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❍❛r❞♥❡ss ♦❢ ❈♦♠♣✉t✐♥❣ t❤❡ ❇✐❝❧✐q✉❡ P❛rt✐t✐♦♥ ◆✉♠❜❡r

❆❧❡①❛♥❞❡r ❘✳ ❇❧♦❝❦ ❏✉❧② ✶✼✱ ✷✵✶✼

✶ ✴ ✶✻

■♥tr♦❞✉❝t✐♦♥

❇✐❝❧✐q✉❡ P❛rt✐t✐♦♥ ◆✉♠❜❡r ✭✐♥ s❤♦rt✱ bp✮

◮ ▼✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ❜✐❝❧✐q✉❡s ♥❡❡❞❡❞ t♦ ♣❛rt✐t✐♦♥ t❤❡ ❡❞❣❡s ♦❢ ❛

❣r❛♣❤ G✱ ❞❡♥♦t❡❞ bp (G)

◮ ◆♦t❡ t❤❛t G ❝❛♥ ❜❡ ❛♥② ❣r❛♣❤ ◮ ❇✐❝❧✐q✉❡s ❛r❡ ❝♦♠♣❧❡t❡ ❜✐♣❛rt✐t❡ ❣r❛♣❤s✱ ❞❡♥♦t❡❞ Kn,m ✷ ✴ ✶✻

■♥tr♦❞✉❝t✐♦♥ ✭❊①❛♠♣❧❡s✮

K4

= + +

K1,3 K1,2 K1,1

=

K2,2

+

K2,1 K1,1

✸ ✴ ✶✻

■♥tr♦❞✉❝t✐♦♥

• r❛❤❛♠ ❛♥❞ P♦❧❧❛❦ ✐♥tr♦❞✉❝❡❞ t❤❡ ❜✐❝❧✐q✉❡ ♣❛rt✐t✐♦♥ ♥✉♠❜❡r ✐♥

✶✾✼✷ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ♥❡t✇♦r❦ ❛❞❞r❡ss✐♥❣ ❛♥❞ ❣r❛♣❤ st♦r❛❣❡ ♣r♦❜❧❡♠s ❬●P✼✶✱ ●P✼✷❪ ■♥tr♦❞✉❝❡❞ ❛♥ ❡①tr❡♠❡❧② ♣r♦❧✐✜❝ r❡s❡❛r❝❤ ❛r❡❛ ✐♥ ▼❛t❤❡♠❛t✐❝s

• r❛❤❛♠✲P♦❧❧❛❦ ❚❤❡♦r❡♠✿ bp (Kn) = (n − 1)✳ ❆❧❧ ♣r♦♦❢s ❛r❡

❛❧❣❡❜r❛✐❝ ❛♥❞ ♥♦ ♣✉r❡❧② ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦♦❢ ✐s ❦♥♦✇♥ ❬●P✼✷✱ ❚✈❡✽✷✱ P❡❝✽✹✱ ❱✐s✵✽✱ ❱✐s✶✸❪

◮ ❙❤♦✇❡❞ bp (G) max{n+ (A(G)) , n− (A(G))} ❬❲✐ts❡♥❤❛✉s❡♥✱

✶✾✽✵s❪

◮ ❑♥♦✇♥ t❤❛t n− (A(Kn)) = (n − 1) ◮ ❈❛♥ ♣❛rt✐t✐♦♥ Kn ✐♥t♦ (n − 1) st❛rs ⋆ ❆ st❛r ✐s ❛ ❜✐❝❧✐q✉❡ ♦❢ t❤❡ ❢♦r♠ K1,i ❢♦r s♦♠❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r i ✹ ✴ ✶✻

❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❇✐❝❧✐q✉❡ P❛rt✐t✐♦♥

• r❛❤❛♠ ❛♥❞ P♦❧❧❛❦ t❤❛t ❛ ♣r♦❜❧❡♠ ♦♥ ❧♦♦♣ s✇✐t❝❤✐♥❣ ✐♥

♥❡t✇♦r❦✐♥❣ ✐s ❡q✉✐✈❛❧❡♥t t♦ ♣❛rt✐t✐♦♥✐♥❣ ❛ ♠✉❧t✐❣r❛♣❤✱ ②✐❡❧❞✐♥❣ t❤❡✐r ❝❡❧❡❜r❛t❡❞ r❡s✉❧t ❬●P✼✶✱ ●P✼✷✱ ❚❛✐✶✸❪ ❍❛s ❛♣♣❧✐❝❛t✐♦♥s ❢♦r ♣❡r❢❡❝t ❤❛s❤✐♥❣s ❬❚❛✐✶✸❪

■◆P❯❚✿ n, r, k ∈ N ✇✐t❤ k r n ▼■◆■▼■❩❊✿ s✐③❡ ♦❢ F := {fi : [n] → [r]} ❈❖◆❙❚❘❆■◆❚✿ ∀K ⊆ [n] ✇✐t❤ |K| = k✱ ∃i s✉❝❤ t❤❛t fi|K ✐s ✐♥❥❡❝t✐✈❡

