢ t❤❡ ❡❞❣❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ t❤❡ ❣r❛♣❤ ✐s c✱ t❤❡r❡ ✐s ❛♥

♦r❞❡r✐♥❣ ♦❢ t❤❡ ❡❞❣❡s s♦ t❤❛t ♥♦ r❡❛❝❤❛❜✐❧✐t② s❡t ❤❛s s✐③❡ ♠♦r❡ t❤❛♥ ✷c✿

◮ ✇❡ ❝❛♥ ❞❡❝♦♠♣♦s❡ t❤❡ ❡❞❣❡s ✐♥t♦ ❛t ♠♦st c s❡ts ♦❢ ❞✐s❥♦✐♥t

❡❞❣❡s

◮ ❛❞❞✐♥❣ ❛ s❡t ♦❢ ❞✐s❥♦✐♥t ❡❞❣❡s✱ ✇✐t❤ ❛❝t✐✈❡ t✐♠❡s ❧❛t❡r t❤❛♥ ❛♥②

♦t❤❡r ❡❞❣❡s✱ ❛t ♠♦st ❞♦✉❜❧❡s t❤❡ s✐③❡ ♦❢ ❛♥② r❡❛❝❤❛❜✐❧✐t② s❡t

• ❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ❛♥❞ ❝♦♥s❡q✉❡♥❝❡s ✭❝♦♥t✳✮

❚❤❡♦r❡♠

❚❤❡ s♠❛❧❧❡st ❛❝❤✐❡✈❛❜❧❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ❢♦r ❛ ✭❧❛r❣❡ ❡♥♦✉❣❤✮ ❜✐♥❛r② tr❡❡ ✐s ❡①❛❝t❧② ❡✐❣❤t✳

❆♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

❚❤❡♦r❡♠

• ✐✈❡♥ ❛♥② ❣r❛♣❤ G✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ ❛ ✷∆(G)+✶

∆(G)+✶✲❛♣♣r♦①✐♠❛t✐♦♥ t♦

t❤❡ s♠❛❧❧❡st ❛❝❤✐❡✈❛❜❧❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ✐♥ ❧✐♥❡❛r t✐♠❡✱ ✇❤❡r❡ ∆(G) ❞❡♥♦t❡s t❤❡ ♠❛①✐♠✉♠ ❞❡❣r❡❡ ♦❢ G✳ ▼♦r❡♦✈❡r✱ ✇❡ ❝❛♥ ❛❧s♦ ❝♦♠♣✉t❡ ❛♥ ❛ss✐❣♥♠❡♥t ♦❢ t✐♠❡s t♦ ❡❞❣❡s ✇❤✐❝❤ ❛❝❤✐❡✈❡s t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

❈♦r♦❧❧❛r②

• ✐✈❡♥ ❛♥② ❣r❛♣❤

♦❢ ❜♦✉♥❞❡❞ ♠❛①✐♠✉♠ ❞❡❣r❡❡✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ ❛ ❝♦♥st❛♥t ❢❛❝t♦r ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ ❧✐♥❡❛r t✐♠❡✳

❆♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

❚❤❡♦r❡♠

• ✐✈❡♥ ❛♥② ❣r❛♣❤ G✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ ❛ ✷∆(G)+✶

∆(G)+✶✲❛♣♣r♦①✐♠❛t✐♦♥ t♦

t❤❡ s♠❛❧❧❡st ❛❝❤✐❡✈❛❜❧❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ✐♥ ❧✐♥❡❛r t✐♠❡✱ ✇❤❡r❡ ∆(G) ❞❡♥♦t❡s t❤❡ ♠❛①✐♠✉♠ ❞❡❣r❡❡ ♦❢ G✳ ▼♦r❡♦✈❡r✱ ✇❡ ❝❛♥ ❛❧s♦ ❝♦♠♣✉t❡ ❛♥ ❛ss✐❣♥♠❡♥t ♦❢ t✐♠❡s t♦ ❡❞❣❡s ✇❤✐❝❤ ❛❝❤✐❡✈❡s t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

❈♦r♦❧❧❛r②

• ✐✈❡♥ ❛♥② ❣r❛♣❤ G ♦❢ ❜♦✉♥❞❡❞ ♠❛①✐♠✉♠ ❞❡❣r❡❡✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡

❛ ❝♦♥st❛♥t ❢❛❝t♦r ❛♣♣r♦①✐♠❛t✐♦♥ ✐♥ ❧✐♥❡❛r t✐♠❡✳

• ❡♥❡r❛❧✐s❛t✐♦♥s

❙♣❡❝✐✜❡❞ ❝❧❛ss❡s ♦❢ ❡❞❣❡s ❤❛✈❡ t♦ ❜❡ ❛❝t✐✈❡ s✐♠✉❧t❛♥❡♦✉s❧②✳

• ❡♥❡r❛❧✐s❛t✐♦♥s

◮ ❙♣❡❝✐✜❡❞ ❝❧❛ss❡s ♦❢ ❡❞❣❡s ❤❛✈❡ t♦ ❜❡ ❛❝t✐✈❡ s✐♠✉❧t❛♥❡♦✉s❧②✳

• ❡♥❡r❛❧✐s❛t✐♦♥s

❚❤❡♦r❡♠

❲✐t❤ t❤✐s ❣❡♥❡r❛❧✐s❛t✐♦♥✱ t❤❡ ♣r♦❜❧❡♠ ✐s ◆P✲❤❛r❞ ❡✈❡♥ ✐❢ t❤❡ ✐♥♣✉t ❣r❛♣❤ ✐s ✭❛✮ ❛ ❞✐r❡❝t❡❞ ❛❝②❝❧✐❝ ❣r❛♣❤✱ ♦r ✭❜✮ ❛ tr❡❡✳

❙❡ts ♦❢ s✐♠✉❧t❛♥❡♦✉s ❡❞❣❡s

❚❤❡♦r❡♠

❲❤❡♥ ✇❡ ❝❛♥ s♣❡❝✐❢② ❝❧❛ss❡s ♦❢ ❡❞❣❡s ✇❤✐❝❤ ♠✉st ❜❡ ❛❝t✐✈❡ s✐♠✉❧t❛♥❡♦✉s❧②✱ t❤❡ ♣r♦❜❧❡♠ ✐s ❲❬✶❪✲❤❛r❞ ♣❛r❛♠❡t❡r✐s❡❞ ❜② t❤❡ ✈❡rt❡① ❝♦✈❡r ♥✉♠❜❡r✱ ❡✈❡♥ ✇❤❡♥ r❡str✐❝t❡❞ t♦ tr❡❡s✳ ❘❡❞✉❝t✐♦♥ ❢r♦♠ ❈❧✐q✉❡✳ ❲❡ ❤❛✈❡ ♦♥❡ ❡❞❣❡✲❝❧❛ss ❢♦r ❡❛❝❤ ✈❡rt❡① ✐♥ t❤❡ ❈❧✐q✉❡ ✐♥st❛♥❝❡✳

❙❡ts ♦❢ s✐♠✉❧t❛♥❡♦✉s ❡❞❣❡s

❚❤❡♦r❡♠

❲❤❡♥ ✇❡ ❝❛♥ s♣❡❝✐❢② ❝❧❛ss❡s ♦❢ ❡❞❣❡s ✇❤✐❝❤ ♠✉st ❜❡ ❛❝t✐✈❡ s✐♠✉❧t❛♥❡♦✉s❧②✱ t❤❡ ♣r♦❜❧❡♠ ✐s ❲❬✶❪✲❤❛r❞ ♣❛r❛♠❡t❡r✐s❡❞ ❜② t❤❡ ✈❡rt❡① ❝♦✈❡r ♥✉♠❜❡r✱ ❡✈❡♥ ✇❤❡♥ r❡str✐❝t❡❞ t♦ tr❡❡s✳

◮ ❘❡❞✉❝t✐♦♥ ❢r♦♠ ❈❧✐q✉❡✳ ◮ ❲❡ ❤❛✈❡ ♦♥❡ ❡❞❣❡✲❝❧❛ss ❢♦r ❡❛❝❤ ✈❡rt❡① ✐♥ t❤❡ ❈❧✐q✉❡ ✐♥st❛♥❝❡✳ One for each edge in the CLIQUE instance r edges, each belonging to every edge-class The edge corresponding to e = vivj belongs to edge-classes Ei and Ej

s

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❙❡ts ♦❢ s✐♠✉❧t❛♥❡♦✉s ❡❞❣❡s

❚❤❡♦r❡♠

❲❤❡♥ ✇❡ ❝❛♥ s♣❡❝✐❢② ❝❧❛ss❡s ♦❢ ❡❞❣❡s ✇❤✐❝❤ ♠✉st ❜❡ ❛❝t✐✈❡ s✐♠✉❧t❛♥❡♦✉s❧②✱ t❤❡ ♣r♦❜❧❡♠ ✐s ❲❬✶❪✲❤❛r❞ ♣❛r❛♠❡t❡r✐s❡❞ ❜② t❤❡ ✈❡rt❡① ❝♦✈❡r ♥✉♠❜❡r✱ ❡✈❡♥ ✇❤❡♥ r❡str✐❝t❡❞ t♦ tr❡❡s✳

