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Oct 27, 2023 343 likes •988 views

t rt r r rt Prs st t sst strts

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SLIDE 1

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ❢♦r ❇✐♥❛r② ◗✉❛❞r❛t✐❝ Pr♦❜❧❡♠s s✉❜❥❡❝t t♦ ❆ss✐❣♥♠❡♥t ❈♦♥str❛✐♥ts

❙✈❡♥ ▼❛❧❧❛❝❤

■♥st✐t✉t ❢ür ■♥❢♦r♠❛t✐❦ ❯♥✐✈❡rs✐tät ③✉ ❑ö❧♥ ✷✶st ❈♦♠❜✐♥❛t♦r✐❛❧ ❖♣t✐♠✐③❛t✐♦♥ ❲♦r❦s❤♦♣ ❆✉ss♦✐s✱ ✾t❤✲✶✸t❤ ❏❛♥✉❛r② ✷✵✶✼

❏❛♥✉❛r② ✶✶✱ ✷✵✶✼

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SLIDE 2

❇✐❧✐♥❡❛r ✵✲✶ Pr♦❜❧❡♠s ✇✐t❤ ❆ss✐❣♥♠❡♥t ❈♦♥str❛✐♥ts

♠✐♥ cTx + dTy s✳t✳

  • i∈Ak

xi = ✶ ❢♦r ❛❧❧ k ∈ K ✭✶✮ (❇◗P❆) Cx + Dy ≥ b yij = xixj ❢♦r ❛❧❧ (i, j) ∈ E ✭✷✮ xi ∈ {✵, ✶} ❢♦r ❛❧❧ i ∈ N ✱ ✇❤❡r❡ ✶ ❢♦r ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ✳ s❡t ♦❢ ✐♥❞❡① s❡ts ✱ ✭❵❛ss✐❣♥♠❡♥t ❣r♦✉♣s✬✮ s✉❝❤ t❤❛t ✳ ❜✐❧✐♥❡❛r t❡r♠s ✱ ✐♥ ♦❜❥❡❝t✐✈❡ ❛♥❞ ❝♦♥str❛✐♥ts✳ ❝♦❧❧❡❝t❡❞ ✐♥ ❛♥ ♦r❞❡r❡❞ s❡t ✱ s✳t✳ ❢♦r ✳

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SLIDE 3

❇✐❧✐♥❡❛r ✵✲✶ Pr♦❜❧❡♠s ✇✐t❤ ❆ss✐❣♥♠❡♥t ❈♦♥str❛✐♥ts

♠✐♥ cTx + dTy s✳t✳

  • i∈Ak

xi = ✶ ❢♦r ❛❧❧ k ∈ K ✭✶✮ (❇◗P❆) Cx + Dy ≥ b yij = xixj ❢♦r ❛❧❧ (i, j) ∈ E ✭✷✮ xi ∈ {✵, ✶} ❢♦r ❛❧❧ i ∈ N

◮ xi✱ i ∈ N ✇❤❡r❡ N = {✶, . . . , n} ❢♦r ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✳

s❡t ♦❢ ✐♥❞❡① s❡ts ✱ ✭❵❛ss✐❣♥♠❡♥t ❣r♦✉♣s✬✮ s✉❝❤ t❤❛t ✳ ❜✐❧✐♥❡❛r t❡r♠s ✱ ✐♥ ♦❜❥❡❝t✐✈❡ ❛♥❞ ❝♦♥str❛✐♥ts✳ ❝♦❧❧❡❝t❡❞ ✐♥ ❛♥ ♦r❞❡r❡❞ s❡t ✱ s✳t✳ ❢♦r ✳

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SLIDE 4

❇✐❧✐♥❡❛r ✵✲✶ Pr♦❜❧❡♠s ✇✐t❤ ❆ss✐❣♥♠❡♥t ❈♦♥str❛✐♥ts

♠✐♥ cTx + dTy s✳t✳

  • i∈Ak

xi = ✶ ❢♦r ❛❧❧ k ∈ K ✭✶✮ (❇◗P❆) Cx + Dy ≥ b yij = xixj ❢♦r ❛❧❧ (i, j) ∈ E ✭✷✮ xi ∈ {✵, ✶} ❢♦r ❛❧❧ i ∈ N

◮ xi✱ i ∈ N ✇❤❡r❡ N = {✶, . . . , n} ❢♦r ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✳ ◮ s❡t K ♦❢ ✐♥❞❡① s❡ts Ak✱ k ∈ K ✭❵❛ss✐❣♥♠❡♥t ❣r♦✉♣s✬✮

s✉❝❤ t❤❛t ✳ ❜✐❧✐♥❡❛r t❡r♠s ✱ ✐♥ ♦❜❥❡❝t✐✈❡ ❛♥❞ ❝♦♥str❛✐♥ts✳ ❝♦❧❧❡❝t❡❞ ✐♥ ❛♥ ♦r❞❡r❡❞ s❡t ✱ s✳t✳ ❢♦r ✳

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SLIDE 5

❇✐❧✐♥❡❛r ✵✲✶ Pr♦❜❧❡♠s ✇✐t❤ ❆ss✐❣♥♠❡♥t ❈♦♥str❛✐♥ts

♠✐♥ cTx + dTy s✳t✳

  • i∈Ak

xi = ✶ ❢♦r ❛❧❧ k ∈ K ✭✶✮ (❇◗P❆) Cx + Dy ≥ b yij = xixj ❢♦r ❛❧❧ (i, j) ∈ E ✭✷✮ xi ∈ {✵, ✶} ❢♦r ❛❧❧ i ∈ N

◮ xi✱ i ∈ N ✇❤❡r❡ N = {✶, . . . , n} ❢♦r ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✳ ◮ s❡t K ♦❢ ✐♥❞❡① s❡ts Ak✱ k ∈ K ✭❵❛ss✐❣♥♠❡♥t ❣r♦✉♣s✬✮ ◮ s✉❝❤ t❤❛t k∈K Ak = N✳

❜✐❧✐♥❡❛r t❡r♠s ✱ ✐♥ ♦❜❥❡❝t✐✈❡ ❛♥❞ ❝♦♥str❛✐♥ts✳ ❝♦❧❧❡❝t❡❞ ✐♥ ❛♥ ♦r❞❡r❡❞ s❡t ✱ s✳t✳ ❢♦r ✳

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SLIDE 6

❇✐❧✐♥❡❛r ✵✲✶ Pr♦❜❧❡♠s ✇✐t❤ ❆ss✐❣♥♠❡♥t ❈♦♥str❛✐♥ts

♠✐♥ cTx + dTy s✳t✳

  • i∈Ak

xi = ✶ ❢♦r ❛❧❧ k ∈ K ✭✶✮ (❇◗P❆) Cx + Dy ≥ b yij = xixj ❢♦r ❛❧❧ (i, j) ∈ E ✭✷✮ xi ∈ {✵, ✶} ❢♦r ❛❧❧ i ∈ N

◮ xi✱ i ∈ N ✇❤❡r❡ N = {✶, . . . , n} ❢♦r ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✳ ◮ s❡t K ♦❢ ✐♥❞❡① s❡ts Ak✱ k ∈ K ✭❵❛ss✐❣♥♠❡♥t ❣r♦✉♣s✬✮ ◮ s✉❝❤ t❤❛t k∈K Ak = N✳ ◮ ❜✐❧✐♥❡❛r t❡r♠s yij = xixj✱ i, j ∈ N ✐♥ ♦❜❥❡❝t✐✈❡ ❛♥❞ ❝♦♥str❛✐♥ts✳