◮ ❆s❦✐♥❣ ❢♦r ✉♥✐q✉❡ i ❛♥❞ r = k = 2 ❛s❦s ❢♦r bp (Kn)

❈♦♥♥❡❝t✐♦♥s t♦ t❤❡ ♥♦♥❞❡t❡r♠✐s♥✐st✐❝ st❛t❡ ❝♦♠♣❧❡①✐t② ♦❢ ✜♥✐t❡ ❛✉t♦♠❛t❛✱ ♥❛♠❡❧② ✉s❡❞ ❛s ❛ ❧♦✇❡r ❜♦✉♥❞ ♠❡t❤♦❞ ❬●❍✵✻❪ P❧❛② ❛ r♦❧❧ ✐♥ ❛♥❛❧②s✐s ♦❢ ❍▲❆ r❡❛❝t✐♦♥ ♠❛tr✐❝❡s ✉s❡❞ ✐♥ ❜✐♦❧♦❣② ❬◆▼❲❆✼✽❪

✺ ✴ ✶✻

❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❇✐❝❧✐q✉❡ P❛rt✐t✐♦♥

❆ r❡❧❛①❛t✐♦♥✿ bpt (G)✱ ❛ ❝♦✈❡r✐♥❣ ♦❢ ❡❞❣❡s ✇✐t❤ ❛t ♠♦st t✲❜✐❝❧✐q✉❡s

◮ ❊①❛♠✐♥❡❞ ❜② ◆♦❣❛ ❆❧♦♥ ❬❆❧♦✾✼❪ ◮ ❙❤♦✇❡❞ t❤❛t ✇✐t❤ G = Kn✱ bpt (G) ✐s ❡q✉✐✈❛❧❡♥t t♦ ✜♥❞✐♥❣ t❤❡ ♠❛①

♥✉♠❜❡r ♦❢ ❜♦①❡s ✐♥ Rn t❤❛t ❛r❡ t✲♥❡✐❣❤❜♦r❧②

◮ ❆❧s♦ s❤♦✇❡❞ t❤❛t bpt (Kn) Θ

• tn1/t

❲②♥❡r✬s ❝♦♠♠♦♥ ✐♥❢♦r♠❛t✐♦♥ J(RA, RB) ✿

◮ ▼✐♥✐♠✉♠ ✭❛♠♦✉♥t ♦❢✮ ❧❡❛❦❛❣❡ t❤❛t ❦✐❧❧s t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢ ❦❡②

❛❣r❡❡♠❡♥t

◮ min H(L) s✉❝❤ t❤❛t I(RA, RB|L) = 0 ◮ ❇✐❝❧✐q✉❡s ❛r❡ ✉s❡❧❡ss ❢♦r ❑❆ ✭❜❡❝❛✉s❡ ✵ ♠✉t✉❛❧ ✐♥❢♦r♠❛t✐♦♥✮ ◮ ❘♦✉❣❤❧② ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❜✐❝❧✐q✉❡ ♣❛rt✐t✐♦♥ ♥✉♠❜❡r

❈❧♦s❡ ❝♦♥♥❡❝t✐♦♥s t♦ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❝♦♠♣❧❡①✐t② ❛♥❞ ❝✐r❝✉✐t ❧♦✇❡r ❜♦✉♥❞s ❬❍❙✶✷✱ ◆❲✾✺✱ ❘❛③✾✷❪

✻ ✴ ✶✻

❋✐♥❞✐♥❣ bp ✐s ❍❛r❞

❙✉♣♣♦s❡ ✇❡ ❛r❡ ❣✐✈❡♥ ❛ ✭❜✐♣❛rt✐t❡✮ ❣r❛♣❤ G ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❜✐❝❧✐q✉❡ ♣❛rt✐t✐♦♥ ♦❢ G ♦❢ s✐③❡ k❄ ❚❤❡ ♣r♦❜❧❡♠ ✐s ◆P✲❈♦♠♣❧❡t❡ ❢♦r ❜♦t❤ ❜✐♣❛rt✐t❡ ❛♥❞ ❣❡♥❡r❛❧ ❣r❛♣❤s ❬❖r❧✼✼✱ ❈✐♦✵✺❪ ❲❡ s❤♦✇ ❛ ♣r♦♦❢ ❢♦r ❜✐♣❛rt✐t❡ ❣r❛♣❤s ❛♥❞ ❛♥♦t❤❡r ❢♦r ❣❡♥❡r❛❧ ❣r❛♣❤s✳ Pr♦♦❢ ❢♦r ❜✐♣❛rt✐t❡ ❣r❛♣❤s ✐s ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ✈❡rt❡① ❝❧✐q✉❡ ♣r♦❜❧❡♠

• ■❱❊◆✿ ●r❛♣❤ G ✇✐t❤ V (G) = {v1, . . . , vn}

❉❊❚❊❘▼■◆❊✿ ❋❡✇❡st ♥✉♠❜❡r ♦❢ ❝❧✐q✉❡s ✇❤✐❝❤ ✐♥❝❧✉❞❡ ❛❧❧ ♦❢ V (G)