◮ ❘❡❞✉❝t✐♦♥ ❢r♦♠ ❈❧✐q✉❡✳ ◮ ❲❡ ❤❛✈❡ ♦♥❡ ❡❞❣❡✲❝❧❛ss ❢♦r ❡❛❝❤ ✈❡rt❡① ✐♥ t❤❡ ❈❧✐q✉❡ ✐♥st❛♥❝❡✳ One for each edge in the CLIQUE instance r edges, each belonging to every edge-class The edge corresponding to e = vivj belongs to edge-classes Ei and Ej

s

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❚❤❡ ❡❞❣❡✲❝❧❛ss ✐♥t❡r❛❝t✐♦♥ ❣r❛♣❤

❚❤❡ ❡❞❣❡✲❝❧❛ss ✐♥t❡r❛❝t✐♦♥ ❣r❛♣❤ ❤❛s t❤❡ s❡t ♦❢ ❡❞❣❡✲❝❧❛ss❡s ❛s ✐ts ✈❡rt❡① s❡t✱ ❛♥❞ t✇♦ ❡❞❣❡✲❝❧❛ss❡s ❛r❡ ❛❞❥❛❝❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ s♦♠❡ ✈❡rt❡① ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❣r❛♣❤ ✐s ✐♥❝✐❞❡♥t ✇✐t❤ ❛♥ ❡❞❣❡ ❢r♦♠ ❜♦t❤ ❝❧❛ss❡s✳

❚❤❡♦r❡♠

❚❤❡r❡ ✐s ❛ ♦r❞❡r✐♥❣ ♦❢ t❤❡ ❡❞❣❡✲❝❧❛ss❡s s✉❝❤ t❤❛t t❤❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ✐s ❛t ♠♦st ✶ ✱ ✇❤❡r❡ ✐s t❤❡ ♠❛①✐♠✉♠ ❞❡❣r❡❡ ✐♥ ❛♥② s✐♥❣❧❡ ❡❞❣❡ ❝❧❛ss✱ ❛♥❞ ✐s t❤❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ t❤❡ ❡❞❣❡✲❝❧❛ss ✐♥t❡r❛❝t✐♦♥ ❣r❛♣❤✳

❈♦r♦❧❧❛r②

❲❡ ❝❛♥ ❝♦♠♣✉t❡ ❛ ✲❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ s♠❛❧❧❡st ❛❝❤✐❡✈❛❜❧❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✐❢ t❤❡ ❡❞❣❡✲❝❧❛ss ✐♥t❡r❛❝t✐♦♥ ❣r❛♣❤ ✐s ❜✐♣❛rt✐t❡✱ ✇❤❡r❡ ✐s t❤❡ ♠❛①✐♠✉♠ ❞❡❣r❡❡ ♦❢ ❛♥② s✐♥❣❧❡ ❡❞❣❡ ❝❧❛ss✳

❚❤❡ ❡❞❣❡✲❝❧❛ss ✐♥t❡r❛❝t✐♦♥ ❣r❛♣❤

❚❤❡ ❡❞❣❡✲❝❧❛ss ✐♥t❡r❛❝t✐♦♥ ❣r❛♣❤ ❤❛s t❤❡ s❡t ♦❢ ❡❞❣❡✲❝❧❛ss❡s ❛s ✐ts ✈❡rt❡① s❡t✱ ❛♥❞ t✇♦ ❡❞❣❡✲❝❧❛ss❡s ❛r❡ ❛❞❥❛❝❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ s♦♠❡ ✈❡rt❡① ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❣r❛♣❤ ✐s ✐♥❝✐❞❡♥t ✇✐t❤ ❛♥ ❡❞❣❡ ❢r♦♠ ❜♦t❤ ❝❧❛ss❡s✳

❚❤❡♦r❡♠

❚❤❡r❡ ✐s ❛ ♦r❞❡r✐♥❣ ♦❢ t❤❡ ❡❞❣❡✲❝❧❛ss❡s s✉❝❤ t❤❛t t❤❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ✐s ❛t ♠♦st (d + ✶)χ✱ ✇❤❡r❡

