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❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s

❍♦♥❣✇❡✐ ●❛♦ 1 ▲❡♦♥ P❡tr♦s②❛♥ 2 ❆rt❡♠ ❙❡❞❛❦♦✈ 2

1❈♦❧❧❡❣❡ ♦❢ ▼❛t❤❡♠❛t✐❝s

◗✐♥❣❞❛♦ ❯♥✐✈❡rs✐t②

2❉❡♣❛rt♠❡♥t ♦❢ ●❛♠❡ ❚❤❡♦r② ❛♥❞ ❙t❛t✐st✐❝❛❧ ❉❡❝✐s✐♦♥s

❙❛✐♥t P❡t❡rs❜✉r❣ ❙t❛t❡ ❯♥✐✈❡rs✐t②

❏✉♥❡ ✸✱ ✷✵✶✹

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✶✴✸✵

❖✉t❧✐♥❡

✶ ❚❤❡ ♠♦❞❡❧ ✷ ❈♦♦♣❡r❛t✐♦♥ ✐♥ t✇♦✲st❛❣❡ ♥❡t✇♦r❦ ❣❛♠❡ ✸ ❙tr♦♥❣❧② t✐♠❡✲❝♦♥s✐st❡♥t s♦❧✉t✐♦♥ ✹ ◆✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡ ❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✷✴✸✵

❚❤❡ ▼♦❞❡❧

N = {1, . . . , n}✿ ❛ ✜♥✐t❡ s❡t ♦❢ ♣❧❛②❡rs ✇❤♦ ❝❛♥ ✐♥t❡r❛❝t ✇✐t❤ ❡❛❝❤ ♦t❤❡r✳ g✿ ❛ ✜♥✐t❡ s❡t ♦❢ ♣❛✐rs (i, j) ∈ N × N✱ ♦r ❛ ♥❡t✇♦r❦✳ ■❢ (i, j) ∈ g✱ ✇❡ s❛② t❤❛t t❤❡r❡ ✐s ❛ ❧✐♥❦ ❝♦♥♥❡❝t✐♥❣ ♣❧❛②❡rs i ❛♥❞ j✱ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ❝♦♠♠✉♥✐❝❛t✐♦♥ ♦❢ t❤❡ ♣❧❛②❡rs✳ ■♥ ♦✉r s❡tt✐♥❣ ✇❡ s✉♣♣♦s❡ t❤❛t ❛❧❧ ❧✐♥❦s ❛r❡ ✉♥❞✐r❡❝t❡❞✱ ✐✳❡✳ (i, j) = (j, i)✳ ❋♦r t❤❡ s✐♠♣❧✐❝✐t② ❞❡♥♦t❡ (i, j) ❛s ij✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✸✴✸✵

❈♦♥s✐❞❡r ❛ t✇♦✲st❛❣❡ ♣r♦❜❧❡♠✿ ❆t t❤❡ ✜rst st❛❣❡ ❡❛❝❤ ♣❧❛②❡r ❝❤♦♦s❡s ❤✐s ♣❛rt♥❡rs✖♦t❤❡r ♣❧❛②❡rs ✇✐t❤ ✇❤♦♠ ❤❡ ✇❛♥ts t♦ ❢♦r♠ ❧✐♥❦s✳ ❆❢t❡r ❝❤♦♦s✐♥❣ ♣❛rt♥❡rs ❛♥❞ ❡st❛❜❧✐s❤✐♥❣ ❧✐♥❦s✱ ♣❧❛②❡rs✱ t❤❡r❡❜②✱ ❢♦r♠ ❛ ♥❡t✇♦r❦✳ ❆t t❤❡ s❡❝♦♥❞ st❛❣❡ ❤❛✈✐♥❣ t❤❡ ♥❡t✇♦r❦ ❢♦r♠❡❞✱ ❡❛❝❤ ♣❧❛②❡r ❝❤♦♦s❡s ❛ ❝♦♥tr♦❧ ✐♥✢✉❡♥❝✐♥❣ ❤✐s ♣❛②♦✛✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✹✴✸✵

❋✐rst ❙t❛❣❡✿ ◆❡t✇♦r❦ ❋♦r♠❛t✐♦♥

Mi ⊆ N \ {i}✿ t❤❡ s❡t ♦❢ ♣❧❛②❡rs ✇❤♦♠ ♣❧❛②❡r i ∈ N ❝❛♥ ♦✛❡r ❛ ♠✉t✉❛❧ ❧✐♥❦✳ ai ∈ {0, . . . , n − 1}✿ t❤❡ ♠❛①✐♠❛❧ ♥✉♠❜❡r ♦❢ ❧✐♥❦s ✇❤✐❝❤ ♣❧❛②❡r i ❝❛♥ ♦✛❡r✳ ❇❡❤❛✈✐♦r ♦❢ ♣❧❛②❡r i ∈ N ❛t t❤❡ ✜rst st❛❣❡ ✐s ❛ ♣r♦✜❧❡ gi = (gi1, . . . , gin) ✇❤✐❝❤ ❝♦♠♣♦♥❡♥ts ❛r❡ ❞❡✜♥❡❞ ❛s✿ gij = 1, ✐❢ ♣❧❛②❡r i ♦✛❡rs ❛ ❧✐♥❦ t♦ j ∈ Mi, 0, ♦t❤❡r✇✐s❡, ✭✶✮ s✉❜❥❡❝t t♦ t❤❡ ❝♦♥str❛✐♥✿

• j∈N

gij

• ai,

✭✷✮ gii = 0, i ∈ N. Gi = {gi : (1) − (2) ❛r❡ tr✉❡}✱ i ∈ N✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✺✴✸✵