Pr♦♦❢ ❢♦r ❣❡♥❡r❛❧ ❣r❛♣❤s ✐s ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ✈❡rt❡① ❝♦✈❡r ♣r♦❜❧❡♠

✼ ✴ ✶✻

Pr♦♦❢ ♦❢ ◆P✲❈♦♠♣❧❡t❡♥❡ss ♦❢ bp ❢♦r ❜✐♣❛rt✐t❡ ❣r❛♣❤s

❙✉♣♣♦s❡ ✇❡ ❛r❡ ❣✐✈❡♥ ❛ G ✐♥ t❤❡ ✐♥st❛♥❝❡ ♦❢ t❤❡ ✈❡rt❡① ❝❧✐q✉❡ ♣r♦❜❧❡♠ ❞❡s❝r✐❜❡❞ ❜❡❢♦r❡ ❛♥❞ ✇❡ ✇❛♥t t♦ ❛♥s✇❡r t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥

• ■❱❊◆✿ ❇✐♣❛rt✐t❡ ❣r❛♣❤ G

❉❊❚❊❘▼■◆❊✿ ❋❡✇❡st ♥✉♠❜❡r ♦❢ ❜✐❝❧✐q✉❡s ✇❤✐❝❤ ♣❛rt✐t✐♦♥ ❛ s✉❜s❡t H ⊆ E(G)

❈♦♥str✉❝t G′ = (L, R, E′) ✇✐t❤ L = {x1, . . . , xn}✱ R = {y1, . . . , yn} ❛♥❞ E′ = {(xi, yi): ∀i} ∪ {(xi, yj): (vi, vj) ∈ E(G)} ▲❡t H′ = {(xi, yj): i = 1, . . . , n} ❜❡ t❤❡ s❡t ♦❢ ❡❞❣❡s t♦ ❜❡ ❝♦✈❡r❡❞✳ ❆♥② ❝❧✐q✉❡ C ✐♥ G ✇❤✐❝❤ ✐♥❝❧✉❞❡s vi ✐♥❞✉❝❡s ❛ ❜✐❝❧✐q✉❡ ✐♥ G′ ✇❤✐❝❤ ✐♥❝❧✉❞❡s t❤❡ ❡❞❣❡ (xi, yi)✳ ■❢ C′ ✐s ❛ ❜✐❝❧✐q✉❡ ♦❢ G′ ✇❤✐❝❤ ✐♥❝❧✉❞❡s ❡❞❣❡s (xj1, yj1), (xj2, yj2), . . . , (xjk, yjk)✱ t❤❡♥ ❜② ❝♦♥str✉❝t✐♦♥ ✐t ♠✉st ❜❡ t❤❡ ❝❛s❡ t❤❛t {vj1, . . . , vjk} ✐s ❛ ❝❧✐q✉❡ ✐♥ G✳ ❙♦ t❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ❝❧✐q✉❡s t❤❛t ❝♦✈❡r ❛❧❧ ✈❡rt✐❝❡s ✐♥ G ✐s ❡q✉❛❧ t♦ t❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ❜✐❝❧✐q✉❡s ♦❢ G′ ♥❡❡❞❡❞ t♦ ❝♦✈❡r t❤❡ ❡❞❣❡s ✐♥ H′✳

✽ ✴ ✶✻

Pr♦♦❢ ♦❢ ◆P✲❈♦♠♣❧❡t❡♥❡ss ♦❢ bp ❢♦r ❣❡♥❡r❛❧ ❣r❛♣❤s

❙✉♣♣♦s❡ ✇❡ ❛r❡ ❣✐✈❡♥ ❛ ❣r❛♣❤ G ❛♥❞ ♥❡❡❞ t♦ ✜♥❞ ❛ ✈❡rt❡① ❝♦✈❡r ♦❢ s✐③❡ k |V (G)| ❚r❛♥s❢♦r♠ G ✐♥t♦ G′ ❜② r❡♣❧❛❝✐♥❣ ❡✈❡r② ❡❞❣❡ ✇✐t❤ ❛ ♣❛t❤ ♦❢ 3 ❡❞❣❡s G′ ❝♦♥t❛✐♥s ♥♦ 4✲❝②❝❧❡s✱ s♦ ♦♥❧② st❛rs ❛r❡ ❜✐❝❧✐q✉❡s ✐♥ G′ ❚❤✐s ✐♠♣❧✐❡s t❤❛t bp (G′) = α(G′)✱ ✇❤❡r❡ α(G′) ✐s t❤❡ s✐③❡ ♦❢ t❤❡ ♠✐♥✐♠❛❧ ✈❡rt❡① ❝♦✈❡r ♦❢ G′ ◆♦t✐❝❡ t❤❛t α(G′) = α(G) + |E| ❚❤✉s✱ bp (G′) = α(G′) = α(G) + |E| ❙♦ α(G) k ✐❢ ❛♥❞ ♦♥❧② ✐❢ bp (G′) k + |E|

