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t rt r r t Prr s PPt rst

  1. ❉❡❝✐❞✐♥❣ t❤❡ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ♦❢ ❛ ●r❛♣❤ ✇✐t❤ Pr❡✲❈♦❧♦r✐♥❣ ✐s P❙P❆❈❊✲❈♦♠♣❧❡t❡ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ❯♥✐✈❡rs✐t② ♦❢ ❙♦✉t❤❡r♥ ❉❡♥♠❛r❦ ❦✉❞❛❤❧❅✐♠❛❞❛✳s❞✉✳❞❦ ❏✉❧② ✼✱ ✷✵✶✹ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✶ ✴ ✷✵

  2. ❲❤❡♥ ✈❡rt❡①✱ ✱ ✐s r❡✈❡❛❧❡❞✱ ✐t ✐s r❡✈❡❛❧❡❞ ✇❤✐❝❤ ✈❡rt✐❝❡s ✐t ✐s ❛❞❥❛❝❡♥t t♦ ✭❛♠♦♥❣ t❤♦s❡ t❤❛t ✇❡r❡ ♣r❡✈✐♦✉s❧② r❡✈❡❛❧❡❞✮✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ❛ss✐❣♥s ❛ ❝♦❧♦r t♦ ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❝♦❧♦rs ✐t ✐s ❛❧r❡❛❞② ❛❞❥❛❝❡♥t t♦✳ ❚❤❡ ❝♦❧♦r ❝❛♥♥♦t ❜❡ ❝❤❛♥❣❡❞ ❧❛t❡r✳ ❚❤❡ ❣♦❛❧ ✐s t♦ ✉s❡ ❛s ❢❡✇ ❝♦❧♦rs ❛s ♣♦ss✐❜❧❡✳ ❖♥✲❧✐♥❡ ❣r❛♣❤ ❝♦❧♦r✐♥❣ ❚❤❡ ✈❡rt✐❝❡s ♦❢ ❛ ❣r❛♣❤ ❛r❡ r❡✈❡❛❧❡❞ ♦♥❡ ❜② ♦♥❡✳ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✷ ✴ ✷✵

  3. ❚❤❡ ❛❧❣♦r✐t❤♠ ❛ss✐❣♥s ❛ ❝♦❧♦r t♦ ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❝♦❧♦rs ✐t ✐s ❛❧r❡❛❞② ❛❞❥❛❝❡♥t t♦✳ ❚❤❡ ❝♦❧♦r ❝❛♥♥♦t ❜❡ ❝❤❛♥❣❡❞ ❧❛t❡r✳ ❚❤❡ ❣♦❛❧ ✐s t♦ ✉s❡ ❛s ❢❡✇ ❝♦❧♦rs ❛s ♣♦ss✐❜❧❡✳ ❖♥✲❧✐♥❡ ❣r❛♣❤ ❝♦❧♦r✐♥❣ ❚❤❡ ✈❡rt✐❝❡s ♦❢ ❛ ❣r❛♣❤ ❛r❡ r❡✈❡❛❧❡❞ ♦♥❡ ❜② ♦♥❡✳ ❲❤❡♥ ✈❡rt❡①✱ v ✱ ✐s r❡✈❡❛❧❡❞✱ ✐t ✐s r❡✈❡❛❧❡❞ ✇❤✐❝❤ ✈❡rt✐❝❡s ✐t ✐s ❛❞❥❛❝❡♥t t♦ ✭❛♠♦♥❣ t❤♦s❡ t❤❛t ✇❡r❡ ♣r❡✈✐♦✉s❧② r❡✈❡❛❧❡❞✮✳ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✷ ✴ ✷✵

  4. ❚❤❡ ❣♦❛❧ ✐s t♦ ✉s❡ ❛s ❢❡✇ ❝♦❧♦rs ❛s ♣♦ss✐❜❧❡✳ ❖♥✲❧✐♥❡ ❣r❛♣❤ ❝♦❧♦r✐♥❣ ❚❤❡ ✈❡rt✐❝❡s ♦❢ ❛ ❣r❛♣❤ ❛r❡ r❡✈❡❛❧❡❞ ♦♥❡ ❜② ♦♥❡✳ ❲❤❡♥ ✈❡rt❡①✱ v ✱ ✐s r❡✈❡❛❧❡❞✱ ✐t ✐s r❡✈❡❛❧❡❞ ✇❤✐❝❤ ✈❡rt✐❝❡s ✐t ✐s ❛❞❥❛❝❡♥t t♦ ✭❛♠♦♥❣ t❤♦s❡ t❤❛t ✇❡r❡ ♣r❡✈✐♦✉s❧② r❡✈❡❛❧❡❞✮✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ❛ss✐❣♥s ❛ ❝♦❧♦r t♦ v ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❝♦❧♦rs ✐t ✐s ❛❧r❡❛❞② ❛❞❥❛❝❡♥t t♦✳ ❚❤❡ ❝♦❧♦r ❝❛♥♥♦t ❜❡ ❝❤❛♥❣❡❞ ❧❛t❡r✳ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✷ ✴ ✷✵