P❧❛②❡rs ❝❤♦♦s❡ t❤❡✐r ❜❡❤❛✈✐♦rs ❛t t❤❡ ✜rst st❛❣❡ s✐♠✉❧t❛♥❡♦✉s❧② ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t❧② ❢r♦♠ ❡❛❝❤ ♦t❤❡r✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ♣❧❛②❡r i ∈ N ❝❤♦♦s❡s gi ∈ Gi✱ ❛♥❞ ❛s ❛ r❡s✉❧t t❤❡ ❜❡❤❛✈✐♦r ♣r♦✜❧❡ (g1, . . . , gn) ✐s ❢♦r♠❡❞✳ ❍❛✈✐♥❣ t❤❡ ❜❡❤❛✈✐♦r ♣r♦✜❧❡ (g1, . . . , gn) ❢♦r♠❡❞✱ ❛♥ ✉♥❞✐r❡❝t❡❞ ❧✐♥❦ ij = ji ✐s ❡st❛❜❧✐s❤❡❞ ✐♥ ♥❡t✇♦r❦ g ✐❢ ❛♥❞ ♦♥❧② ✐❢ gij = gji = 1, ✐✳❡✳ g ❝♦♥s✐sts ♦❢ ♠✉t✉❛❧ ❧✐♥❦s ✇❤✐❝❤ ✇❡r❡ ♦✛❡r❡❞ ♦♥❧② ❜② ❜♦t❤ ♣❧❛②❡rs✳ ❊①❛♠♣❧❡ ▲❡t N = {1, 2, 3, 4} ❛♥❞ ♣❧❛②❡rs ❝❤♦♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❜❡❤❛✈✐♦rs ❛t t❤❡ ✜rst st❛❣❡✿ g1 = (0, 1, 1, 1)✱ g2 = (1, 0, 1, 0)✱ g3 = (1, 1, 0, 0)✱ g4 = (0, 0, 1, 0)✳ ❚❤❡ r❡s✉❧t✐♥❣ ♥❡t✇♦r❦ g ❝♦♥t❛✐♥s t❤r❡❡ ❧✐♥❦s {12, 13, 23}✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✻✴✸✵

❙❡❝♦♥❞ ❙t❛❣❡✿ ❈♦♥tr♦❧s

Ni(g) = {j ∈ N \ {i} : ij ∈ g}✿ ♥❡✐❣❤❜♦rs ♦❢ ♣❧❛②❡r i ✐♥ ♥❡t✇♦r❦ g✳ ▲❡t di(g) = (di1(g), . . . , din(g)) ❜❡ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿ dij(g) =        1, ✐❢ i ❞♦❡s ♥♦t ❜r❡❛❦ t❤❡ ❧✐♥❦ ❢♦r♠❡❞ ❛t t❤❡ ✜rst st❛❣❡ ✇✐t❤ ♣❧❛②❡r j ∈ Ni(g) ✐♥ ♥❡t✇♦r❦ g, 0, ♦t❤❡r✇✐s❡. ✭✸✮ Di(g) = {di(g) : (3) ✐s tr✉❡}✳ Pr♦✜❧❡ (d1(g), . . . , dn(g)) ❝❤❛♥❣❡s ♥❡t✇♦r❦ g ❛♥❞ ❢♦r♠s ❛ ♥❡✇ ♥❡t✇♦r❦✱ ❞❡♥♦t❡❞ ❜② gd✳ ■♥ ♥❡t✇♦r❦ gd ❛❧❧ ❧✐♥❦s ij s✉❝❤ t❤❛t ❡✐t❤❡r dij(g) = 0 ♦r dji(g) = 0 ❛r❡ r❡♠♦✈❡❞✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✼✴✸✵

❆t t❤❡ s❡❝♦♥❞ st❛❣❡ ♣❧❛②❡r i ∈ N ❝❤♦♦s❡s ❝♦♥tr♦❧ ui ❢r♦♠ ❛ ✜♥✐t❡ s❡t Ui✳ ❇❡❤❛✈✐♦r ♦❢ ♣❧❛②❡r i ∈ N ❛t t❤❡ s❡❝♦♥❞ st❛❣❡ ✐s ❛ ♣❛✐r (di(g), ui)✿ ✐t ❞❡✜♥❡s✱ ♦♥ t❤❡ ♦♥❡ ❤❛♥❞✱ ❧✐♥❦s t♦ ❜❡ r❡♠♦✈❡❞ di(g)✱ ❛♥❞✱ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❝♦♥tr♦❧ ui✳ P❛②♦✛ ❢✉♥❝t✐♦♥ Ki ♦❢ ♣❧❛②❡r i✿ ✐t r✉❧❡s✱ ❞❡♣❡♥❞s ♦♥ ♣❧❛②❡r✬s ❜❡❤❛✈✐♦r ❛t t❤❡ s❡❝♦♥❞ st❛❣❡ ❛s ✇❡❧❧ ❛s ❜❡❤❛✈✐♦r ♦❢ ❤✐s ♥❡✐❣❤❜♦rs ✐♥ ♥❡t✇♦r❦ gd✳ Ki(ui, uNi(gd))✱ i ∈ N✱ ✐s ♥♦♥✲♥❡❣❛t✐✈❡ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ❞❡✜♥❡❞ ♦♥ t❤❡ s❡t Ui ×

j∈Ni(gd) Uj ❛♥❞ s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt②✿

✭P✮✿ ❢♦r ❛♥② t✇♦ ♥❡t✇♦r❦s g ❛♥❞ g′ ❛♥❞ ♣❧❛②❡r i ✐❢ |Ni(g)| |Ni(g′)|✱ t❤❡ ✐♥❡q✉❛❧✐t② Ki(ui, uNi(g)) Ki(ui, uNi(g′)) ❤♦❧❞s ❢♦r ❛❧❧ (ui, uNi(g)) ∈ Ui ×

j∈Ni(g) Uj ❛♥❞

(ui, uNi(g′)) ∈ Ui ×

j∈Ni(g′) Uj✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✽✴✸✵

❈♦♦♣❡r❛t✐♦♥ ✐♥ ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡

❲❡ st✉❞② t❤❡ ❝♦♦♣❡r❛t✐✈❡ ❝❛s❡ ❛♥❞ ❛♥s✇❡r t❤r❡❡ ♠❛✐♥ q✉❡st✐♦♥s✿ ❲❤❛t ✐s ❛ ❝♦♦♣❡r❛t✐✈❡ s♦❧✉t✐♦♥ ✐♥ t❤❡ ❣❛♠❡❄ ❈❛♥ ✐t ❜❡ r❡❛❧✐③❡❞ ✐♥ t❤❡ ❣❛♠❡❄ ■s ✐t str♦♥❣❧② t✐♠❡✲❝♦♥s✐st❡♥t❄