✾ ✴ ✶✻

❆♣♣r♦①✐♠❛t✐♥❣ bp ✐s ❍❛r❞

❙✐♥❝❡ ❞❡t❡r♠✐♥✐♥❣ bp ✐s ◆P✲❍❛r❞✱ ❝❛♥ ✇❡ ❛♣♣r♦①✐♠❛t❡❄ ❯♥❢♦rt✉♥❛t❡❧②✱ bp ✐s ❛❧s♦ ◆P✲❍❛r❞ t♦ ❛♣♣r♦①✐♠❛t❡ ❬❙✐♠✾✵✱ ❇▼❇+✵✽✱ ❈❍❍❑✶✹❪ ❙✐♠♦♥ ❬❙✐♠✾✵❪ ❡①❛♠✐♥❡❞ r❡❞✉❝t✐♦♥s ✇❤✐❝❤ ♣r❡s❡r✈❡❞ ❛♣♣r♦①✐♠❛❜✐❧✐t② ♦❢ ❤❛r❞ ♣r♦❜❧❡♠s

◮ ▼❛♥② t✐♠❡s✱ ♥❡❛r ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ✐♥ ♦♥❡ ♣r♦❜❧❡♠ r❡❞✉❝❡s t♦ ❛ ♣♦♦r

s♦❧✉t✐♦♥ ✐♥ ❛♥♦t❤❡r

◮ ●✐✈❡s ♣r♦♦❢ t❤❛t bp ✐s ◆P✲❍❛r❞ t♦ ❛♣♣r♦①✐♠❛t❡ ❜② ❛ ❝♦♥t✐♥✉♦✉s

r❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ✈❡rt❡① ❝❧✐q✉❡ ♣r♦❜❧❡♠ ❞✐s❝✉ss❡❞ ❡❛r❧✐❡r

◮ ❚❤❡ ♣r♦♦❢ ✐s ♥♦t ✈❡r② ✐♥s✐❣❤t❢✉❧✱ s♦ ✐t ✇✐❧❧ ❜❡ s❦✐♣♣❡❞ ✐♥ t❤✐s t❛❧❦ ✶✵ ✴ ✶✻

◆❡❛r❧② ❚✐❣❤t ❆♣♣r♦①✐♠❛❜✐❧✐t② ❢♦r bp

❈❤❛❧❡r♠s♦♦❦✱ ❍❡②❞r✐❝❤✱ ❍♦❧♠✱ ❛♥❞ ❑❛rr❡♥❜❛✉❡r ❬❈❍❍❑✶✹❪ ♣r♦✈❡s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r bp ✇✐t❤ ❛♣♣r♦①✐♠❛t✐♦♥ ❣✉❛r❛♥t❡❡ ♦❢ O

• nL/
• log(nL)
• ✱ ✇❤❡r❡ |L| = nL ❛♥❞ t❤❡ ✐♥♣✉t ❣r❛♣❤ ✐s

❜✐♣❛rt✐t❡ ❋♦r ♦✉r ♣✉r♣♦s❡s✱ ❛ss✉♠❡ |L| = |R| = n✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ s❝❤❡♠❡ ✐s ❛s ❢♦❧❧♦✇s

◮ ❈❤♦♦s❡ ♣❛r❛♠❡t❡r r ✭t♦ ❜❡ ✜①❡❞ ❧❛t❡r✮ ❛♥❞ ♣❛rt✐t✐♦♥ L ✐♥t♦ n/r

s✉❜s❡ts ♦❢ s✐③❡ r ✭L1, . . . , Ln/r✮

◮ ❋♦r ❡❛❝❤ Li✱ r✉♥ ❛♥ α(r)✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠ t♦ ✜♥❞ ❛ ❜✐❝❧✐q✉❡

❝♦✈❡r ✐♥ ❡❛❝❤ s✉❜❣r❛♣❤ ✐♥❞✉❝❡❞ ❜② Li ✭♥♦t❡ ❡❛❝❤ Li ✐s ❡❞❣❡✲❞✐s❥♦✐♥t✮

◮ ❊❛❝❤ ❜✐❝❧✐q✉❡ ❢r♦♠ ❡❛❝❤ Li ❛r❡ ♣✉t t♦❣❡t❤❡r ❛♥❞ ❢r♦♠ ❛ ❜✐❝❧✐q✉❡

❝♦✈❡r ♦❢ t❤❡ ✇❤♦❧❡ ❣r❛♣❤

⋆ ◆♦t❡ t❤❛t s✐♥❝❡ Li ✇❡r❡ ❡❞❣❡✲❞✐s❥♦✐♥t✱ t❤✐s ✐s ❛❧s♦ ❛ ❜✐❝❧✐q✉❡ ♣❛rt✐t✐♦♥ ✶✶ ✴ ✶✻