◮ d ✐s t❤❡ ♠❛①✐♠✉♠ ❞❡❣r❡❡ ✐♥ ❛♥② s✐♥❣❧❡ ❡❞❣❡ ❝❧❛ss✱ ❛♥❞ ◮ χ ✐s t❤❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ t❤❡ ❡❞❣❡✲❝❧❛ss ✐♥t❡r❛❝t✐♦♥ ❣r❛♣❤✳

❈♦r♦❧❧❛r②

❲❡ ❝❛♥ ❝♦♠♣✉t❡ ❛ ✲❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ s♠❛❧❧❡st ❛❝❤✐❡✈❛❜❧❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✐❢ t❤❡ ❡❞❣❡✲❝❧❛ss ✐♥t❡r❛❝t✐♦♥ ❣r❛♣❤ ✐s ❜✐♣❛rt✐t❡✱ ✇❤❡r❡ ✐s t❤❡ ♠❛①✐♠✉♠ ❞❡❣r❡❡ ♦❢ ❛♥② s✐♥❣❧❡ ❡❞❣❡ ❝❧❛ss✳

❚❤❡ ❡❞❣❡✲❝❧❛ss ✐♥t❡r❛❝t✐♦♥ ❣r❛♣❤

❚❤❡ ❡❞❣❡✲❝❧❛ss ✐♥t❡r❛❝t✐♦♥ ❣r❛♣❤ ❤❛s t❤❡ s❡t ♦❢ ❡❞❣❡✲❝❧❛ss❡s ❛s ✐ts ✈❡rt❡① s❡t✱ ❛♥❞ t✇♦ ❡❞❣❡✲❝❧❛ss❡s ❛r❡ ❛❞❥❛❝❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ s♦♠❡ ✈❡rt❡① ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❣r❛♣❤ ✐s ✐♥❝✐❞❡♥t ✇✐t❤ ❛♥ ❡❞❣❡ ❢r♦♠ ❜♦t❤ ❝❧❛ss❡s✳

❚❤❡♦r❡♠

❚❤❡r❡ ✐s ❛ ♦r❞❡r✐♥❣ ♦❢ t❤❡ ❡❞❣❡✲❝❧❛ss❡s s✉❝❤ t❤❛t t❤❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ✐s ❛t ♠♦st (d + ✶)χ✱ ✇❤❡r❡

◮ d ✐s t❤❡ ♠❛①✐♠✉♠ ❞❡❣r❡❡ ✐♥ ❛♥② s✐♥❣❧❡ ❡❞❣❡ ❝❧❛ss✱ ❛♥❞ ◮ χ ✐s t❤❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ t❤❡ ❡❞❣❡✲❝❧❛ss ✐♥t❡r❛❝t✐♦♥ ❣r❛♣❤✳

❈♦r♦❧❧❛r②

❲❡ ❝❛♥ ❝♦♠♣✉t❡ ❛ d✲❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ s♠❛❧❧❡st ❛❝❤✐❡✈❛❜❧❡ ♠❛①✐♠✉♠ r❡❛❝❤❛❜✐❧✐t② ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✐❢ t❤❡ ❡❞❣❡✲❝❧❛ss ✐♥t❡r❛❝t✐♦♥ ❣r❛♣❤ ✐s ❜✐♣❛rt✐t❡✱ ✇❤❡r❡ d ✐s t❤❡ ♠❛①✐♠✉♠ ❞❡❣r❡❡ ♦❢ ❛♥② s✐♥❣❧❡ ❡❞❣❡ ❝❧❛ss✳

❋✉t✉r❡ ❞✐r❡❝t✐♦♥s

◮ ❊①♣❧♦r❡ ♥❡✇ t❡❝❤♥✐q✉❡s ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ✭✇✐t❤

❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦s✮ ■♥❝❧✉❞❡ ❧✐sts ♦❢ ♣❡r♠✐tt❡❞ t✐♠❡s ❢♦r ❡❛❝❤ ❡❞❣❡✭✲❝❧❛ss✮✱ ♣♦ss✐❜❧② ✇✐t❤ ❛ss♦❝✐❛t❡❞ ❝♦sts ❈♦♥s✐❞❡r ♠❛①✐♠✐s✐♥❣ t❤❡ s✐③❡ ♦❢ t❤❡ s♠❛❧❧❡st r❡❛❝❤❛❜✐❧✐t② s❡t ❲❤❛t ❛❜♦✉t ♠✐♥✐♠✐s✐♥❣✴♠❛①✐♠✐s✐♥❣ ✐♥st❡❛❞✿

t❤❡ ❛✈❡r❛❣❡ s✐③❡ ♦❢ r❡❛❝❤❛❜✐❧✐t② s❡ts✱ ♦r t❤❡ ❡①♣❡❝t❡❞ ✇♦rst✲❝❛s❡ ♦r ❛✈❡r❛❣❡ r❡❛❝❤❛❜✐❧✐t② s❡t ✐♥ ❛ ♠♦❞❡❧ ✇❤❡r❡ ❡❛❝❤ ❡❞❣❡ ❤❛s ❛♥ ❛ss♦❝✐❛t❡❞ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❛❜✐❧✐t②❄

❙②st❡♠s ✇✐t❤ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ st❛t✐❝ ❛♥❞ r❡♦r❞❡r❛❜❧❡ ❡❞❣❡s

❋✉t✉r❡ ❞✐r❡❝t✐♦♥s

◮ ❊①♣❧♦r❡ ♥❡✇ t❡❝❤♥✐q✉❡s ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ✭✇✐t❤

❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦s✮

◮ ■♥❝❧✉❞❡ ❧✐sts ♦❢ ♣❡r♠✐tt❡❞ t✐♠❡s ❢♦r ❡❛❝❤ ❡❞❣❡✭✲❝❧❛ss✮✱ ♣♦ss✐❜❧②

✇✐t❤ ❛ss♦❝✐❛t❡❞ ❝♦sts ❈♦♥s✐❞❡r ♠❛①✐♠✐s✐♥❣ t❤❡ s✐③❡ ♦❢ t❤❡ s♠❛❧❧❡st r❡❛❝❤❛❜✐❧✐t② s❡t ❲❤❛t ❛❜♦✉t ♠✐♥✐♠✐s✐♥❣✴♠❛①✐♠✐s✐♥❣ ✐♥st❡❛❞✿

t❤❡ ❛✈❡r❛❣❡ s✐③❡ ♦❢ r❡❛❝❤❛❜✐❧✐t② s❡ts✱ ♦r t❤❡ ❡①♣❡❝t❡❞ ✇♦rst✲❝❛s❡ ♦r ❛✈❡r❛❣❡ r❡❛❝❤❛❜✐❧✐t② s❡t ✐♥ ❛ ♠♦❞❡❧ ✇❤❡r❡ ❡❛❝❤ ❡❞❣❡ ❤❛s ❛♥ ❛ss♦❝✐❛t❡❞ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❛❜✐❧✐t②❄

❙②st❡♠s ✇✐t❤ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ st❛t✐❝ ❛♥❞ r❡♦r❞❡r❛❜❧❡ ❡❞❣❡s

❋✉t✉r❡ ❞✐r❡❝t✐♦♥s

◮ ❊①♣❧♦r❡ ♥❡✇ t❡❝❤♥✐q✉❡s ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ✭✇✐t❤

❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦s✮

◮ ■♥❝❧✉❞❡ ❧✐sts ♦❢ ♣❡r♠✐tt❡❞ t✐♠❡s ❢♦r ❡❛❝❤ ❡❞❣❡✭✲❝❧❛ss✮✱ ♣♦ss✐❜❧②

✇✐t❤ ❛ss♦❝✐❛t❡❞ ❝♦sts

◮ ❈♦♥s✐❞❡r ♠❛①✐♠✐s✐♥❣ t❤❡ s✐③❡ ♦❢ t❤❡ s♠❛❧❧❡st r❡❛❝❤❛❜✐❧✐t② s❡t

❲❤❛t ❛❜♦✉t ♠✐♥✐♠✐s✐♥❣✴♠❛①✐♠✐s✐♥❣ ✐♥st❡❛❞✿

t❤❡ ❛✈❡r❛❣❡ s✐③❡ ♦❢ r❡❛❝❤❛❜✐❧✐t② s❡ts✱ ♦r t❤❡ ❡①♣❡❝t❡❞ ✇♦rst✲❝❛s❡ ♦r ❛✈❡r❛❣❡ r❡❛❝❤❛❜✐❧✐t② s❡t ✐♥ ❛ ♠♦❞❡❧ ✇❤❡r❡ ❡❛❝❤ ❡❞❣❡ ❤❛s ❛♥ ❛ss♦❝✐❛t❡❞ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❛❜✐❧✐t②❄