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✾✴✸✵

❙✉♣♣♦s❡ ♥♦✇ t❤❛t ♣❧❛②❡rs ❥♦✐♥t❧② ❝❤♦♦s❡ t❤❡✐r ❜❡❤❛✈✐♦rs ❛t ❜♦t❤ st❛❣❡s ♦❢ t❤❡ ❣❛♠❡✳ ❆❝t✐♥❣ ❛s ♦♥❡ ♣❧❛②❡r ❛♥❞ ❝❤♦♦s✐♥❣ gi ∈ Gi✱ ui ∈ Ui✱ i ∈ N✱ t❤❡ ❣r❛♥❞ ❝♦❛❧✐t✐♦♥✱ N✱ ♠❛①✐♠✐③❡s t❤❡ ✈❛❧✉❡✿

• i∈N

Ki(ui, uNi(g)). ✭✹✮ ▲❡t t❤❡ ♠❛①✐♠✉♠ ❜❡ ❛tt❛✐♥❡❞ ✇❤❡♥ ♣❧❛②❡rs✬ ❜❡❤❛✈✐♦r ♣r♦✜❧❡s g∗

i ✱ u∗ i ✱

i ∈ N ❛r❡ ❝❤♦s❡♥✱ ❛♥❞ ♣r♦✜❧❡ (g∗

1 , . . . , g∗ n) ❢♦r♠s ♥❡t✇♦r❦ g∗✳

Pr♦♣♦s✐t✐♦♥ d∗

i (g∗) = g∗ i ❢♦r ❛❧❧ i ∈ N✱ ✐✳❡✳ ♣❧❛②❡rs s❤♦✉❧❞ ♥♦t r❡♠♦✈❡ ❧✐♥❦s ❢r♦♠

♥❡t✇♦r❦ g∗✳ ▲❡t

• i∈N

Ki(u∗

i , u∗ Ni(g∗))

= max

gi∈Gi,i∈N

max

ui∈Ui,i∈N

• i∈N

Ki(ui, uNi(g)).

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✶✵✴✸✵

❚♦ ❛❧❧♦❝❛t❡ t❤❡ ♠❛①✐♠❛❧ s✉♠ ♦❢ ♣❧❛②❡rs✬ ♣❛②♦✛s ❛❝❝♦r❞✐♥❣ t♦ s♦♠❡ ✐♠♣✉t❛t✐♦♥✱ ✇❡ ❝♦♥str✉❝t ❛♥ ❛✉①✐❧✐❛r② ❝♦♦♣❡r❛t✐✈❡ ❚❯✲❣❛♠❡ (N, V )✳ ❚❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ V ✐s ❝♦♥str✉❝t❡❞ ❛s ❢♦❧❧♦✇s✳ Pr♦♣♦s✐t✐♦♥ ■♥ t❤❡ ❝♦♦♣❡r❛t✐✈❡ t✇♦✲st❛❣❡ ♥❡t✇♦r❦ ❣❛♠❡ t❤❡ s✉♣❡r❛❞❞✐t✐✈❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ V (·) ✐♥ t❤❡ s❡♥s❡ ♦❢ ✈♦♥ ◆❡✉♠❛♥♥ ❛♥❞ ▼♦r❣❡♥st❡r♥ ✐s ❞❡✜♥❡❞ ❛s✿ V (N) =

• i∈N

Ki(u∗

i , u∗ Ni(g∗)),

V (S) = max

gi∈Gi,i∈S

max

ui∈Ui,i∈S

• i∈S

Ki(ui, uNi(g)∩S), V (∅) = 0. ❋♦r ❛ s✐♥❣❧❡t♦♥ {i}✱ ✐ts ✈❛❧✉❡ ✐s ❞❡✜♥❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✿ V ({i}) = max

ui∈Ui Ki(ui).

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✶✶✴✸✵

❆♥ ✐♠♣✉t❛t✐♦♥ ✐♥ t❤❡ ❝♦♦♣❡r❛t✐✈❡ t✇♦✲st❛❣❡ ♥❡t✇♦r❦ ❣❛♠❡ ✐s ❛♥ n✲ ❞✐♠❡♥s✐♦♥❛❧ ♣r♦✜❧❡ ξ = (ξ1, . . . , ξn)✱ s❛t✐s❢②✐♥❣

i∈N ξi = V (N) ❛♥❞

ξi V ({i}) ❢♦r ❛❧❧ i ∈ N✳ ❚❤❡ s❡t ♦❢ ❛❧❧ ✐♠♣✉t❛t✐♦♥s ✐♥ t❤❡ ❣❛♠❡ (N, V ) ✇❡ ❞❡♥♦t❡ ❜② I(V )✳ ❆ ❝♦♦♣❡r❛t✐✈❡ s♦❧✉t✐♦♥ ❝♦♥❝❡♣t ✐♥ t❤❡ ❛✉①✐❧✐❛r② ❝♦♦♣❡r❛t✐✈❡ ❚❯✲❣❛♠❡ (N, V ) ✐s ❛ r✉❧❡ t❤❛t ✉♥✐q✉❡❧② ❛ss✐❣♥s ❛ s✉❜s❡t CSC(V ) ⊆ I(V ) t♦ t❤❡ ❣❛♠❡ (N, V )✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ t❤❡ ❝♦♦♣❡r❛t✐✈❡ s♦❧✉t✐♦♥ ❝♦♥❝❡♣t ✐s t❤❡ ❝♦r❡ C(V )✱ t❤❡♥ CSC(V ) = C(V ) =

• ξ = (ξ1, . . . , ξn) :
• i∈S

ξi V (S), S ⊆ N

• .