◆❡❛r❧② ❚✐❣❤t ❆♣♣r♦①✐♠❛❜✐❧✐t② ❢♦r bp

❚❤✐s s❝❤❡♠❡ ❣✐✈❡s ❛♣♣r♦①✐♠❛t✐♦♥ ❣✉❛r❛♥t❡❡ n

r α(r)

❈❤♦♦s❡ t❤❡ α(r)✲❛♣♣r♦①✐♠❛t✐♦♥ s❝❤❡♠❡ ❛s ❢♦❧❧♦✇s✿

◮ ●✐✈❡♥ Li✱ r✉♥ ❛ ❜r✉t❡ ❢♦r❝❡ ❛❧❣♦r✐t❤♠ ♦✈❡r ❛❧❧ 2r s✉❜s❡ts ❛♥❞

❡♥✉♠❡r❛t❡ ❛❧❧ r✲t✉♣❧❡s ♦❢ ❡❛❝❤ s✉❜s❡t

◮ ❙✉❝❤ ❛ ❞❡✜♥❡❞ s✉❜s❡t S ❛♥❞ ✐ts ✐♥t❡rs❡❝t✐♦♥ ✇✐t❤ t❤❡ s❡t

{w : v ∈ S, w ✐s ❛ ♥❡✐❣❤❜♦r ♦❢ v} ✐♥❞✉❝❡s ❛ ❜✐❝❧✐q✉❡

◮ ❘❡t✉r♥ t❤❡ s♠❛❧❧❡st t✉♣❧❡ ♦❢ ✈❡rt❡① s❡ts ✇❤✐❝❤ ❝♦✈❡rs ❛❧❧ ❡❞❣❡s

✭❡♥s✉r❡ t❤❡s❡ ❜✐❝❧✐q✉❡s ❛r❡ ❡❞❣❡✲❞✐s❥♦✐♥t ❢♦r bp✮

◮ ❆♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ❤❛s ❛t ♠♦st r ❜✐❝❧✐q❡s✱ s♦ t❤✐s r❡t✉r♥s ❛♥

♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ✭✐✳❡✳✱ α(r) = 1✮

✶✷ ✴ ✶✻

◆❡❛r❧② ❚✐❣❤t ❆♣♣r♦①✐♠❛❜✐❧✐t② ❢♦r bp

❚❤❡ r✉♥♥✐♥❣ t✐♠❡ ♦❢ t❤✐s ❛❧❣♦r✐t❤♠ ✐s O ((2r)r) ❚❤❡ ❣✉❛r❛♥t❡❡ ♦❢ t❤❡ s❝❤❡♠❡ ✐s n

r α(r) = n r

❈❤♦♦s❡ r =

• log(n) ❣✐✈❡s ✉s ❛ ❣✉❛r❛♥t❡❡ ♦❢ O
• n/
• log(n)
• r =
• log(n) ❣✐✈❡s ✉s ❛ ♣♦❧②♥♦♠✐❛❧ r✉♥t✐♠❡ ♦❢ O( n

r 2r2) = O(n2)

❈❤❛❧❡r♠s♦♦❦ ❡t ❛❧✳ ❛❧s♦ ❣✐✈❡ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♥✉♠❜❡r ♦❢ ❡❞❣❡s m✱ ✇❤✐❝❤ ❤❛s ❣✉❛r❛♥t❡❡ O m log2 log m log3 m

• ✶✸ ✴ ✶✻

❖♣❡♥ Pr♦❜❧❡♠s

❉♦❡s t❤❡r❡ ❡①✐st ❛ ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦♦❢ ❢♦r t❤❡ ●r❛❤❛♠ P♦❧❧❛❦ ❚❤❡♦r❡♠❄

◮ ❖♥❡ ❡①✐sts ✉s✐♥❣ t❤❡ P✐❣❡♦♥ ❍♦❧❡ Pr✐♥❝✐♣❧❡✱ ❜✉t ✉s❡s str✉❝t✉r❡s ✇✐t❤

s✐③❡ ♦♥ t❤❡ ♦r❞❡r ♦❢ nn ❬❱✐s✶✸❪

◮ ❚❛✐t ❬❚❛✐✶✸❪ ❝❧❛✐♠s ❋✳❘✳❑✳ ❈❤✉♥❣ ❝♦♥✜r♠s t❤❡r❡ ❡①✐sts ❛ ✏❜❡tt❡r✑

❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦♦❢ ✭❝✐t❡❞ ✈✐❛ ♣r✐✈❛t❡ ❝♦♠♠✉♥✐❝❛t✐♦♥✮

❲❤❛t ✐s bp2 (Kn)❄

◮ ❇❡st ❦♥♦✇♥ ❜♦✉♥❞s ❛r❡

√n − 1 bp2 (Kn) ⌈√n⌉ + ⌊√n⌋ − 2 ❬❆❧♦✾✼✱ ❍❙✶✷❪

◮ ❊❛s② t♦ ❛s❦✿ ✇❤❛t ✐s bpt (Kn) ❢♦r ❝♦♥st❛♥t t❄ ✶✹ ✴ ✶✻

❖♣❡♥ Pr♦❜❧❡♠s

❆r❡ t❤❡r❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ✇✐t❤ ❜❡tt❡r ❣✉❛r❛♥t❡❡s❄