❝♦❧❧❡❝t❡❞ ✐♥ ❛♥ ♦r❞❡r❡❞ s❡t ✱ s✳t✳ ❢♦r ✳

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SLIDE 7

❇✐❧✐♥❡❛r ✵✲✶ Pr♦❜❧❡♠s ✇✐t❤ ❆ss✐❣♥♠❡♥t ❈♦♥str❛✐♥ts

♠✐♥ cTx + dTy s✳t✳

  • i∈Ak

xi = ✶ ❢♦r ❛❧❧ k ∈ K ✭✶✮ (❇◗P❆) Cx + Dy ≥ b yij = xixj ❢♦r ❛❧❧ (i, j) ∈ E ✭✷✮ xi ∈ {✵, ✶} ❢♦r ❛❧❧ i ∈ N

◮ xi✱ i ∈ N ✇❤❡r❡ N = {✶, . . . , n} ❢♦r ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✳ ◮ s❡t K ♦❢ ✐♥❞❡① s❡ts Ak✱ k ∈ K ✭❵❛ss✐❣♥♠❡♥t ❣r♦✉♣s✬✮ ◮ s✉❝❤ t❤❛t k∈K Ak = N✳ ◮ ❜✐❧✐♥❡❛r t❡r♠s yij = xixj✱ i, j ∈ N ✐♥ ♦❜❥❡❝t✐✈❡ ❛♥❞ ❝♦♥str❛✐♥ts✳ ◮ ❝♦❧❧❡❝t❡❞ ✐♥ ❛♥ ♦r❞❡r❡❞ s❡t E ⊂ N × N✱ s✳t✳ i ≤ j ❢♦r (i, j) ∈ E✳

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SLIDE 8

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥

▲❡♦ ▲✐❜❡rt✐✿ ❈♦♠♣❛❝t ❧✐♥❡❛r✐③❛t✐♦♥ ❢♦r ❜✐♥❛r② q✉❛❞r❛t✐❝ ♣r♦❜❧❡♠s✳ ✹❖❘ ✺✭✸✮✿✷✸✶✕✷✹✺✱ ✷✵✵✼ ▲✐♥❡❛r✐③❛t✐♦♥ ♠❡t❤♦❞ t❤❛t ❡①♣❧♦✐ts t❤❡ s♣❡❝✐❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❛ss✐❣♥♠❡♥t ❝♦♥str❛✐♥ts✳ ❋♦r ❡❛❝❤ s❡t ❜✉✐❧❞ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ s❡t ♦❢ ✈❛r✐❛❜❧❡ ✐♥❞✐❝❡s✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ✱ ❛❞❞ t❤❡ ❡q✉❛t✐♦♥ t❤❛t r❡s✉❧ts ✇❤❡♥ ♠✉❧t✐♣❧②✐♥❣ t❤❡ ❛ss✐❣♥♠❡♥t ❝♦♥str❛✐♥ts ✇✳r✳t✳ ✇✐t❤ ✿ ✶ ❋✐♥❛❧❧②✱ r❡♣❧❛❝❡ ✇✐t❤ ❛ ❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r✐③❛t✐♦♥ ✈❛r✐❛❜❧❡ ✐❢ ♦r ✐❢ ✳ ❈❛♥ ❜❡ s❡❡♥ ❛s ❛ s♣❡❝✐❛❧ ✜rst✲❧❡✈❡❧ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❘❡❢♦r♠✉❧❛t✐♦♥✲▲✐♥❡❛r✐③❛t✐♦♥✲❚❡❝❤♥✐q✉❡ ✭❘▲❚✮✳

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SLIDE 9

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥

▲❡♦ ▲✐❜❡rt✐✿ ❈♦♠♣❛❝t ❧✐♥❡❛r✐③❛t✐♦♥ ❢♦r ❜✐♥❛r② q✉❛❞r❛t✐❝ ♣r♦❜❧❡♠s✳ ✹❖❘ ✺✭✸✮✿✷✸✶✕✷✹✺✱ ✷✵✵✼ ▲✐♥❡❛r✐③❛t✐♦♥ ♠❡t❤♦❞ t❤❛t ❡①♣❧♦✐ts t❤❡ s♣❡❝✐❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❛ss✐❣♥♠❡♥t ❝♦♥str❛✐♥ts✳ ❋♦r ❡❛❝❤ s❡t Ak ❜✉✐❧❞ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ s❡t Bk ♦❢ ✈❛r✐❛❜❧❡ ✐♥❞✐❝❡s✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ j ∈ Bk✱ ❛❞❞ t❤❡ ❡q✉❛t✐♦♥ t❤❛t r❡s✉❧ts ✇❤❡♥ ♠✉❧t✐♣❧②✐♥❣ t❤❡ ❛ss✐❣♥♠❡♥t ❝♦♥str❛✐♥ts ✇✳r✳t✳ Ak ✇✐t❤ xj✿ ✶ ❋✐♥❛❧❧②✱ r❡♣❧❛❝❡ ✇✐t❤ ❛ ❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r✐③❛t✐♦♥ ✈❛r✐❛❜❧❡ ✐❢ ♦r ✐❢ ✳ ❈❛♥ ❜❡ s❡❡♥ ❛s ❛ s♣❡❝✐❛❧ ✜rst✲❧❡✈❡❧ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❘❡❢♦r♠✉❧❛t✐♦♥✲▲✐♥❡❛r✐③❛t✐♦♥✲❚❡❝❤♥✐q✉❡ ✭❘▲❚✮✳

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SLIDE 10

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥

▲❡♦ ▲✐❜❡rt✐✿ ❈♦♠♣❛❝t ❧✐♥❡❛r✐③❛t✐♦♥ ❢♦r ❜✐♥❛r② q✉❛❞r❛t✐❝ ♣r♦❜❧❡♠s✳ ✹❖❘ ✺✭✸✮✿✷✸✶✕✷✹✺✱ ✷✵✵✼ ▲✐♥❡❛r✐③❛t✐♦♥ ♠❡t❤♦❞ t❤❛t ❡①♣❧♦✐ts t❤❡ s♣❡❝✐❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❛ss✐❣♥♠❡♥t ❝♦♥str❛✐♥ts✳ ❋♦r ❡❛❝❤ s❡t Ak ❜✉✐❧❞ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ s❡t Bk ♦❢ ✈❛r✐❛❜❧❡ ✐♥❞✐❝❡s✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ j ∈ Bk✱ ❛❞❞ t❤❡ ❡q✉❛t✐♦♥ t❤❛t r❡s✉❧ts ✇❤❡♥ ♠✉❧t✐♣❧②✐♥❣ t❤❡ ❛ss✐❣♥♠❡♥t ❝♦♥str❛✐♥ts ✇✳r✳t✳ Ak ✇✐t❤ xj✿

  • i∈Ak

xi = ✶ ❋✐♥❛❧❧②✱ r❡♣❧❛❝❡ ✇✐t❤ ❛ ❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r✐③❛t✐♦♥ ✈❛r✐❛❜❧❡ ✐❢ ♦r ✐❢ ✳ ❈❛♥ ❜❡ s❡❡♥ ❛s ❛ s♣❡❝✐❛❧ ✜rst✲❧❡✈❡❧ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❘❡❢♦r♠✉❧❛t✐♦♥✲▲✐♥❡❛r✐③❛t✐♦♥✲❚❡❝❤♥✐q✉❡ ✭❘▲❚✮✳

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SLIDE 11

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥

▲❡♦ ▲✐❜❡rt✐✿ ❈♦♠♣❛❝t ❧✐♥❡❛r✐③❛t✐♦♥ ❢♦r ❜✐♥❛r② q✉❛❞r❛t✐❝ ♣r♦❜❧❡♠s✳ ✹❖❘ ✺✭✸✮✿✷✸✶✕✷✹✺✱ ✷✵✵✼ ▲✐♥❡❛r✐③❛t✐♦♥ ♠❡t❤♦❞ t❤❛t ❡①♣❧♦✐ts t❤❡ s♣❡❝✐❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❛ss✐❣♥♠❡♥t ❝♦♥str❛✐♥ts✳ ❋♦r ❡❛❝❤ s❡t Ak ❜✉✐❧❞ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ s❡t Bk ♦❢ ✈❛r✐❛❜❧❡ ✐♥❞✐❝❡s✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ j ∈ Bk✱ ❛❞❞ t❤❡ ❡q✉❛t✐♦♥ t❤❛t r❡s✉❧ts ✇❤❡♥ ♠✉❧t✐♣❧②✐♥❣ t❤❡ ❛ss✐❣♥♠❡♥t ❝♦♥str❛✐♥ts ✇✳r✳t✳ Ak ✇✐t❤ xj✿