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✶✷✴✸✵

❙✉♣♣♦s❡ t❤❛t ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❣❛♠❡ ♣❧❛②❡rs ❥♦✐♥t❧② ❞❡❝✐❞❡ t♦ ❝❤♦♦s❡ ❜❡❤❛✈✐♦r ♣r♦✜❧❡s g∗

i ✱ u∗ i ✱ i ∈ N ❛♥❞ t❤❡♥ ❛❧❧♦❝❛t❡ ✐t ❛❝❝♦r❞✐♥❣

t♦ ❛ s♣❡❝✐✜❡❞ ❝♦♦♣❡r❛t✐✈❡ s♦❧✉t✐♦♥ ❝♦♥❝❡♣t CSC(V ) ✇❤✐❝❤ r❡❛❧✐③❡s t❤❡ ✐♠♣✉t❛t✐♦♥ ξ = (ξ1, . . . , ξn)✳ ■t ♠❡❛♥s t❤❛t ✐♥ t❤❡ ❝♦♦♣❡r❛t✐✈❡ t✇♦✲st❛❣❡ ♥❡t✇♦r❦ ❣❛♠❡ ♣❧❛②❡r i ∈ N s❤♦✉❧❞ r❡❝❡✐✈❡ t❤❡ ❛♠♦✉♥t ♦❢ ξi ❛s ❤✐s ♣❛②♦✛✳ ❲❤❛t ✇✐❧❧ ❤❛♣♣❡♥ ✐❢ ❛❢t❡r t❤❡ ✜rst st❛❣❡ ✭❛❢t❡r ❝❤♦♦s✐♥❣ t❤❡ ♣r♦✜❧❡s g∗

1 , . . . , g∗ n✮ ♣❧❛②❡rs r❡❝❛❧❝✉❧❛t❡ t❤❡ ✐♠♣✉t❛t✐♦♥ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ s❛♠❡

❝♦♦♣❡r❛t✐✈❡ s♦❧✉t✐♦♥ ❝♦♥❝❡♣t❄ ❘❡❝❛❧❝✉❧❛t❡❞ ■♠♣✉t❛t✐♦♥ ✐s ❞✐✛❡r❡♥t✦

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✶✸✴✸✵

▲❡t g∗ ❜❡ ❢♦r♠❡❞ ❛t t❤❡ ✜rst st❛❣❡✳ ■❢ ♣❧❛②❡rs r❡❝❛❧❝✉❧❛t❡ t❤❡ ✐♠♣✉t❛t✐♦♥✱ ♦♥❡ ♥❡❡❞s t♦ ❝♦♥str✉❝t ❛ ♥❡✇ ❝♦♦♣❡r❛t✐✈❡ ❣❛♠❡ (N, v(g∗))✱ ♣r♦✈✐❞❡❞ ♥❡t✇♦r❦ g∗ ✐s ✜①❡❞✳ Pr♦♣♦s✐t✐♦♥ ❚❤❡ s✉♣❡r❛❞❞✐t✐✈❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ v(g∗, ·) ✐♥ t❤❡ s❡♥s❡ ♦❢ ✈♦♥ ◆❡✉♠❛♥♥ ❛♥❞ ▼♦r❣❡♥st❡r♥ ✐s ❞❡✜♥❡❞ ❛s✿ v(g∗, N) =

• i∈N

Ki(u∗

i , u∗ Ni(g∗)),

v(g∗, S) = max

ui∈Ui,i∈S

• i∈S

Ki(ui, uNi(g∗)∩S), v(g∗, ∅) = 0, ❋♦r ❛ s✐♥❣❧❡t♦♥ {i}✱ ✐ts ✈❛❧✉❡ ✐s ❞❡✜♥❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✿ v(g∗, {i}) = max

ui∈Ui Ki(ui) = V ({i}),

❛♥❞ ✐t ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ♥❡t✇♦r❦✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✶✹✴✸✵

❆♥ ✐♠♣✉t❛t✐♦♥ ✐s ❛♥ n✲❞✐♠❡♥s✐♦♥❛❧ ♣r♦✜❧❡ ξ(g∗) = (ξ1(g∗), . . . , ξn(g∗))✱ s❛t✐s❢②✐♥❣ ❜♦t❤ t❤❡ ❡✣❝✐❡♥❝② ❝♦♥❞✐t✐♦♥ ❛♥❞ t❤❡ ✐♥❞✐✈✐❞✉❛❧ r❛t✐♦♥❛❧✐t② ❝♦♥❞✐t✐♦♥✿

• i∈N

ξi(g∗) = v(g∗, N), ξi(g∗)

• v(g∗, {i}), i ∈ N.

❚❤❡ s❡t ♦❢ ❛❧❧ ✐♠♣✉t❛t✐♦♥s ✐♥ t❤❡ ❣❛♠❡ (N, v(g∗)) ✇❡ ❞❡♥♦t❡ ❜② I(v(g∗))✳ ❆ ❝♦♦♣❡r❛t✐✈❡ s♦❧✉t✐♦♥ ❝♦♥❝❡♣t ✐♥ t❤❡ ❛✉①✐❧✐❛r② ❝♦♦♣❡r❛t✐✈❡ ❚❯✲❣❛♠❡ (N, v(g∗)) ✇✐t❤ ✜①❡❞ ♥❡t✇♦r❦ g∗ ✐s ❛ r✉❧❡ t❤❛t ✉♥✐q✉❡❧② ❛ss✐❣♥s ❛ s✉❜s❡t CSC(v(g∗)) ⊆ I(v(g∗)) t♦ t❤❡ ❣❛♠❡ (N, v(g∗))✳ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ t❤❡ ❝♦♦♣❡r❛t✐✈❡ s♦❧✉t✐♦♥ ❝♦♥❝❡♣t ✐s t❤❡ ❝♦r❡ C(v(g∗))✱ t❤❡♥ C(v(g∗)) =

• ξ(g∗) = (ξ1(g∗), . . . , ξn(g∗)) :
• i∈S

ξi(g∗) v(g∗, S), S ⊆ N

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✶✺✴✸✵

❉❡✜♥✐t✐♦♥ ❆♥ ✐♠♣✉t❛t✐♦♥ ξ ∈ CSC(V ) ✐s s❛✐❞ t♦ ❜❡ t✐♠❡✲❝♦♥s✐st❡♥t ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ✐♠♣✉t❛t✐♦♥ ξ(g∗) ∈ CSC(v(g∗)) s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛❧✐t② ❤♦❧❞s ❢♦r ❛❧❧ ♣❧❛②❡rs✿ ξi = ξi(g∗), i ∈ N. ✭✺✮ ❆ ❝♦♦♣❡r❛t✐✈❡ s♦❧✉t✐♦♥ ❝♦♥❝❡♣t CSC(V ) ✐s t✐♠❡ ❝♦♥s✐st❡♥t ✐❢ ❛♥② ✐♠♣✉t❛t✐♦♥ ξ ∈ CSC(V ) ✐s t✐♠❡✲❝♦♥s✐st❡♥t✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✶✻✴✸✵