◮ ❈❤❛❧❡r♠s♦♦❦ ❡t ❛❧✳ ❬❈❍❍❑✶✹❪ ❣✐✈❡ ❜❡tt❡r ❣✉❛r❛♥t❡❡s ✐❢

NP ⊆ BPTIME

• 2polylog n

✭❇♦✉♥❞❡❞ ❊rr♦r Pr♦❜❛❜✐❧✐st✐❝ ❚✐♠❡✮

❍♦✇ ❝❧♦s❡ ✐s bp t♦ ❲②♥❡r✬s ❈♦♠♠♦♥ ■♥❢♦r♠❛t✐♦♥❄

◮ ❍♦✇ ❣♦♦❞ ♦❢ ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ♦♥❡ t♦ t❤❡ ♦t❤❡r❄

❍♦✇ ❝❧♦s❡ ❛r❡ bp ❛♥❞ t❤❡ ❜✐❝❧✐q✉❡ ❝♦✈❡r ♥✉♠❜❡r ✭bc✮❄

◮ ❑♥♦✇♥ t❤❛t bc bp ◮ ❬P✐♥✶✹❪ ❚❤✐s r❡❧❛t✐♦♥ ♠❛② ❜❡ q✉✐t❡ ❧♦♦s❡✿ ⋆ bp (Kn) 2bc(Kn)−1 − 1 ✭♥♦t❡ t❤❛t bc (Kn) = ⌈log n⌉✮ ⋆ bp (G) 1 2

• 3bc(G) − 1
• ✶✺ ✴ ✶✻

❈♦♥❝❧✉s✐♦♥s

❚❤❡ ❜✐❝❧✐q✉❡ ♣❛rt✐t✐♦♥ ♥✉♠❜❡r ✐s ❛ ❢❡rt✐❧❡✱ r✐❝❤ ❛r❡❛ ♦❢ r❡s❡❛r❝❤ ✐♥ ♠❛t❤❡♠❛t✐❝s ✇✐t❤ ♠❛♥② ❝♦♥♥❡❝t✐♦♥s t♦ ♦t❤❡r ✜❡❧❞s ❉❡t❡r♠✐♥✐♥❣ bp ❛♥❞ bc ✐s ❛♥ ◆P✲❍❛r❞ ♣r♦❜❧❡♠

◮ ❊✈❡♥ ❢♦r ❜✐♣❛rt✐t❡ ❣r❛♣❤s

bp ❛♥❞ bc ❛r❡ ◆P✲❍❛r❞ t♦ ❛♣♣r♦①✐♠❛t❡ ❛s ✇❡❧❧

◮ ❊✈❡♥ ❢♦r ❜✐♣❛rt✐t❡ ❣r❛♣❤s

❙t✐❧❧ ♠❛♥② ♦♣❡♥ ♣r♦❜❧❡♠s ✐♥ r❡❧❛t✐♦♥ t♦ bp ❛♥❞ bc

✶✻ ✴ ✶✻

❬❆❧♦✾✼❪ ◆♦❣❛ ❆❧♦♥✳ ◆❡✐❣❤❜♦r❧② ❋❛♠✐❧✐❡s ♦❢ ❇♦①❡s ❛♥❞ ❇✐♣❛rt✐t❡ ❈♦✈❡r✐♥❣s✱ ♣❛❣❡s ✷✼✕✸✶✳ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❇❡r❧✐♥✱ ❍❡✐❞❡❧❜❡r❣✱ ✶✾✾✼✳ ❬❇▼❇+✵✽❪ ❉♦✐♥❛ ❇❡✐♥✱ ▲✐♥❞❛ ▼♦r❛❧❡s✱ ❲♦❧❢❣❛♥❣ ❲ ❇❡✐♥✱ ❈❖ ❙❤✐❡❧❞s ❏r✱ ❩ ▼❡♥❣✱ ❛♥❞ ■✈❛♥ ❍❛❧ ❙✉❞❜♦r♦✉❣❤✳ ❈❧✉st❡r✐♥❣ ❛♥❞ t❤❡ ❜✐❝❧✐q✉❡ ♣❛rt✐t✐♦♥ ♣r♦❜❧❡♠✳ ■♥ ❍■❈❙❙✱ ♣❛❣❡ ✹✼✺✱ ✷✵✵✽✳ ❬❈❍❍❑✶✹❪ P❛r✐♥②❛ ❈❤❛❧❡r♠s♦♦❦✱ ❙❛♥❞② ❍❡②❞r✐❝❤✱ ❊✉❣❡♥✐❛ ❍♦❧♠✱ ❛♥❞ ❆♥❞r❡❛s ❑❛rr❡♥❜❛✉❡r✳ ◆❡❛r❧② ❚✐❣❤t ❆♣♣r♦①✐♠❛❜✐❧✐t② ❘❡s✉❧ts ❢♦r ▼✐♥✐♠✉♠ ❇✐❝❧✐q✉❡ ❈♦✈❡r ❛♥❞ P❛rt✐t✐♦♥✱ ♣❛❣❡s ✷✸✺✕✷✹✻✳ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❇❡r❧✐♥✱ ❍❡✐❞❡❧❜❡r❣✱ ✷✵✶✹✳ ❬❈✐♦✵✺❪ ❙❡❜❛st✐❛♥ ▼✳ ❈✐♦❛❜➔✳ ❚❤❡ ♥♣✲❝♦♠♣❧❡t❡♥❡ss ♦❢ s♦♠❡ ❡❞❣❡✲♣❛rt✐t✐♦♥✐♥❣ ♣r♦❜❧❡♠s✳ ▼❛st❡r✬s t❤❡s✐s✱ ◗✉❡❡♥✬s ❯♥✐✈❡rs✐t②✱ ✷✵✵✺✳ ❬●❍✵✻❪ ❍❡r♠❛♥♥ ●r✉❜❡r ❛♥❞ ▼❛r❦✉s ❍♦❧③❡r✳