  • i∈Ak

xixj = xj ❋✐♥❛❧❧②✱ r❡♣❧❛❝❡ ✇✐t❤ ❛ ❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r✐③❛t✐♦♥ ✈❛r✐❛❜❧❡ ✐❢ ♦r ✐❢ ✳ ❈❛♥ ❜❡ s❡❡♥ ❛s ❛ s♣❡❝✐❛❧ ✜rst✲❧❡✈❡❧ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❘❡❢♦r♠✉❧❛t✐♦♥✲▲✐♥❡❛r✐③❛t✐♦♥✲❚❡❝❤♥✐q✉❡ ✭❘▲❚✮✳

slide-12
SLIDE 12

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥

▲❡♦ ▲✐❜❡rt✐✿ ❈♦♠♣❛❝t ❧✐♥❡❛r✐③❛t✐♦♥ ❢♦r ❜✐♥❛r② q✉❛❞r❛t✐❝ ♣r♦❜❧❡♠s✳ ✹❖❘ ✺✭✸✮✿✷✸✶✕✷✹✺✱ ✷✵✵✼ ▲✐♥❡❛r✐③❛t✐♦♥ ♠❡t❤♦❞ t❤❛t ❡①♣❧♦✐ts t❤❡ s♣❡❝✐❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❛ss✐❣♥♠❡♥t ❝♦♥str❛✐♥ts✳ ❋♦r ❡❛❝❤ s❡t Ak ❜✉✐❧❞ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ s❡t Bk ♦❢ ✈❛r✐❛❜❧❡ ✐♥❞✐❝❡s✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ j ∈ Bk✱ ❛❞❞ t❤❡ ❡q✉❛t✐♦♥ t❤❛t r❡s✉❧ts ✇❤❡♥ ♠✉❧t✐♣❧②✐♥❣ t❤❡ ❛ss✐❣♥♠❡♥t ❝♦♥str❛✐♥ts ✇✳r✳t✳ Ak ✇✐t❤ xj✿

  • i∈Ak

xixj = xj ❋✐♥❛❧❧②✱ r❡♣❧❛❝❡ xixj ✇✐t❤ ❛ ❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r✐③❛t✐♦♥ ✈❛r✐❛❜❧❡ yij ✐❢ i ≤ j ♦r yji ✐❢ j < i✳ ❈❛♥ ❜❡ s❡❡♥ ❛s ❛ s♣❡❝✐❛❧ ✜rst✲❧❡✈❡❧ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❘❡❢♦r♠✉❧❛t✐♦♥✲▲✐♥❡❛r✐③❛t✐♦♥✲❚❡❝❤♥✐q✉❡ ✭❘▲❚✮✳

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SLIDE 13

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥

▲❡♦ ▲✐❜❡rt✐✿ ❈♦♠♣❛❝t ❧✐♥❡❛r✐③❛t✐♦♥ ❢♦r ❜✐♥❛r② q✉❛❞r❛t✐❝ ♣r♦❜❧❡♠s✳ ✹❖❘ ✺✭✸✮✿✷✸✶✕✷✹✺✱ ✷✵✵✼ ▲✐♥❡❛r✐③❛t✐♦♥ ♠❡t❤♦❞ t❤❛t ❡①♣❧♦✐ts t❤❡ s♣❡❝✐❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❛ss✐❣♥♠❡♥t ❝♦♥str❛✐♥ts✳ ❋♦r ❡❛❝❤ s❡t Ak ❜✉✐❧❞ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ s❡t Bk ♦❢ ✈❛r✐❛❜❧❡ ✐♥❞✐❝❡s✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ j ∈ Bk✱ ❛❞❞ t❤❡ ❡q✉❛t✐♦♥ t❤❛t r❡s✉❧ts ✇❤❡♥ ♠✉❧t✐♣❧②✐♥❣ t❤❡ ❛ss✐❣♥♠❡♥t ❝♦♥str❛✐♥ts ✇✳r✳t✳ Ak ✇✐t❤ xj✿

  • i∈Ak

xixj = xj ❋✐♥❛❧❧②✱ r❡♣❧❛❝❡ xixj ✇✐t❤ ❛ ❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r✐③❛t✐♦♥ ✈❛r✐❛❜❧❡ yij ✐❢ i ≤ j ♦r yji ✐❢ j < i✳ ❈❛♥ ❜❡ s❡❡♥ ❛s ❛ s♣❡❝✐❛❧ ✜rst✲❧❡✈❡❧ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❘❡❢♦r♠✉❧❛t✐♦♥✲▲✐♥❡❛r✐③❛t✐♦♥✲❚❡❝❤♥✐q✉❡ ✭❘▲❚✮✳

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SLIDE 14

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥

♠✐♥ cTx + dTy s✳t✳

  • i∈Ak

xi = ✶ ❢♦r ❛❧❧ k ∈ K

  • i∈Ak,i≤j

yij +

  • i∈Ak,j<i

yji = xj ❢♦r ❛❧❧ k ∈ K, j ∈ Bk Cx + Dy ≥ b yij = xixj ❢♦r ❛❧❧ (i, j) ∈ E xi ∈ {✵, ✶} ❢♦r ❛❧❧ i ∈ N ❛♥❞ ♦r ❈r✉❝✐❛❧ ❛s♣❡❝ts ✴ ♦❜❥❡❝t✐✈❡s✿ ❍♦✇ t♦ ❝❤♦♦s❡ t❤❡ s❡ts s✉❝❤ t❤❛t ✭✶✮ ✭❝♦✈❡r❛❣❡✮✱ ✭✷✮ ✱ ✱ ❛♥❞ ✶ ❤♦❧❞ ❢♦r ❛❧❧ ✭❝♦♥s✐st❡♥❝②✮✱ ✭✸✮ ❛♥❞ ❛r❡ ♠✐♥✐♠✉♠ ✭s✐③❡✮✳

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SLIDE 15

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥

♠✐♥ cTx + dTy s✳t✳

  • i∈Ak

xi = ✶ ❢♦r ❛❧❧ k ∈ K

  • i∈Ak,i≤j

yij +

  • i∈Ak,j<i

yji = xj ❢♦r ❛❧❧ k ∈ K, j ∈ Bk Cx + Dy ≥ b yij = xixj ❢♦r ❛❧❧ (i, j) ∈ F xi ∈ {✵, ✶} ❢♦r ❛❧❧ i ∈ N F = {(i, j) | i ≤ j ❛♥❞ ∃k ∈ K : i ∈ Ak, j ∈ Bk ♦r j ∈ Ak, i ∈ Bk} ❈r✉❝✐❛❧ ❛s♣❡❝ts ✴ ♦❜❥❡❝t✐✈❡s✿ ❍♦✇ t♦ ❝❤♦♦s❡ t❤❡ s❡ts s✉❝❤ t❤❛t ✭✶✮ ✭❝♦✈❡r❛❣❡✮✱ ✭✷✮ ✱ ✱ ❛♥❞ ✶ ❤♦❧❞ ❢♦r ❛❧❧ ✭❝♦♥s✐st❡♥❝②✮✱ ✭✸✮ ❛♥❞ ❛r❡ ♠✐♥✐♠✉♠ ✭s✐③❡✮✳