❉❡✜♥✐t✐♦♥ ■♠♣✉t❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ ♣r♦❝❡❞✉r❡ ❢♦r ξ ✐♥ t❤❡ ❝♦♦♣❡r❛t✐✈❡ t✇♦✲st❛❣❡ ♥❡t✇♦r❦ ❣❛♠❡ ✐s ❛ ♠❛tr✐① β =    β11 β12 ✳ ✳ ✳ ✳ ✳ ✳ βn1 βn2    , ✇❤❡r❡ ξi = βi1 + βi2, i ∈ N. ❚❤❡ ♣❛②♠❡♥t s❝❤❡♠❡ ✐s ❛♣♣❧✐❡❞✿ ♣❧❛②❡r i ∈ N ❛t t❤❡ ✜rst st❛❣❡ ♦❢ t❤❡ ❣❛♠❡ r❡❝❡✐✈❡s t❤❡ ♣❛②♠❡♥t βi1✱ ❛t t❤❡ s❡❝♦♥❞ st❛❣❡ ♦❢ t❤❡ ❣❛♠❡ ❤❡ r❡❝❡✐✈❡s t❤❡ ♣❛②♠❡♥t βi2✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✶✼✴✸✵

❉❡✜♥✐t✐♦♥ ■♠♣✉t❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ ♣r♦❝❡❞✉r❡ β ❢♦r ξ ✐s t✐♠❡✲❝♦♥s✐st❡♥t ✐❢ ξi − βi1 = ξi(g∗), ❢♦r ❛❧❧ i ∈ N. ■t ✐s ♦❜✈✐♦✉s t❤❛t t✐♠❡✲❝♦♥s✐st❡♥t ✐♠♣✉t❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ ♣r♦❝❡❞✉r❡ ❢♦r ξ = (ξ1, . . . , ξn) ✐♥ t❤❡ ❝♦♦♣❡r❛t✐✈❡ t✇♦✲st❛❣❡ ♥❡t✇♦r❦ ❣❛♠❡ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿ βi1 = ξi − ξi(g∗), ✭✻✮ βi2 = ξi(g∗), i ∈ N. ■♥ ❝❛s❡ ♦❢ t❤❡ ❝♦♦♣❡r❛t✐✈❡ s♦❧✉t✐♦♥ ❝♦♥❝❡♣t CSC(V ) ❛ss✐❣♥s ♠✉❧t✐♣❧❡ ❛❧❧♦❝❛t✐♦♥s ✭❢♦r ❡①❛♠♣❧❡✱ t❤❡ ❝♦r❡✮✱ ♠♦r❡ str✐❝t ♣r♦♣❡rt② ♦❢ ✐♠♣✉t❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ ♣r♦❝❡❞✉r❡ ❝❛♥ ❜❡ ✉s❡❞✖t❤❡ str♦♥❣ t✐♠❡✲❝♦♥s✐st❡♥❝② ♣r♦♣❡rt②✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✶✽✴✸✵

❉❡✜♥✐t✐♦♥ ❆♥ ✐♠♣✉t❛t✐♦♥ ξ ∈ CSC(V ) ✐s s❛✐❞ t♦ ❜❡ str♦♥❣❧② t✐♠❡✲❝♦♥s✐st❡♥t ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❝❧✉s✐♦♥ ✐s s❛t✐s✜❡❞✿ CSC(v(g∗)) ⊆ CSC(V ). ✭✼✮ ❆ ❝♦♦♣❡r❛t✐✈❡ s♦❧✉t✐♦♥ ❝♦♥❝❡♣t CSC(V ) ✐s str♦♥❣❧② t✐♠❡✲❝♦♥s✐st❡♥t ✐❢ ❛♥② ✐♠♣✉t❛t✐♦♥ ξ ∈ CSC(V ) ✐s str♦♥❣❧② t✐♠❡✲❝♦♥s✐st❡♥t✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❝♦r❡ C(V ) ✐s str♦♥❣❧② t✐♠❡✲❝♦♥s✐st❡♥t ✐❢ C(v(g∗)) ⊆ C(V )✳ ❯♥❢♦rt✉♥❛t❡❧②✱ ❢♦r str♦♥❣❧② t✐♠❡✲❝♦♥s✐st❡♥t ✐♠♣✉t❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ ♣r♦❝❡❞✉r❡s ✐t ✐s ✐♠♣♦ss✐❜❧❡ ❡✈❡♥ t♦ ❞❡r✐✈❡ ❢♦r♠✉❧❛s s✐♠✐❧❛r t♦ ✭✻✮✳ Pr♦♣♦s✐t✐♦♥ ■♥ t✇♦✲st❛❣❡ ❝♦♦♣❡r❛t✐✈❡ ❣❛♠❡s t❤❡ ❝♦r❡ C(V ) ✐s t✐♠❡✲❝♦♥s✐st❡♥t ❜✉t ♥♦t str♦♥❣❧② t✐♠❡✲❝♦♥s✐st❡♥t✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✶✾✴✸✵

❉❡✜♥✐t✐♦♥ ■♠♣✉t❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ ♣r♦❝❡❞✉r❡ β ❢♦r ξ ✐s str♦♥❣❧② t✐♠❡✲❝♦♥s✐st❡♥t ✐❢ (β11, . . . , βn1) ⊕ CSC(v(g∗)) ⊆ CSC(V ), ✭✽✮ ✇❤❡r❡ a ⊕ A = {a + a′ : a′ ∈ A, a ∈ Rn, A ⊂ Rn}✳ ◆♦t❡✱ t❤❛t str♦♥❣❧② t✐♠❡✲❝♦♥s✐st❡♥t ✐♠♣✉t❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ ♣r♦❝❡❞✉r❡ β ❢♦r ❛♥ ✐♠♣✉t❛t✐♦♥ ❢r♦♠ t❤❡ ❝♦r❡ C(V ) s❛t✐s✜❡s t❤❡ ✐♥❝❧✉s✐♦♥✿ (β11, . . . , βn1) ⊕ C(v(g∗)) ⊆ C(V ). ✭✾✮