✶✻ ✴ ✶✻

❋✐♥❞✐♥❣ ▲♦✇❡r ❇♦✉♥❞s ❢♦r ◆♦♥❞❡t❡r♠✐♥✐st✐❝ ❙t❛t❡ ❈♦♠♣❧❡①✐t② ■s ❍❛r❞✱ ♣❛❣❡s ✸✻✸✕✸✼✹✳ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❇❡r❧✐♥✱ ❍❡✐❞❡❧❜❡r❣✱ ✷✵✵✻✳ ❬●P✼✶❪ ❘♦♥❛❧❞ ▲ ●r❛❤❛♠ ❛♥❞ ❍❡♥r② ❖ P♦❧❧❛❦✳ ❖♥ t❤❡ ❛❞❞r❡ss✐♥❣ ♣r♦❜❧❡♠ ❢♦r ❧♦♦♣ s✇✐t❝❤✐♥❣✳ ❇❡❧❧ ❙②st❡♠ ❚❡❝❤♥✐❝❛❧ ❏♦✉r♥❛❧✱ ✺✵✭✽✮✿✷✹✾✺✕✷✺✶✾✱ ✶✾✼✶✳ ❬●P✼✷❪ ❘♦♥❛❧❞ ▲ ●r❛❤❛♠ ❛♥❞ ❍❡♥r② ❖ P♦❧❧❛❦✳ ❖♥ ❡♠❜❡❞❞✐♥❣ ❣r❛♣❤s ✐♥ sq✉❛s❤❡❞ ❝✉❜❡s✳ ■♥ ●r❛♣❤ t❤❡♦r② ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ♣❛❣❡s ✾✾✕✶✶✵✳ ❙♣r✐♥❣❡r✱ ✶✾✼✷✳ ❬❍❙✶✷❪ ❍❛♦ ❍✉❛♥❣ ❛♥❞ ❇❡♥♥② ❙✉❞❛❦♦✈✳ ❆ ❝♦✉♥t❡r❡①❛♠♣❧❡ t♦ t❤❡ ❛❧♦♥✲s❛❦s✲s❡②♠♦✉r ❝♦♥❥❡❝t✉r❡ ❛♥❞ r❡❧❛t❡❞ ♣r♦❜❧❡♠s✳ ❈♦♠❜✐♥❛t♦r✐❝❛✱ ✸✷✭✷✮✿✷✵✺✕✷✶✾✱ ✷✵✶✷✳ ❬◆▼❲❆✼✽❪ ❉❛♥❛ ❙✳ ◆❛✉✱ ●❡♦r❣❡ ▼❛r❦♦✇s❦②✱ ▼❛① ❆✳ ❲♦♦❞❜✉r②✱ ❛♥❞ ❉✳ ❇❡r♥❛r❞ ❆♠♦s✳

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❆ ♠❛t❤❡♠❛t✐❝❛❧ ❛♥❛❧②s✐s ♦❢ ❤✉♠❛♥ ❧❡✉❦♦❝②t❡ ❛♥t✐❣❡♥ s❡r♦❧♦❣②✳ ▼❛t❤❡♠❛t✐❝❛❧ ❇✐♦s❝✐❡♥❝❡s✱ ✹✵✭✸✮✿✷✹✸ ✕ ✷✼✵✱ ✶✾✼✽✳ ❬◆❲✾✺❪ ◆♦❛♠ ◆✐s❛♥ ❛♥❞ ❆✈✐ ❲✐❣❞❡rs♦♥✳ ❖♥ r❛♥❦ ✈s✳ ❝♦♠♠✉♥✐❝❛t✐♦♥ ❝♦♠♣❧❡①✐t②✳ ❈♦♠❜✐♥❛t♦r✐❝❛✱ ✶✺✭✹✮✿✺✺✼✕✺✻✺✱ ✶✾✾✺✳ ❬❖r❧✼✼❪ ❏❛♠❡s ❖r❧✐♥✳ ❈♦♥t❡♥t♠❡♥t ✐♥ ❣r❛♣❤ t❤❡♦r②✿ ❈♦✈❡r✐♥❣ ❣r❛♣❤s ✇✐t❤ ❝❧✐q✉❡s✳ ■♥ ■♥❞❛❣t✐♦♥❡s ▼❛t❤❡♠❛t✐❝❛❡ ✭Pr♦❝❡❡❞✐♥❣s✮✱ ✶✾✼✼✳ ❬P❡❝✽✹❪