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SLIDE 16

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥

♠✐♥ cTx + dTy s✳t✳

  • i∈Ak

xi = ✶ ❢♦r ❛❧❧ k ∈ K

  • i∈Ak,i≤j

yij +

  • i∈Ak,j<i

yji = xj ❢♦r ❛❧❧ k ∈ K, j ∈ Bk Cx + Dy ≥ b yij = xixj ❢♦r ❛❧❧ (i, j) ∈ F xi ∈ {✵, ✶} ❢♦r ❛❧❧ i ∈ N F = {(i, j) | i ≤ j ❛♥❞ ∃k ∈ K : i ∈ Ak, j ∈ Bk ♦r j ∈ Ak, i ∈ Bk} ❈r✉❝✐❛❧ ❛s♣❡❝ts ✴ ♦❜❥❡❝t✐✈❡s✿ ❍♦✇ t♦ ❝❤♦♦s❡ t❤❡ s❡ts Bk s✉❝❤ t❤❛t ✭✶✮ E ⊆ F ✭❝♦✈❡r❛❣❡✮✱ ✭✷✮ yij ≤ xj✱ yij ≤ xi✱ ❛♥❞ yij ≥ xi + xj − ✶ ❤♦❧❞ ❢♦r ❛❧❧ (i, j) ∈ F ✭❝♦♥s✐st❡♥❝②✮✱ ✭✸✮ |F| ❛♥❞

k∈K |Bk| ❛r❡ ♠✐♥✐♠✉♠ ✭s✐③❡✮✳

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SLIDE 17

❆❝❤✐❡✈✐♥❣ ❈♦♥s✐st❡♥❝② ✲ ◆❡❝❡ss❛r② ❈♦♥❞✐t✐♦♥s

❋♦r ❡❛❝❤ (i, j) ∈ F✱ ✇❡ ♥❡❡❞ t❤❛t✿ ✭✷❛✮ yij ≤ xj ✭✷❜✮ yij ≤ xi ✭✷❝✮ yij ≥ xi +xj −✶ ✭✷❛✮ ❛♥❞ ✭✷❜✮ s✉❣❣❡st t❤❛t t❤❡r❡ ♠✉st ❜❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠✿ ✇❤❡r❡ ✇❤❡r❡ ❚❤✐s tr❛♥s❧❛t❡s ✐♥t♦✿ ❋♦r ❡❛❝❤ ✱ t❤❡r❡ ♠✉st ❜❡ ✭❈✶✮ ❛ s✉❝❤ t❤❛t ❛♥❞ ✱ ❛♥❞ ✭❈✷✮ ❛♥ s✉❝❤ t❤❛t ❛♥❞ ✳ ■s t❤✐s ❛❧s♦ s✉✣❝✐❡♥t❄

slide-18
SLIDE 18

❆❝❤✐❡✈✐♥❣ ❈♦♥s✐st❡♥❝② ✲ ◆❡❝❡ss❛r② ❈♦♥❞✐t✐♦♥s

❋♦r ❡❛❝❤ (i, j) ∈ F✱ ✇❡ ♥❡❡❞ t❤❛t✿ ✭✷❛✮ yij ≤ xj ✭✷❜✮ yij ≤ xi ✭✷❝✮ yij ≥ xi +xj −✶ ✭✷❛✮ ❛♥❞ ✭✷❜✮ s✉❣❣❡st t❤❛t t❤❡r❡ ♠✉st ❜❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠✿

  • a∈Ak,(a,j)∈F

yaj +

  • a∈Ak,(j,a)∈F

yja = xj ✇❤❡r❡ k : i ∈ Ak

  • a∈Aℓ,(a,i)∈F

yai +

  • a∈Aℓ,(i,a)∈F

yia = xi ✇❤❡r❡ l : j ∈ Aℓ ❚❤✐s tr❛♥s❧❛t❡s ✐♥t♦✿ ❋♦r ❡❛❝❤ ✱ t❤❡r❡ ♠✉st ❜❡ ✭❈✶✮ ❛ s✉❝❤ t❤❛t ❛♥❞ ✱ ❛♥❞ ✭❈✷✮ ❛♥ s✉❝❤ t❤❛t ❛♥❞ ✳ ■s t❤✐s ❛❧s♦ s✉✣❝✐❡♥t❄

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SLIDE 19

❆❝❤✐❡✈✐♥❣ ❈♦♥s✐st❡♥❝② ✲ ◆❡❝❡ss❛r② ❈♦♥❞✐t✐♦♥s

❋♦r ❡❛❝❤ (i, j) ∈ F✱ ✇❡ ♥❡❡❞ t❤❛t✿ ✭✷❛✮ yij ≤ xj ✭✷❜✮ yij ≤ xi ✭✷❝✮ yij ≥ xi +xj −✶ ✭✷❛✮ ❛♥❞ ✭✷❜✮ s✉❣❣❡st t❤❛t t❤❡r❡ ♠✉st ❜❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠✿

  • a∈Ak,(a,j)∈F

yaj +

  • a∈Ak,(j,a)∈F

yja ≤ xj ✇❤❡r❡ k : i ∈ Ak

  • a∈Aℓ,(a,i)∈F

yai +

  • a∈Aℓ,(i,a)∈F

yia ≤ xi ✇❤❡r❡ l : j ∈ Aℓ ❚❤✐s tr❛♥s❧❛t❡s ✐♥t♦✿ ❋♦r ❡❛❝❤ ✱ t❤❡r❡ ♠✉st ❜❡ ✭❈✶✮ ❛ s✉❝❤ t❤❛t ❛♥❞ ✱ ❛♥❞ ✭❈✷✮ ❛♥ s✉❝❤ t❤❛t ❛♥❞ ✳ ■s t❤✐s ❛❧s♦ s✉✣❝✐❡♥t❄

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SLIDE 20

❆❝❤✐❡✈✐♥❣ ❈♦♥s✐st❡♥❝② ✲ ◆❡❝❡ss❛r② ❈♦♥❞✐t✐♦♥s

❋♦r ❡❛❝❤ (i, j) ∈ F✱ ✇❡ ♥❡❡❞ t❤❛t✿ ✭✷❛✮ yij ≤ xj ✭✷❜✮ yij ≤ xi ✭✷❝✮ yij ≥ xi +xj −✶ ✭✷❛✮ ❛♥❞ ✭✷❜✮ s✉❣❣❡st t❤❛t t❤❡r❡ ♠✉st ❜❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠✿

  • a∈Ak,(a,j)∈F

yaj +

  • a∈Ak,(j,a)∈F

yja ≥ xj ✇❤❡r❡ k : i ∈ Ak

  • a∈Aℓ,(a,i)∈F

yai +

  • a∈Aℓ,(i,a)∈F

yia ≥ xi ✇❤❡r❡ l : j ∈ Aℓ ❚❤✐s tr❛♥s❧❛t❡s ✐♥t♦✿ ❋♦r ❡❛❝❤ ✱ t❤❡r❡ ♠✉st ❜❡ ✭❈✶✮ ❛ s✉❝❤ t❤❛t ❛♥❞ ✱ ❛♥❞ ✭❈✷✮ ❛♥ s✉❝❤ t❤❛t ❛♥❞ ✳ ■s t❤✐s ❛❧s♦ s✉✣❝✐❡♥t❄

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SLIDE 21

❆❝❤✐❡✈✐♥❣ ❈♦♥s✐st❡♥❝② ✲ ◆❡❝❡ss❛r② ❈♦♥❞✐t✐♦♥s

❋♦r ❡❛❝❤ (i, j) ∈ F✱ ✇❡ ♥❡❡❞ t❤❛t✿ ✭✷❛✮ yij ≤ xj ✭✷❜✮ yij ≤ xi ✭✷❝✮ yij ≥ xi +xj −✶ ✭✷❛✮ ❛♥❞ ✭✷❜✮ s✉❣❣❡st t❤❛t t❤❡r❡ ♠✉st ❜❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠✿

  • a∈Ak,(a,j)∈F

yaj +

  • a∈Ak,(j,a)∈F

yja = xj ✇❤❡r❡ k : i ∈ Ak

  • a∈Aℓ,(a,i)∈F

yai +

  • a∈Aℓ,(i,a)∈F

yia = xi ✇❤❡r❡ l : j ∈ Aℓ ❚❤✐s tr❛♥s❧❛t❡s ✐♥t♦✿ ❋♦r ❡❛❝❤ (i, j) ∈ F✱ t❤❡r❡ ♠✉st ❜❡ ✭❈✶✮ ❛ k ∈ K s✉❝❤ t❤❛t i ∈ Ak ❛♥❞ j ∈ Bk✱ ❛♥❞ ✭❈✷✮ ❛♥ ℓ ∈ K s✉❝❤ t❤❛t j ∈ Aℓ ❛♥❞ i ∈ Bℓ✳ ■s t❤✐s ❛❧s♦ s✉✣❝✐❡♥t❄

slide-22
SLIDE 22

❆❝❤✐❡✈✐♥❣ ❈♦♥s✐st❡♥❝② ✲ ◆❡❝❡ss❛r② ❈♦♥❞✐t✐♦♥s

❋♦r ❡❛❝❤ (i, j) ∈ F✱ ✇❡ ♥❡❡❞ t❤❛t✿ ✭✷❛✮ yij ≤ xj ✭✷❜✮ yij ≤ xi ✭✷❝✮ yij ≥ xi +xj −✶ ✭✷❛✮ ❛♥❞ ✭✷❜✮ s✉❣❣❡st t❤❛t t❤❡r❡ ♠✉st ❜❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠✿

  • a∈Ak,(a,j)∈F

yaj +

  • a∈Ak,(j,a)∈F

yja = xj ✇❤❡r❡ k : i ∈ Ak

  • a∈Aℓ,(a,i)∈F

yai +

  • a∈Aℓ,(i,a)∈F

yia = xi ✇❤❡r❡ l : j ∈ Aℓ ❚❤✐s tr❛♥s❧❛t❡s ✐♥t♦✿ ❋♦r ❡❛❝❤ (i, j) ∈ F✱ t❤❡r❡ ♠✉st ❜❡ ✭❈✶✮ ❛ k ∈ K s✉❝❤ t❤❛t i ∈ Ak ❛♥❞ j ∈ Bk✱ ❛♥❞ ✭❈✷✮ ❛♥ ℓ ∈ K s✉❝❤ t❤❛t j ∈ Aℓ ❛♥❞ i ∈ Bℓ✳ ■s t❤✐s ❛❧s♦ s✉✣❝✐❡♥t❄

slide-23
SLIDE 23

❆❝❤✐❡✈✐♥❣ ❈♦♥s✐st❡♥❝② ✲ ◆❡❝❡ss❛r② ❛♥❞ ❙✉✣❝✐❡♥t ❈♦♥❞✐t✐♦♥s

❚❤❡♦r❡♠

▲❡t (i, j) ∈ F✳ ■❢ ❝♦♥❞✐t✐♦♥s ✭❈✶✮ ❛♥❞ ✭❈✷✮ ❛r❡ s❛t✐s✜❡❞✱ t❤❡♥ ✐t ❤♦❧❞s t❤❛t yij ≤ xi✱ yij ≤ xj ❛♥❞ yij ≥ xi + xj − ✶✳

Pr♦♦❢✳ ❲❡ ❛❧r❡❛❞② ❤❛✈❡ t❤❛t

✵ ✇❤❡♥❡✈❡r ✵ ♦r ✵✳ ■♥ t❤✐s ❝❛s❡✱ ✶ ✐s tr✐✈✐❛❧❧② s❛t✐s✜❡❞✳ ◆♦✇ ❧❡t ✶✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ✶ ✶ ■❢ ✶✱ t❤✐s ✐s ❝♦♥s✐st❡♥t ❛♥❞ ❝♦rr❡❝t✳ ❙♦ s✉♣♣♦s❡ t❤❛t ✶✳ ❚❤❡♥ ✐♥ t❤❡r❡ ✐s s♦♠❡ ✭♦r ✮✱ ✱ ✇✐t❤ ✵ ✭ ✵✮✳ ❚❤✐s ✐♠♣❧✐❡s ✶ ✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts ✶✳ ❛♥❛❧♦❣♦✉s✳

slide-24
SLIDE 24

❆❝❤✐❡✈✐♥❣ ❈♦♥s✐st❡♥❝② ✲ ◆❡❝❡ss❛r② ❛♥❞ ❙✉✣❝✐❡♥t ❈♦♥❞✐t✐♦♥s

❚❤❡♦r❡♠

▲❡t (i, j) ∈ F✳ ■❢ ❝♦♥❞✐t✐♦♥s ✭❈✶✮ ❛♥❞ ✭❈✷✮ ❛r❡ s❛t✐s✜❡❞✱ t❤❡♥ ✐t ❤♦❧❞s t❤❛t yij ≤ xi✱ yij ≤ xj ❛♥❞ yij ≥ xi + xj − ✶✳

Pr♦♦❢✳ ❲❡ ❛❧r❡❛❞② ❤❛✈❡ t❤❛t yij = ✵ ✇❤❡♥❡✈❡r xi = ✵ ♦r xj = ✵✳ ■♥

t❤✐s ❝❛s❡✱ yij ≥ xi + xj − ✶ ✐s tr✐✈✐❛❧❧② s❛t✐s✜❡❞✳ ◆♦✇ ❧❡t ✶✳ ❚❤❡♥ ✇❡ ❤❛✈❡ ✶ ✶ ■❢ ✶✱ t❤✐s ✐s ❝♦♥s✐st❡♥t ❛♥❞ ❝♦rr❡❝t✳ ❙♦ s✉♣♣♦s❡ t❤❛t ✶✳ ❚❤❡♥ ✐♥ t❤❡r❡ ✐s s♦♠❡ ✭♦r ✮✱ ✱ ✇✐t❤ ✵ ✭ ✵✮✳ ❚❤✐s ✐♠♣❧✐❡s ✶ ✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts ✶✳ ❛♥❛❧♦❣♦✉s✳

slide-25
SLIDE 25

❆❝❤✐❡✈✐♥❣ ❈♦♥s✐st❡♥❝② ✲ ◆❡❝❡ss❛r② ❛♥❞ ❙✉✣❝✐❡♥t ❈♦♥❞✐t✐♦♥s

❚❤❡♦r❡♠

▲❡t (i, j) ∈ F✳ ■❢ ❝♦♥❞✐t✐♦♥s ✭❈✶✮ ❛♥❞ ✭❈✷✮ ❛r❡ s❛t✐s✜❡❞✱ t❤❡♥ ✐t ❤♦❧❞s t❤❛t yij ≤ xi✱ yij ≤ xj ❛♥❞ yij ≥ xi + xj − ✶✳

Pr♦♦❢✳ ❲❡ ❛❧r❡❛❞② ❤❛✈❡ t❤❛t yij = ✵ ✇❤❡♥❡✈❡r xi = ✵ ♦r xj = ✵✳ ■♥

t❤✐s ❝❛s❡✱ yij ≥ xi + xj − ✶ ✐s tr✐✈✐❛❧❧② s❛t✐s✜❡❞✳ ◆♦✇ ❧❡t xi = xj = ✶✳ ❚❤❡♥ ✇❡ ❤❛✈❡ yij +