❙②st❡♠s ✇✐t❤ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ st❛t✐❝ ❛♥❞ r❡♦r❞❡r❛❜❧❡ ❡❞❣❡s

❋✉t✉r❡ ❞✐r❡❝t✐♦♥s

◮ ❊①♣❧♦r❡ ♥❡✇ t❡❝❤♥✐q✉❡s ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ✭✇✐t❤

❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦s✮

◮ ■♥❝❧✉❞❡ ❧✐sts ♦❢ ♣❡r♠✐tt❡❞ t✐♠❡s ❢♦r ❡❛❝❤ ❡❞❣❡✭✲❝❧❛ss✮✱ ♣♦ss✐❜❧②

✇✐t❤ ❛ss♦❝✐❛t❡❞ ❝♦sts

◮ ❈♦♥s✐❞❡r ♠❛①✐♠✐s✐♥❣ t❤❡ s✐③❡ ♦❢ t❤❡ s♠❛❧❧❡st r❡❛❝❤❛❜✐❧✐t② s❡t ◮ ❲❤❛t ❛❜♦✉t ♠✐♥✐♠✐s✐♥❣✴♠❛①✐♠✐s✐♥❣ ✐♥st❡❛❞✿

◮ t❤❡ ❛✈❡r❛❣❡ s✐③❡ ♦❢ r❡❛❝❤❛❜✐❧✐t② s❡ts✱ ♦r ◮ t❤❡ ❡①♣❡❝t❡❞ ✇♦rst✲❝❛s❡ ♦r ❛✈❡r❛❣❡ r❡❛❝❤❛❜✐❧✐t② s❡t ✐♥ ❛ ♠♦❞❡❧

✇❤❡r❡ ❡❛❝❤ ❡❞❣❡ ❤❛s ❛♥ ❛ss♦❝✐❛t❡❞ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❛❜✐❧✐t②❄

❙②st❡♠s ✇✐t❤ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ st❛t✐❝ ❛♥❞ r❡♦r❞❡r❛❜❧❡ ❡❞❣❡s

❋✉t✉r❡ ❞✐r❡❝t✐♦♥s

◮ ❊①♣❧♦r❡ ♥❡✇ t❡❝❤♥✐q✉❡s ❢♦r ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ✭✇✐t❤

❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦s✮

◮ ■♥❝❧✉❞❡ ❧✐sts ♦❢ ♣❡r♠✐tt❡❞ t✐♠❡s ❢♦r ❡❛❝❤ ❡❞❣❡✭✲❝❧❛ss✮✱ ♣♦ss✐❜❧②

✇✐t❤ ❛ss♦❝✐❛t❡❞ ❝♦sts

◮ ❈♦♥s✐❞❡r ♠❛①✐♠✐s✐♥❣ t❤❡ s✐③❡ ♦❢ t❤❡ s♠❛❧❧❡st r❡❛❝❤❛❜✐❧✐t② s❡t ◮ ❲❤❛t ❛❜♦✉t ♠✐♥✐♠✐s✐♥❣✴♠❛①✐♠✐s✐♥❣ ✐♥st❡❛❞✿

◮ t❤❡ ❛✈❡r❛❣❡ s✐③❡ ♦❢ r❡❛❝❤❛❜✐❧✐t② s❡ts✱ ♦r ◮ t❤❡ ❡①♣❡❝t❡❞ ✇♦rst✲❝❛s❡ ♦r ❛✈❡r❛❣❡ r❡❛❝❤❛❜✐❧✐t② s❡t ✐♥ ❛ ♠♦❞❡❧

✇❤❡r❡ ❡❛❝❤ ❡❞❣❡ ❤❛s ❛♥ ❛ss♦❝✐❛t❡❞ tr❛♥s♠✐ss✐♦♥ ♣r♦❜❛❜✐❧✐t②❄

◮ ❙②st❡♠s ✇✐t❤ ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ st❛t✐❝ ❛♥❞ r❡♦r❞❡r❛❜❧❡ ❡❞❣❡s

❋✉t✉r❡ ❞✐r❡❝t✐♦♥s ❚❍❆◆❑ ❨❖❯

❛r①✐✈✳♦r❣✴❛❜s✴✶✽✵✷✳✵✺✾✵✺

❋✉t✉r❡ ❞✐r❡❝t✐♦♥s ❚❍❆◆❑ ❨❖❯

❛r①✐✈✳♦r❣✴❛❜s✴✶✽✵✷✳✵✺✾✵✺