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✷✵✴✸✵

◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡

N = {1, 2, 3}✳ ❙✉❜s❡ts ♦❢ ♣❧❛②❡rs t♦ ✇❤♦♠ ❡❛❝❤ ♣❧❛②❡r ❝❛♥ ♦✛❡r ❛ ❧✐♥❦ ❛r❡✿ M1 = {2, 3}, M2 = {1, 3}, M3 = {1}✳ ❆ ♥✉♠❜❡r ♦❢ ❧✐♥❦s ❡❛❝❤ ♣❧❛②❡rs ❝❛♥ ♦✛❡r✿ a1 = a2 = a3 = 1✳ ❚❤❡r❡❢♦r❡✱ ❛t t❤❡ ✜rst st❛❣❡ s❡ts ♦❢ ♣❧❛②❡rs✬ ❜❡❤❛✈✐♦rs ❛r❡✿ G1 = {(0, 0, 0); (0, 1, 0); (0, 0, 1)}✱ G2 = {(0, 0, 0); (1, 0, 0); (0, 0, 1)}✱ G3 = {(0, 0, 0); (1, 0, 0)}✱ ❛♥❞ ♦♥❧② t❤r❡❡ ♥❡t✇♦r❦s ❝❛♥ ❜❡ ❢♦r♠❡❞ ❛t t❤❡ ✜rst st❛❣❡ ♦❢ t❤❡ ❣❛♠❡✿ t❤❡ ❡♠♣t② ♥❡t✇♦r❦ ✭t❤❡ ♥❡t✇♦r❦ ✇✐t❤♦✉t ❧✐♥❦s✱ g = ∅✮✱ g = {12}✱ ❛♥❞ g = {13}✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✷✶✴✸✵

❙✉♣♣♦s❡ t❤❛t s❡ts ♦❢ ❝♦♥tr♦❧s Ui ❛t t❤❡ s❡❝♦♥❞ st❛❣❡ ❢♦r ❛♥② ♥❡t✇♦r❦ g✱ r❡❛❧✐③❡❞ ❛t t❤❡ ✜rst st❛❣❡✱ ❛r❡ t❤❡ s❛♠❡ U1 = U2 = U3 = {A, B}✱ ❛♥❞ ♣❛②♦✛ ❢✉♥❝t✐♦♥s ❛r❡ ❞❡✜♥❡❞ ❛s✿

Ki(ui) : Ki(A) = 1, Ki(B) = 0, i = 1, 2, 3, K1(u1, u2) : K1(A, A) = 2, K1(A, B) = 4, K1(B, A) = 1, K1(B, B) = 3, K1(u1, u3) : K1(A, A) = 3, K1(A, B) = 5, K1(B, A) = 1, K1(B, B) = 3, K2(u2, u1) : K2(A, A) = 2, K2(A, B) = 4, K2(B, A) = 1, K2(B, B) = 3, K3(u3, u1) : K3(A, A) = 2, K3(A, B) = 5, K3(B, A) = 1, K3(B, B) = 4.

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✷✷✴✸✵

❈♦♥s✐❞❡r t❤❡ ❝❛s❡ ♦❢ ❝♦♦♣❡r❛t✐♦♥ ❛t ❜♦t❤ st❛❣❡s✳ ■♥ t❤✐s ❝❛s❡ t❤❡ ♠❛①✐♠❛❧ ✈❛❧✉❡

• i∈N

Ki(u∗

i , u∗ Ni(g∗))

= 8, ❛♥❞ ✐t ❝❛♥ ❜❡ r❡❛❝❤❡❞ ✐❢ ♣❧❛②❡rs ❝❤♦♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❜❡❤❛✈✐♦rs✿ g∗

1

= (0, 0, 1), g∗

2 = (0, 0, 0), g∗ 3 = (1, 0, 0),

u∗

1

= B, u∗

2 = A, u∗ 3 = B.

◆♦t❡ t❤❛t ❜❡❤❛✈✐♦r ♣r♦✜❧❡ g∗

1 , g∗ 2 , g∗ 3 ❛t t❤❡ ✜rst st❛❣❡ ❢♦r♠s t❤❡

♥❡t✇♦r❦ g∗ = {13}✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✷✸✴✸✵

❈❛❧❝✉❧❛t❡ ✈❛❧✉❡s ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ V (S) ❢♦r ❛❧❧ ❝♦❛❧✐t✐♦♥s S ⊆ N✿ V (N) = 8✱ V ({1, 2}) = 6✱ V ({1, 3}) = 7✱ V ({2, 3}) = 2✱ V ({1}) = V ({2}) = V ({3}) = 1✳ ❚❤❡ ❝♦r❡ C(V )✿ ξ1 + ξ3 = 7, ξ1

• 5,

ξ2 = 1, ξ3

• 1.

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✷✹✴✸✵

❈♦♥s✐❞❡r ♥♦✇ ❝♦♦♣❡r❛t✐♦♥ ❛t t❤❡ s❡❝♦♥❞ st❛❣❡ ♦❢ t❤❡ ❣❛♠❡✱ ♣r♦✈✐❞❡❞ t❤❛t ♥❡t✇♦r❦ g∗ = {13} ❛t t❤❡ ✜rst st❛❣❡ ✐s ✜①❡❞✳ ❈❛❧❝✉❧❛t❡ ✈❛❧✉❡s ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ v(g∗, S) ❢♦r ❛❧❧ ❝♦❛❧✐t✐♦♥s S ⊆ N✿ v({13}, N) = 8✱ v({13}, {1, 2}) = v({13}, {2, 3}) = 2✱ v({13}, {1, 3}) = 7✱ v({13}, {1}) = v({13}, {2}) = v({13}, {3}) = 1✳ ❚❤❡ ❝♦r❡ C(v({13}))✿ ξ1({13}) + ξ3({13}) = 7, ξ1({13})

• 1,

ξ2({13}) = 1, ξ3({13})

• 1.