  • a=i∈Ak,(a,j)∈F

yaj +

  • a=i∈Ak,(j,a)∈F

yja = xj = ✶ (∗) yij +

  • a=j∈Aℓ,(a,i)∈F

yai +

  • a=j∈Aℓ,(i,a)∈F

yia = xi = ✶ (∗∗) ■❢ ✶✱ t❤✐s ✐s ❝♦♥s✐st❡♥t ❛♥❞ ❝♦rr❡❝t✳ ❙♦ s✉♣♣♦s❡ t❤❛t ✶✳ ❚❤❡♥ ✐♥ t❤❡r❡ ✐s s♦♠❡ ✭♦r ✮✱ ✱ ✇✐t❤ ✵ ✭ ✵✮✳ ❚❤✐s ✐♠♣❧✐❡s ✶ ✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts ✶✳ ❛♥❛❧♦❣♦✉s✳

slide-26
SLIDE 26

❆❝❤✐❡✈✐♥❣ ❈♦♥s✐st❡♥❝② ✲ ◆❡❝❡ss❛r② ❛♥❞ ❙✉✣❝✐❡♥t ❈♦♥❞✐t✐♦♥s

❚❤❡♦r❡♠

▲❡t (i, j) ∈ F✳ ■❢ ❝♦♥❞✐t✐♦♥s ✭❈✶✮ ❛♥❞ ✭❈✷✮ ❛r❡ s❛t✐s✜❡❞✱ t❤❡♥ ✐t ❤♦❧❞s t❤❛t yij ≤ xi✱ yij ≤ xj ❛♥❞ yij ≥ xi + xj − ✶✳

Pr♦♦❢✳ ❲❡ ❛❧r❡❛❞② ❤❛✈❡ t❤❛t yij = ✵ ✇❤❡♥❡✈❡r xi = ✵ ♦r xj = ✵✳ ■♥

t❤✐s ❝❛s❡✱ yij ≥ xi + xj − ✶ ✐s tr✐✈✐❛❧❧② s❛t✐s✜❡❞✳ ◆♦✇ ❧❡t xi = xj = ✶✳ ❚❤❡♥ ✇❡ ❤❛✈❡ yij +

  • a=i∈Ak,(a,j)∈F

yaj +

  • a=i∈Ak,(j,a)∈F

yja = xj = ✶ (∗) yij +

  • a=j∈Aℓ,(a,i)∈F

yai +

  • a=j∈Aℓ,(i,a)∈F

yia = xi = ✶ (∗∗) ■❢ yij = ✶✱ t❤✐s ✐s ❝♦♥s✐st❡♥t ❛♥❞ ❝♦rr❡❝t✳ ❙♦ s✉♣♣♦s❡ t❤❛t yij < ✶✳ ❚❤❡♥ ✐♥ (∗) t❤❡r❡ ✐s s♦♠❡ yaj ✭♦r yja✮✱ a = i✱ ✇✐t❤ yaj > ✵ ✭yja > ✵✮✳ ❚❤✐s ✐♠♣❧✐❡s xa = ✶ ✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts xi = ✶✳ (∗∗) ❛♥❛❧♦❣♦✉s✳

slide-27
SLIDE 27

❲❤✐❝❤ ✈❛r✐❛❜❧❡s t♦ ❝r❡❛t❡❄

❚❤❡ ❝♦♥❝❡♣t✉❛❧ ❛❧❣♦r✐t❤♠ ✐s ❝❧❡❛r ♥♦✇✿ ❙t❛rt ✇✐t❤ ✶✱

✶

✱ ❛♥❞ ❢♦r ❛❧❧ ✳ r❡♣❡❛t

❆♣♣❡♥❞ s❡ts ❢♦r ✭❈✶✮ ❛♥❞ ✭❈✷✮ t♦ ❜❡ s❛t✐s✜❡❞ ❢♦r ✳ ❖❜t❛✐♥

✶✳

■❢

✶

✱ ❙❚❖P✳ ❖t❤❡r✇✐s❡ s❡t ✶✳

slide-28
SLIDE 28

❲❤✐❝❤ ✈❛r✐❛❜❧❡s t♦ ❝r❡❛t❡❄

❚❤❡ ❝♦♥❝❡♣t✉❛❧ ❛❧❣♦r✐t❤♠ ✐s ❝❧❡❛r ♥♦✇✿

◮ ❙t❛rt ✇✐t❤ i = ✶✱ F✶ = E✱ ❛♥❞ Bk = ∅ ❢♦r ❛❧❧ k ∈ K✳

r❡♣❡❛t

❆♣♣❡♥❞ s❡ts ❢♦r ✭❈✶✮ ❛♥❞ ✭❈✷✮ t♦ ❜❡ s❛t✐s✜❡❞ ❢♦r ✳ ❖❜t❛✐♥

✶✳

■❢

✶

✱ ❙❚❖P✳ ❖t❤❡r✇✐s❡ s❡t ✶✳

slide-29
SLIDE 29

❲❤✐❝❤ ✈❛r✐❛❜❧❡s t♦ ❝r❡❛t❡❄

❚❤❡ ❝♦♥❝❡♣t✉❛❧ ❛❧❣♦r✐t❤♠ ✐s ❝❧❡❛r ♥♦✇✿

◮ ❙t❛rt ✇✐t❤ i = ✶✱ F✶ = E✱ ❛♥❞ Bk = ∅ ❢♦r ❛❧❧ k ∈ K✳ ◮ r❡♣❡❛t

◮ ❆♣♣❡♥❞ s❡ts Bk ❢♦r ✭❈✶✮ ❛♥❞ ✭❈✷✮ t♦ ❜❡ s❛t✐s✜❡❞ ❢♦r Fi✳ ◮ ❖❜t❛✐♥ Fi+✶✳ ◮ ■❢ Fi+✶ = Fi✱ ❙❚❖P✳ ◮ ❖t❤❡r✇✐s❡ s❡t i = i + ✶✳

slide-30
SLIDE 30

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2

slide-31
SLIDE 31

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5}

slide-32
SLIDE 32

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5}

slide-33
SLIDE 33

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} F \ E B1 = {} B2 = {} B3 = {} B1 = {} B2 = {} B3 = {}

slide-34
SLIDE 34

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} B1 = {x4} B2 = {} B3 = {} F \ E

slide-35
SLIDE 35

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} B1 = {x4} B2 = {} B3 = {} F \ E

slide-36
SLIDE 36

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} B1 = {x4} B2 = {} B3 = {x2} F \ E

slide-37
SLIDE 37

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} B1 = {x4} B2 = {} B3 = {x2} F \ E

slide-38
SLIDE 38

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} B1 = {x4} B2 = {x5} B3 = {x2} F \ E

slide-39
SLIDE 39

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} B1 = {x4} B2 = {x5} B3 = {x2, x3} F \ E

slide-40
SLIDE 40

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} B1 = {x4} B2 = {x5} B3 = {x2, x3} F \ E

slide-41
SLIDE 41

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} B1 = {x4} B2 = {x5} B3 = {x2, x3, x1} F \ E

slide-42
SLIDE 42

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} B1 = {x4} B2 = {x5} B3 = {x2, x3, x1} F \ E

slide-43
SLIDE 43

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} F \ E B1 = {x4} B2 = {x5, x4} B3 = {x2, x3, x1}

slide-44
SLIDE 44

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} F \ E B1 = {x4, x5} B2 = {x5, x4} B3 = {x2, x3, x1}

slide-45
SLIDE 45

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5}

slide-46
SLIDE 46

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} F \ E B1 = {} B2 = {} B3 = {} B1 = {} B2 = {} B3 = {}

slide-47
SLIDE 47

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} F \ E B1 = {} B2 = {x4} B3 = {}

slide-48
SLIDE 48

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} F \ E B1 = {} B2 = {x4} B3 = {}

slide-49
SLIDE 49

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} F \ E B1 = {} B2 = {x4} B3 = {x2}

slide-50
SLIDE 50

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} F \ E B1 = {} B2 = {x4} B3 = {x2}

slide-51
SLIDE 51

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} F \ E B1 = {} B2 = {x4, x5} B3 = {x2}