  5. ❖♥✲❧✐♥❡ ❣r❛♣❤ ❝♦❧♦r✐♥❣ ❚❤❡ ✈❡rt✐❝❡s ♦❢ ❛ ❣r❛♣❤ ❛r❡ r❡✈❡❛❧❡❞ ♦♥❡ ❜② ♦♥❡✳ ❲❤❡♥ ✈❡rt❡①✱ v ✱ ✐s r❡✈❡❛❧❡❞✱ ✐t ✐s r❡✈❡❛❧❡❞ ✇❤✐❝❤ ✈❡rt✐❝❡s ✐t ✐s ❛❞❥❛❝❡♥t t♦ ✭❛♠♦♥❣ t❤♦s❡ t❤❛t ✇❡r❡ ♣r❡✈✐♦✉s❧② r❡✈❡❛❧❡❞✮✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ❛ss✐❣♥s ❛ ❝♦❧♦r t♦ v ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❝♦❧♦rs ✐t ✐s ❛❧r❡❛❞② ❛❞❥❛❝❡♥t t♦✳ ❚❤❡ ❝♦❧♦r ❝❛♥♥♦t ❜❡ ❝❤❛♥❣❡❞ ❧❛t❡r✳ ❚❤❡ ❣♦❛❧ ✐s t♦ ✉s❡ ❛s ❢❡✇ ❝♦❧♦rs ❛s ♣♦ss✐❜❧❡✳ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✷ ✴ ✷✵

  6. ❆♥ ❡①❛♠♣❧❡ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

  7. ❆♥ ❡①❛♠♣❧❡ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

  8. ❆♥ ❡①❛♠♣❧❡ ✶ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

  9. ❆♥ ❡①❛♠♣❧❡ ✶ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

  10. ❆♥ ❡①❛♠♣❧❡ ✶ ✶ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

  11. ❆♥ ❡①❛♠♣❧❡ ✶ ✶ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

  12. ❆♥ ❡①❛♠♣❧❡ ✶ ✷ ✶ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

  13. ❆♥ ❡①❛♠♣❧❡ ✶ ✷ ✶ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

  14. ❆♥ ❡①❛♠♣❧❡ ✶ ✷ ✸ ✶ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✸ ✴ ✷✵

  15. ❍♦✇ t♦ ♠❡❛s✉r❡ t❤❡ q✉❛❧✐t② ♦❢ ❛♥ ❛❧❣♦r✐t❤♠❄ ❈♦♠♣❡t✐t✐✈❡ ❆♥❛❧②s✐s ❬❙❧❡❛t♦r✱ ❚❛r❥❛♥ ✽✺❪ ❆♥ ❛❧❣♦r✐t❤♠ A ✐s c ✲❝♦♠♣❡t✐t✐✈❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ b s✉❝❤ t❤❛t ❢♦r ❛♥② ❣r❛♣❤✱ G ✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ A ( G ) ≤ c · χ ( G ) + b ❍❡r❡✱ χ ( G ) ✐s t❤❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ G ✳ ❚❤❛t ✐s t❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ❝♦❧♦rs ♥❡❡❞❡❞ t♦ ❝♦❧♦r G ✭♦✤✐♥❡✮✳ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✹ ✴ ✷✵

  16. ❍♦✇ t♦ ♠❡❛s✉r❡ t❤❡ q✉❛❧✐t② ♦❢ ❛♥ ❛❧❣♦r✐t❤♠❄ ❚❤❡♦r❡♠ ❬●②❛r❢❛s✱ ❑✐r❛❧②✱ ▲❡❤❡❧ ✶✾✾✵❪ ❋♦r ❛♥② k ∈ N ✱ t❤❡r❡ ❡①✐sts ❛ tr❡❡ T k s✉❝❤ t❤❛t ❛♥② ♦♥✲❧✐♥❡ ❛❧❣♦r✐t❤♠ ❝❛♥ ❜❡ ❢♦r❝❡❞ t♦ ✉s❡ k ❝♦❧♦rs ✇❤❡♥ ♦♥✲❧✐♥❡ ❝♦❧♦r✐♥❣ T k ✳ ❋✐❣✉r❡✿ T ✸ ❛♥❞ T ✹ ✳ ❚❤✐s ♠❡❛♥s t❤❛t ♥♦ c ✲❝♦♠♣❡t✐t✐✈❡ ❛❧❣♦r✐t❤♠s ❡①✐st ❢♦r ❛♥② ❝♦♥st❛♥t c ✲ ❡✈❡♥ ❢♦r t❤❡ ❝❧❛ss ♦❢ tr❡❡s✳ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✺ ✴ ✷✵

  17. ❍♦✇ t♦ ♠❡❛s✉r❡ t❤❡ q✉❛❧✐t② ♦❢ ❛♥ ❛❧❣♦r✐t❤♠❄ ❖♥✲❧✐♥❡ ❈♦♠♣❡t✐t✐✈❡ ❆♥❛❧②s✐s ❬●②❛r❢❛s✱ ❑✐r❛❧②✱ ▲❡❤❡❧❪ ❆♥ ❛❧❣♦r✐t❤♠ A ✐s ♦♥✲❧✐♥❡ c ✲❝♦♠♣❡t✐t✐✈❡ ✐❢ t❤❡r❡ ❡①✐sts ❛♥ b s✉❝❤ t❤❛t ❢♦r ❛♥② ❣r❛♣❤✱ G ✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ A ( G ) ≤ c · χ O ( G ) + b ❍❡r❡✱ χ O ( G ) ✐s t❤❡ ♦♥✲❧✐♥❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ G ✳ ❚❤✐s ✐s t❤❡ s♠❛❧❧❡st ♥✉♠❜❡r s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ❝❛♥ ❝♦❧♦r G ✉s✐♥❣ ❛t ♠♦st χ O ( G ) ❝♦❧♦rs ❢♦r ❡✈❡r② ♦r❞❡r✐♥❣ ♦❢ t❤❡ ✈❡rt✐❝❡s✳ ❈❤r✐st✐❛♥ ❑✉❞❛❤❧ ✭■▼❆❉❆✮ ❖♥✲❧✐♥❡ ❈❤r♦♠❛t✐❝ ◆✉♠❜❡r ❏✉❧② ✼✱ ✷✵✶✹ ✻ ✴ ✷✵

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