• ❲ P❡❝❦✳

❆ ♥❡✇ ♣r♦♦❢ ♦❢ ❛ t❤❡♦r❡♠ ♦❢ ❣r❛❤❛♠ ❛♥❞ ♣♦❧❧❛❦✳ ❉✐s❝r❡t❡ ♠❛t❤❡♠❛t✐❝s✱ ✹✾✭✸✮✿✸✷✼✕✸✷✽✱ ✶✾✽✹✳ ❬P✐♥✶✹❪ ❚r❡✈♦r P✐♥t♦✳ ❇✐❝❧✐q✉❡ ❝♦✈❡rs ❛♥❞ ♣❛rt✐t✐♦♥s✳ ■♥ ❊❧❡❝tr♦♥✐❝ ❏♦✉r♥❛❧ ♦❢ ❈♦♠❜✐♥❛t♦r✐❝s✱ ✷✵✶✹✳ ❬❘❛③✾✷❪ ❆❧❡①❛♥❞❡r ❆ ❘❛③❜♦r♦✈✳

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❚❤❡ ❣❛♣ ❜❡t✇❡❡♥ t❤❡ ❝❤r♦♠❛t✐❝ ♥✉♠❜❡r ♦❢ ❛ ❣r❛♣❤ ❛♥❞ t❤❡ r❛♥❦ ♦❢ ✐ts ❛❞❥❛❝❡♥❝② ♠❛tr✐① ✐s s✉♣❡r❧✐♥❡❛r✳ ❉✐s❝r❡t❡ ♠❛t❤❡♠❛t✐❝s✱ ✶✵✽✭✶✮✿✸✾✸✕✸✾✻✱ ✶✾✾✷✳ ❬❙✐♠✾✵❪ ❍❛♥s ❯❧r✐❝❤ ❙✐♠♦♥✳ ❖♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥s ❢♦r ❝♦♠❜✐♥❛t♦r✐❛❧ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✳ ❙■❆▼ ❏♦✉r♥❛❧ ♦♥ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✱ ✶✾✾✵✳ ❬❚❛✐✶✸❪ ▼✐❝❤❛❡❧ ❚❛✐t✳ ▼② ❢❛✈♦r✐t❡ ❛♣♣❧✐❝❛t✐♦♥ ✉s✐♥❣ ❡✐❣❡♥✈❛❧✉❡s✿ ❊✐❣❡♥✈❛❧✉❡s ❛♥❞ t❤❡ ❣r❛❤❛♠✲♣♦❧❧❛❦ t❤❡♦r❡♠✳ ✷✵✶✸✳ ❬❚✈❡✽✷❪ ❍❡❧❣❡ ❚✈❡r❜❡r❣✳ ❖♥ t❤❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❦♥ ✐♥t♦ ❝♦♠♣❧❡t❡ ❜✐♣❛rt✐t❡ ❣r❛♣❤s✳ ❏♦✉r♥❛❧ ♦❢ ●r❛♣❤ ❚❤❡♦r②✱ ✻✭✹✮✿✹✾✸✕✹✾✹✱ ✶✾✽✷✳ ❬❱✐s✵✽❪ ❙✉♥❞❛r ❱✐s❤✇❛♥❛t❤❛♥✳ ❆ ♣♦❧②♥♦♠✐❛❧ s♣❛❝❡ ♣r♦♦❢ ♦❢ t❤❡ ❣r❛❤❛♠✕♣♦❧❧❛❦ t❤❡♦r❡♠✳

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❏♦✉r♥❛❧ ♦❢ ❈♦♠❜✐♥❛t♦r✐❛❧ ❚❤❡♦r②✱ ❙❡r✐❡s ❆✱ ✶✶✺✭✹✮✿✻✼✹✕✻✼✻✱ ✷✵✵✽✳ ❬❱✐s✶✸❪ ❙✉♥❞❛r ❱✐s❤✇❛♥❛t❤❛♥✳ ❆ ❝♦✉♥t✐♥❣ ♣r♦♦❢ ♦❢ t❤❡ ❣r❛❤❛♠✕♣♦❧❧❛❦ t❤❡♦r❡♠✳ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✱ ✸✶✸✭✻✮✿✼✻✺✕✼✻✻✱ ✷✵✶✸✳

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