slide-52
SLIDE 52

❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼✐♥✐❡①❛♠♣❧❡

x4 x5 x3 x1 x2 E A1 = {x1, x2} A2 = {x2, x3} A3 = {x4, x5} F \ E B1 = {} B2 = {x4, x5} B3 = {x2, x3}

slide-53
SLIDE 53

❚❤❡ ▼♦st ❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼■P

❱❛r✐❛❜❧❡s✿

◮ fij✿ ✶ ✐❢ (i, j) ∈ F✱ ✵ ♦t❤❡r✇✐s❡✳ ◮ zik✿ ✶ ✐❢ i ∈ Bk✱ ✵ ♦t❤❡r✇✐s❡✳

♠✐♥

✶ ✶

s✳t✳ ✶ ❢♦r ❛❧❧ ❢♦r ❛❧❧ ❢♦r ❛❧❧ ❢♦r ❛❧❧ ✶ ❢♦r ❛❧❧ ✶ ✵ ✶ ❢♦r ❛❧❧ ✶ ✵ ✶ ❢♦r ❛❧❧ ✶

slide-54
SLIDE 54

❚❤❡ ▼♦st ❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼■P

❱❛r✐❛❜❧❡s✿

◮ fij✿ ✶ ✐❢ (i, j) ∈ F✱ ✵ ♦t❤❡r✇✐s❡✳ ◮ zik✿ ✶ ✐❢ i ∈ Bk✱ ✵ ♦t❤❡r✇✐s❡✳

♠✐♥ weqn

✶≤i≤n

  • k∈K

zik

  • + wvar

✶≤i≤n

  • i≤j≤n

fij

  • s✳t✳

✶ ❢♦r ❛❧❧ ❢♦r ❛❧❧ ❢♦r ❛❧❧ ❢♦r ❛❧❧ ✶ ❢♦r ❛❧❧ ✶ ✵ ✶ ❢♦r ❛❧❧ ✶ ✵ ✶ ❢♦r ❛❧❧ ✶

slide-55
SLIDE 55

❚❤❡ ▼♦st ❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼■P

❱❛r✐❛❜❧❡s✿

◮ fij✿ ✶ ✐❢ (i, j) ∈ F✱ ✵ ♦t❤❡r✇✐s❡✳ ◮ zik✿ ✶ ✐❢ i ∈ Bk✱ ✵ ♦t❤❡r✇✐s❡✳

♠✐♥ weqn

✶≤i≤n

  • k∈K

zik

  • + wvar

✶≤i≤n

  • i≤j≤n

fij

  • s✳t✳

fij = ✶ ❢♦r ❛❧❧ (i, j) ∈ E fij ≥ zjk ❢♦r ❛❧❧ k ∈ K, i ∈ Ak, j ∈ N, i ≤ j fji ≥ zjk ❢♦r ❛❧❧ k ∈ K, i ∈ Ak, j ∈ N, j < i

  • k:i∈Ak

zjk ≥ fij ❢♦r ❛❧❧ ✶ ≤ i ≤ j ≤ n

  • k:j∈Ak

zik ≥ fij ❢♦r ❛❧❧ ✶ ≤ i ≤ j ≤ n fij ∈ [✵, ✶] ❢♦r ❛❧❧ ✶ ≤ i ≤ j ≤ n zik ∈ {✵, ✶} ❢♦r ❛❧❧ k ∈ K, ✶ ≤ i ≤ n

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SLIDE 56

❚❤❡ ▼♦st ❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ▼■P

❱❛r✐❛❜❧❡s✿

◮ fij✿ ✶ ✐❢ (i, j) ∈ F✱ ✵ ♦t❤❡r✇✐s❡✳ ◮ zik✿ ✶ ✐❢ i ∈ Bk✱ ✵ ♦t❤❡r✇✐s❡✳

♠✐♥ weqn

✶≤i≤n

  • k∈K

zik

  • + wvar

✶≤i≤n

  • i≤j≤n

fij

  • s✳t✳

fij = ✶ ❢♦r ❛❧❧ (i, j) ∈ E fij ≥ zjk ❢♦r ❛❧❧ k ∈ K, i ∈ Ak, j ∈ N, i ≤ j fji ≥ zjk ❢♦r ❛❧❧ k ∈ K, i ∈ Ak, j ∈ N, j < i

  • k:i∈Ak

zjk ≥ fij ❢♦r ❛❧❧ ✶ ≤ i ≤ j ≤ n

  • k:j∈Ak

zik ≥ fij ❢♦r ❛❧❧ ✶ ≤ i ≤ j ≤ n fij ∈ [✵, ✶] ❢♦r ❛❧❧ ✶ ≤ i ≤ j ≤ n zik ∈ {✵, ✶} ❢♦r ❛❧❧ k ∈ K, ✶ ≤ i ≤ n

slide-57
SLIDE 57

❚❤❡ ▼♦st ❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ❉✐s❥♦✐♥t ❈❛s❡

❚❤❡♦r❡♠

■❢ Ak ∪ Aℓ = ∅✱ ❢♦r ❛❧❧ k, ℓ ∈ K✱ ℓ = k✱ t❤❡♥ t❤❡ ❝♦♥str❛✐♥t ♠❛tr✐① ♦❢ t❤❡ ♠✐①❡❞✲✐♥t❡❣❡r ♣r♦❣r❛♠ ✐s t♦t❛❧❧② ✉♥✐♠♦❞✉❧❛r✳

Pr♦♦❢✳

■♥t❡r♣r❡t t❤❡ ❝♦♥str❛✐♥ts

✵ ❢♦r ❛❧❧ ✵ ❢♦r ❛❧❧ ✵ ❢♦r ❛❧❧ ✶ ✵ ❢♦r ❛❧❧ ✶

❛s t❤❡ r♦✇s ♦❢ ✇❤❡r❡ ✐s t❤❡ ✉♣♣❡r tr✐❛♥❣✉❧❛r ♠❛tr✐① ❞❡✜♥❡❞ ❜② ✶ ❛♥❞ ❢♦r ❛♥❞ ✳ ❊❛❝❤ r♦✇ ❤❛s ❡①❛❝t❧② t✇♦ ♥♦♥③❡r♦ ❡♥tr✐❡s ❛♥❞ ✵✳

slide-58
SLIDE 58

❚❤❡ ▼♦st ❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ❉✐s❥♦✐♥t ❈❛s❡

❚❤❡♦r❡♠

■❢ Ak ∪ Aℓ = ∅✱ ❢♦r ❛❧❧ k, ℓ ∈ K✱ ℓ = k✱ t❤❡♥ t❤❡ ❝♦♥str❛✐♥t ♠❛tr✐① ♦❢ t❤❡ ♠✐①❡❞✲✐♥t❡❣❡r ♣r♦❣r❛♠ ✐s t♦t❛❧❧② ✉♥✐♠♦❞✉❧❛r✳

Pr♦♦❢✳

■♥t❡r♣r❡t t❤❡ ❝♦♥str❛✐♥ts

fij − zjk ≥ ✵ ❢♦r ❛❧❧ k ∈ K, i ∈ Ak, j ∈ N, i ≤ j fji − zjk ≥ ✵ ❢♦r ❛❧❧ k ∈ K, i ∈ Ak, j ∈ N, j < i − fij +

k:i∈Ak zjk

≥ ✵ ❢♦r ❛❧❧ ✶ ≤ i ≤ j ≤ n − fij +

k:j∈Ak zik

≥ ✵ ❢♦r ❛❧❧ ✶ ≤ i ≤ j ≤ n

❛s t❤❡ r♦✇s ♦❢ H = [F Z] ✇❤❡r❡ F ✐s t❤❡ ✉♣♣❡r tr✐❛♥❣✉❧❛r ♠❛tr✐① ❞❡✜♥❡❞ ❜② {fij | ✶ ≤ i ≤ j ≤ n} ❛♥❞ Z = (zik) ❢♦r i ∈ N ❛♥❞ k ∈ K✳ ⇒ ❊❛❝❤ r♦✇ hi· ❤❛s ❡①❛❝t❧② t✇♦ ♥♦♥③❡r♦ ❡♥tr✐❡s ❛♥❞

j hij = ✵✳

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SLIDE 59

❚❤❡ ▼♦st ❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ❈♦♠♣❧❡①✐t②

Minimum-Size Linearization Problem ∈ P no Minimum-Size Linearization Problem NP-hard? yes LP Combinatorial Algorithm running time O(n3) MIP Modified Combinatorial Algorithm as Heuristic Are the sets Ak overlapping?