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✷✺✴✸✵

❙✐♥❝❡ C(V ) ⊂ C(v(g∗))✱ t❤❡ ❝♦r❡ C(V ) ✐s t✐♠❡✲❝♦♥s✐st❡♥t ❝♦♦♣❡r❛t✐✈❡ s♦❧✉t✐♦♥ ❝♦♥❝❡♣t ✐♥ t✇♦✲st❛❣❡ ♥❡t✇♦r❦ ❣❛♠❡s ❜✉t ✐t ✐s ♦❜✈✐♦✉s t❤❛t t❤❡ ❝♦r❡ C(V ) ✐s ♥♦t str♦♥❣❧② t✐♠❡✲❝♦♥s✐st❡♥t ✭✐♥❝❧✉s✐♦♥ ✭✼✮ ❞♦❡s ♥♦t ❤♦❧❞✮✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✷✻✴✸✵

❈♦♥s✐❞❡r ❛♥♦t❤❡r ❝♦♦♣❡r❛t✐✈❡ s♦❧✉t✐♦♥ ❝♦♥❝❡♣t✖t❤❡ τ✲✈❛❧✉❡✳ ■♥ t❤❡ ❝♦♦♣❡r❛t✐✈❡ t✇♦✲st❛❣❡ ♥❡t✇♦r❦ ❣❛♠❡ t❤❡ τ✲✈❛❧✉❡ τ = (τ1, . . . , τn) ✐s ❝❛❧❝✉❧❛t❡❞ ❛s ❢♦❧❧♦✇s✿ τi = mi(V ) + αV (Mi(V ) − mi(V )) , ✭✶✵✮ ✇❤❡r❡ Mi(V ) = V (N) − V (N \ {i}), mi(V ) = max

S∋i

 V (S) −

• j∈S\{i}

Mj(V )   , αV =      0, M(V ) = m(V ),

• i∈N

Mi(V )−

i∈N

mi(V ) V (N)−

i∈N

mi(V )

, ♦t❤❡r✇✐s❡. ❯s✐♥❣ ✭✶✵✮✱ ✇❡ ♦❜t❛✐♥✿ τ = (51

2, 1, 11 2)✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✷✼✴✸✵

■♥ t❤❡ s✐♠✐❧❛r ✇❛② ♦♥❡ ❝❛♥ ❝❛❧❝✉❧❛t❡ t❤❡ τ✲✈❛❧✉❡ τ(g∗) = (τ1(g∗), . . . , τn(g∗)) ✐❢ ♥❡t✇♦r❦ g∗ ✐s ❢♦r♠❡❞ ❛t t❤❡ ✜rst st❛❣❡ ♦❢ t❤❡ ❣❛♠❡✿ τi(g∗) = mi(v(g∗)) + αv(g∗) (Mi(v(g∗)) − mi(v(g∗))) , ✭✶✶✮ ✇❤❡r❡ Mi(v(g∗)) = v(g∗, N) − v(g∗, N \ {i}), mi(v(g∗)) = max

S∋i

 v(g∗, S) −

• j∈S\{i}

Mj(v(g∗))   , αv(g∗) =      0, M(v(g∗)) = m(v(g∗)),

• i∈N

Mi(v(g∗))−

i∈N

mi(v(g∗)) v(g∗,N)−

i∈N

mi(v(g∗))

, ♦t❤❡r✇✐s❡. ❯s✐♥❣ ✭✶✶✮✱ ✇❡ ♦❜t❛✐♥✿ τ(g∗) = (3 1

2, 1, 31 2)✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✷✽✴✸✵

◆♦t❡ t❤❛t t❤❡ τ✲✈❛❧✉❡ ✐s t✐♠❡✲✐♥❝♦♥s✐st❡♥t s✐♥❝❡ t❤❡r❡ ❡①✐sts ❛ ♣❧❛②❡r i ∈ N t❤❛t τi = τi(g∗). ◆❡✈❡rt❤❡❧❡ss✱ ♦♥❡ ❝❛♥ ✜♥❞ t✐♠❡✲❝♦♥s✐st❡♥t ✐♠♣✉t❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ ♣r♦❝❡❞✉r❡ β ❢♦r t❤❡ τ✲✈❛❧✉❡ ✭✇❤✐❝❤ ✇✐❧❧ ❛❧s♦ ❜❡ str♦♥❣❧② t✐♠❡✲❝♦♥s✐st❡♥t s✐♥❝❡ t❤❡ τ✲✈❛❧✉❡ ✐s t❤❡ s✐♥❣❧❡✲✈❛❧✉❡❞ ❝♦♦♣❡r❛t✐✈❡ s♦❧✉t✐♦♥ ❝♦♥❝❡♣t✮✿ β12 = τ1(g∗) = 31

2,

β11 = τ1 − τ1(g∗) = 2, β22 = τ2(g∗) = 1, β21 = τ2 − τ2(g∗) = 0, β32 = τ3(g∗) = 31

2,

β31 = τ3 − τ3(g∗) = −2, ♦r ✐♥ t❤❡ ♠❛tr✐① ❢♦r♠✿ β =   2 31

2

1 −2 31

2

  .

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✷✾✴✸✵

❘❡❢❡r❡♥❝❡s

❇❛❧❛ ❱✱ ●♦②❛❧ ❙✳ ❆ ♥♦♥✲❝♦♦♣❡r❛t✐✈❡ ♠♦❞❡❧ ♦❢ ♥❡t✇♦r❦ ❢♦r♠❛t✐♦♥✳ ❊❝♦♥♦♠❡tr✐❝❛ ✷✵✵✵❀✻✽✭✺✮✿✶✶✽✶✲✶✷✸✶✳ ❉✉tt❛ ❇✱ ❱❛♥ ❞❡♥ ◆♦✉✇❡❧❛♥❞ ❆✱ ❚✐❥s ❙✳ ▲✐♥❦ ❢♦r♠❛t✐♦♥ ✐♥ ❝♦♦♣❡r❛t✐✈❡ s✐t✉❛t✐♦♥s✳ ■♥t ❏ ●❛♠❡ ❚❤❡♦r② ✶✾✾✽❀✷✼✿✷✹✺✲✷✺✻✳

• ♦②❛❧ ❙✱ ❱❡❣❛✲❘❡❞♦♥❞♦ ❋✳ ◆❡t✇♦r❦ ❢♦r♠❛t✐♦♥ ❛♥❞ s♦❝✐❛❧ ❝♦♦r❞✐♥❛t✐♦♥✳
• ❛♠❡s ❊❝♦♥ ❇❡❤❛✈ ✷✵✵✺❀✺✵✿✶✼✽✲✷✵✼✳