▲✐st✐♥❣s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❛r❳✐✈✿ ✶✻✶✵✳✵✺✸✼✺ ❬♠❛t❤✳❖❈❪✳

slide-60
SLIDE 60

❚❤❡ ▼♦st ❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ❈♦♠♣❧❡①✐t②

Are the sets Ak overlapping? Minimum-Size Linearization Problem ∈ P no Minimum-Size Linearization Problem NP-hard? yes LP Combinatorial Algorithm running time O(n3) MIP Modified Combinatorial Algorithm as Heuristic

▲✐st✐♥❣s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❛r❳✐✈✿ ✶✻✶✵✳✵✺✸✼✺ ❬♠❛t❤✳❖❈❪✳

slide-61
SLIDE 61

❚❤❡ ▼♦st ❈♦♠♣❛❝t ▲✐♥❡❛r✐③❛t✐♦♥ ✲ ❈♦♠♣❧❡①✐t②

Are the sets Ak overlapping? Minimum-Size Linearization Problem ∈ P no Minimum-Size Linearization Problem NP-hard? yes LP Combinatorial Algorithm running time O(n3) MIP Modified Combinatorial Algorithm as Heuristic

▲✐st✐♥❣s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❛r❳✐✈✿ ✶✻✶✵✳✵✺✸✼✺ ❬♠❛t❤✳❖❈❪✳

slide-62
SLIDE 62

❈♦♥str❛✐♥t✲s✐❞❡ ❈♦♠♣❛❝t♥❡ss

❵❙t❛♥❞❛r❞✬ ▲✐♥❡❛r✐③❛t✐♦♥✿ |E| ✈❛r✐❛❜❧❡s✱ ✸|E| ✐♥❡q✉❛❧✐t✐❡s ❵❈♦♠♣❛❝t✬ ▲✐♥❡❛r✐③❛t✐♦♥✿ |F| ≥ |E| ✈❛r✐❛❜❧❡s✱ ❄ ❡q✉❛t✐♦♥s ❆ s✐♠♣❧❡ ✉♣♣❡r ❜♦✉♥❞ ♦♥ ✐s ❣✐✈❡♥ ❜② ✷ ✳ ❢♦r s✉r❡ ♠♦r❡ ❝♦♥str❛✐♥t✲❝♦♠♣❛❝t ✐❢ ✷✳ ■♥ ♣r❛❝t✐❝❡ t❤✐♥❣s ❛r❡ ❜❡tt❡r ✭❵❛♠♦rt✐③❛t✐♦♥✬✮✿ ❙❛②

✶ ✷

❛♥❞

✶ ✷

✳ P✉tt✐♥❣ ♠❛❦❡s ✶ s❛t✐s✜❡❞ ❢♦r ❛❧❧ ✇✐t❤ ✳ ❚❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ ❞❡♣❡♥❞s ♦♥ t❤❡ s✐③❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ s❡ts ✱ ✳ t❤❡ ♦✈❡r❧❛♣ ♦❢ t❤❡ s❡ts ✱ ✳ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣❛✐rs ♦✈❡r t❤❡ s❡ts ✱ ✳

slide-63
SLIDE 63

❈♦♥str❛✐♥t✲s✐❞❡ ❈♦♠♣❛❝t♥❡ss

❵❙t❛♥❞❛r❞✬ ▲✐♥❡❛r✐③❛t✐♦♥✿ |E| ✈❛r✐❛❜❧❡s✱ ✸|E| ✐♥❡q✉❛❧✐t✐❡s ❵❈♦♠♣❛❝t✬ ▲✐♥❡❛r✐③❛t✐♦♥✿ |F| ≥ |E| ✈❛r✐❛❜❧❡s✱ ❄ ❡q✉❛t✐♦♥s ❆ s✐♠♣❧❡ ✉♣♣❡r ❜♦✉♥❞ ♦♥

k∈K |Bk| ✐s ❣✐✈❡♥ ❜② ✷|F|✳

⇒ ❢♦r s✉r❡ ♠♦r❡ ❝♦♥str❛✐♥t✲❝♦♠♣❛❝t ✐❢ |F| − |E| < |E|/✷✳ ■♥ ♣r❛❝t✐❝❡ t❤✐♥❣s ❛r❡ ❜❡tt❡r ✭❵❛♠♦rt✐③❛t✐♦♥✬✮✿ ❙❛② {(i✶, j), (i✷, j), . . . (ir, j)} ⊂ E ❛♥❞ {i✶, i✷, . . . ir} ∈ Ak✳ P✉tt✐♥❣ j ∈ Bk ♠❛❦❡s (C✶) s❛t✐s✜❡❞ ❢♦r ❛❧❧ (i, j) ✇✐t❤ i ∈ Ak✳ ❚❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ ❞❡♣❡♥❞s ♦♥ t❤❡ s✐③❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ s❡ts ✱ ✳ t❤❡ ♦✈❡r❧❛♣ ♦❢ t❤❡ s❡ts ✱ ✳ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣❛✐rs ♦✈❡r t❤❡ s❡ts ✱ ✳

slide-64
SLIDE 64

❈♦♥str❛✐♥t✲s✐❞❡ ❈♦♠♣❛❝t♥❡ss

❵❙t❛♥❞❛r❞✬ ▲✐♥❡❛r✐③❛t✐♦♥✿ |E| ✈❛r✐❛❜❧❡s✱ ✸|E| ✐♥❡q✉❛❧✐t✐❡s ❵❈♦♠♣❛❝t✬ ▲✐♥❡❛r✐③❛t✐♦♥✿ |F| ≥ |E| ✈❛r✐❛❜❧❡s✱ ❄ ❡q✉❛t✐♦♥s ❆ s✐♠♣❧❡ ✉♣♣❡r ❜♦✉♥❞ ♦♥

k∈K |Bk| ✐s ❣✐✈❡♥ ❜② ✷|F|✳

⇒ ❢♦r s✉r❡ ♠♦r❡ ❝♦♥str❛✐♥t✲❝♦♠♣❛❝t ✐❢ |F| − |E| < |E|/✷✳ ■♥ ♣r❛❝t✐❝❡ t❤✐♥❣s ❛r❡ ❜❡tt❡r ✭❵❛♠♦rt✐③❛t✐♦♥✬✮✿ ❙❛② {(i✶, j), (i✷, j), . . . (ir, j)} ⊂ E ❛♥❞ {i✶, i✷, . . . ir} ∈ Ak✳ P✉tt✐♥❣ j ∈ Bk ♠❛❦❡s (C✶) s❛t✐s✜❡❞ ❢♦r ❛❧❧ (i, j) ✇✐t❤ i ∈ Ak✳ ❚❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ F ❞❡♣❡♥❞s ♦♥

◮ t❤❡ s✐③❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ s❡ts Ak✱ k ∈ K✳ ◮ t❤❡ ♦✈❡r❧❛♣ ♦❢ t❤❡ s❡ts Ak✱ k ∈ K✳ ◮ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♣❛✐rs (i, j) ∈ E ♦✈❡r t❤❡ s❡ts Ak✱ k ∈ K✳