❏❛❝❦s♦♥ ▼✱ ❲❛tts ❆✳ ❖♥ t❤❡ ❢♦r♠❛t✐♦♥ ♦❢ ✐♥t❡r❛❝t✐♦♥ ♥❡t✇♦r❦s ✐♥ s♦❝✐❛❧ ❝♦♦r❞✐♥❛t✐♦♥ ❣❛♠❡s✳ ●❛♠❡s ❊❝♦♥ ❇❡❤❛✈ ✷✵✵✷❀✹✶✭✷✮✿✷✻✺✲✷✾✶✳ ❏❛❝❦s♦♥ ▼✱ ❲♦❧✐♥s❦② ❆✳ ❆ str❛t❡❣✐❝ ♠♦❞❡❧ ♦❢ s♦❝✐❛❧ ❛♥❞ ❡❝♦♥♦♠✐❝ ♥❡t✇♦r❦s✳ ❏ ❊❝♦♥ ❚❤❡♦r② ✶✾✾✻❀✼✶✿✹✹✲✼✹✳ ❑✉❤♥ ❍❲✳ ❊①t❡♥s✐✈❡ ❣❛♠❡s ❛♥❞ t❤❡ ♣r♦❜❧❡♠ ♦❢ ✐♥❢♦r♠❛t✐♦♥✳ ■♥✿ ❑✉❤♥ ❍❲✱ ❚✉❝❦❡r ❆❲✱ ❡❞✐t♦rs✳ ❈♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ t❤❡♦r② ♦❢ ❣❛♠❡s ■■✳ Pr✐♥❝❡t♦♥✿ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss❀ ✶✾✺✸✳ ♣✳ ✶✾✸✲✷✶✻✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✷✾✴✸✵

❘❡❢❡r❡♥❝❡s

P❡tr♦s②❛♥ ▲❆✳ ❙t❛❜✐❧✐t② ♦❢ s♦❧✉t✐♦♥s ✐♥ ❞✐✛❡r❡♥t✐❛❧ ❣❛♠❡s ✇✐t❤ ♠❛♥② ♣❛rt✐❝✐♣❛♥ts✳ ❱❡st♥✐❦ ▲❡♥✐♥❣r❛❞s❦♦❣♦ ❯♥✐✈❡rs✐t❡t❛✳ ❙❡r ✶✳ ▼❛t❡♠❛t✐❦❛ ▼❡❦❤❛♥✐❦❛ ❆str♦♥♦♠✐②❛ ✶✾✼✼❀✶✾✿✹✻✲✺✷✳ P❡tr♦s❥❛♥ ▲❆✳ ❈♦♦♣❡r❛t✐✈❡ ❞✐✛❡r❡♥t✐❛❧ ❣❛♠❡s✳ ■♥✿ ◆♦✇❛❦ ❆❙✱ ❙③❛❥♦✇s❦✐ ❑✱ ❡❞✐t♦rs✳ ❆♥♥❛❧s ♦❢ t❤❡ ■♥t❡r♥❛t✐♦♥❛❧ ❙♦❝✐❡t② ♦❢ ❉②♥❛♠✐❝ ●❛♠❡s✳ ❆♣♣❧✐❝❛t✐♦♥s t♦ ❡❝♦♥♦♠✐❝s✱ ✜♥❛♥❝❡✱ ♦♣t✐♠✐③❛t✐♦♥✱ ❛♥❞ st♦❝❤❛st✐❝ ❝♦♥tr♦❧✳ ❇❛s❡❧✿ ❇✐r❦❤☎ ❛✉s❡r❀ ✷✵✵✺✳ ♣✳ ✶✽✸✲✷✵✵✳ P❡tr♦s②❛♥ ▲❆✱ ❉❛♥✐❧♦✈ ◆◆✳ ❙t❛❜✐❧✐t② ♦❢ s♦❧✉t✐♦♥s ✐♥ ♥♦♥✲③❡r♦ s✉♠ ❞✐✛❡r❡♥t✐❛❧ ❣❛♠❡s ✇✐t❤ tr❛♥s❢❡r❛❜❧❡ ♣❛②♦✛s✳ ❱❡st♥✐❦ ▲❡♥✐♥❣r❛❞s❦♦❣♦ ❯♥✐✈❡rs✐t❡t❛✳ ❙❡r ✶✳ ▼❛t❡♠❛t✐❦❛ ▼❡❦❤❛♥✐❦❛ ❆str♦♥♦♠✐②❛ ✶✾✼✾❀✶✿✺✷✲✺✾✳ P❡tr♦s②❛♥ ▲❆✱ ❙❡❞❛❦♦✈ ❆❆✳ ▼✉❧t✐st❛❣❡ ♥❡t✇♦r❦✐♥❣ ❣❛♠❡s ✇✐t❤ ❢✉❧❧ ✐♥❢♦r♠❛t✐♦♥✳ ▼❛t❡♠❛t✐❝❤❡s❦❛②❛ t❡♦r✐②❛ ✐❣r ✐ ❡❡ ♣r✐❧♦③❤❡♥✐②❛ ✷✵✵✾❀✶✭✷✮✿✻✻✲✽✶✳ P❡tr♦s②❛♥ ▲❆✱ ❙❡❞❛❦♦✈ ❆❆✱ ❇♦❝❤❦❛r❡✈ ❆❖✳ ❚✇♦✲st❛❣❡ ♥❡t✇♦r❦ ❣❛♠❡s✳ ▼❛t❡♠❛t✐❝❤❡s❦❛②❛ t❡♦r✐②❛ ✐❣r ✐ ❡❡ ♣r✐❧♦③❤❡♥✐②❛ ✷✵✶✸❀✺✭✹✮✿✽✹✲✶✵✹✳ ❚✐❥s ❙❍✳ ❆♥ ❛①✐♦♠❛t✐③❛t✐♦♥ ♦❢ t❤❡ τ✲✈❛❧✉❡✳ ▼❛t❤ ❙♦❝ ❙❝✐ ✶✾✽✼❀✶✸✿✶✼✼✲✶✽✶✳

❙tr♦♥❣❧② ❚✐♠❡✲❝♦♥s✐st❡♥t ❙♦❧✉t✐♦♥s ❢♦r ❚✇♦✲st❛❣❡ ◆❡t✇♦r❦ ●❛♠❡s ❍✳ ●❛♦✱ ▲✳ P❡tr♦s②❛♥✱ ❆✳ ❙❡❞❛❦♦✈ ✸✵✴✸✵