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❈♦♠♣❛❝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 ✶✶✱ ✷✵✶✼

✱ ✇❤❡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 ✳ ❇✐❧✐♥❡❛r ✵✲✶ Pr♦❜❧❡♠s ✇✐t❤ ❆ss✐❣♥♠❡♥t ❈♦♥str❛✐♥ts c T x + d T y ♠✐♥ � s✳t✳ = ✶ ❢♦r ❛❧❧ k ∈ K ✭✶✮ x i i ∈ A k ( ❇◗P❆ ) Cx + Dy ≥ b y ij = x i x j ❢♦r ❛❧❧ ( i , j ) ∈ E ✭✷✮ ∈ { ✵ , ✶ } ❢♦r ❛❧❧ i ∈ N x i

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 ✳ ❇✐❧✐♥❡❛r ✵✲✶ Pr♦❜❧❡♠s ✇✐t❤ ❆ss✐❣♥♠❡♥t ❈♦♥str❛✐♥ts c T x + d T y ♠✐♥ � s✳t✳ = ✶ ❢♦r ❛❧❧ k ∈ K ✭✶✮ x i i ∈ A k ( ❇◗P❆ ) Cx + Dy ≥ b y ij = x i x j ❢♦r ❛❧❧ ( i , j ) ∈ E ✭✷✮ ∈ { ✵ , ✶ } ❢♦r ❛❧❧ i ∈ N x i ◮ x i ✱ i ∈ N ✇❤❡r❡ N = { ✶ , . . . , n } ❢♦r ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n ✳

s✉❝❤ t❤❛t ✳ ❜✐❧✐♥❡❛r t❡r♠s ✱ ✐♥ ♦❜❥❡❝t✐✈❡ ❛♥❞ ❝♦♥str❛✐♥ts✳ ❝♦❧❧❡❝t❡❞ ✐♥ ❛♥ ♦r❞❡r❡❞ s❡t ✱ s✳t✳ ❢♦r ✳ ❇✐❧✐♥❡❛r ✵✲✶ Pr♦❜❧❡♠s ✇✐t❤ ❆ss✐❣♥♠❡♥t ❈♦♥str❛✐♥ts c T x + d T y ♠✐♥ � s✳t✳ = ✶ ❢♦r ❛❧❧ k ∈ K ✭✶✮ x i i ∈ A k ( ❇◗P❆ ) Cx + Dy ≥ b y ij = x i x j ❢♦r ❛❧❧ ( i , j ) ∈ E ✭✷✮ ∈ { ✵ , ✶ } ❢♦r ❛❧❧ i ∈ N x i ◮ x i ✱ i ∈ N ✇❤❡r❡ N = { ✶ , . . . , n } ❢♦r ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n ✳ ◮ s❡t K ♦❢ ✐♥❞❡① s❡ts A k ✱ k ∈ K ✭❵❛ss✐❣♥♠❡♥t ❣r♦✉♣s✬✮

❜✐❧✐♥❡❛r t❡r♠s ✱ ✐♥ ♦❜❥❡❝t✐✈❡ ❛♥❞ ❝♦♥str❛✐♥ts✳ ❝♦❧❧❡❝t❡❞ ✐♥ ❛♥ ♦r❞❡r❡❞ s❡t ✱ s✳t✳ ❢♦r ✳ ❇✐❧✐♥❡❛r ✵✲✶ Pr♦❜❧❡♠s ✇✐t❤ ❆ss✐❣♥♠❡♥t ❈♦♥str❛✐♥ts c T x + d T y ♠✐♥ � s✳t✳ = ✶ ❢♦r ❛❧❧ k ∈ K ✭✶✮ x i i ∈ A k ( ❇◗P❆ ) Cx + Dy ≥ b y ij = x i x j ❢♦r ❛❧❧ ( i , j ) ∈ E ✭✷✮ ∈ { ✵ , ✶ } ❢♦r ❛❧❧ i ∈ N x i ◮ x i ✱ i ∈ N ✇❤❡r❡ N = { ✶ , . . . , n } ❢♦r ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n ✳ ◮ s❡t K ♦❢ ✐♥❞❡① s❡ts A k ✱ k ∈ K ✭❵❛ss✐❣♥♠❡♥t ❣r♦✉♣s✬✮ ◮ s✉❝❤ t❤❛t � k ∈ K A k = N ✳

❝♦❧❧❡❝t❡❞ ✐♥ ❛♥ ♦r❞❡r❡❞ s❡t ✱ s✳t✳ ❢♦r ✳ ❇✐❧✐♥❡❛r ✵✲✶ Pr♦❜❧❡♠s ✇✐t❤ ❆ss✐❣♥♠❡♥t ❈♦♥str❛✐♥ts c T x + d T y ♠✐♥ � s✳t✳ = ✶ ❢♦r ❛❧❧ k ∈ K ✭✶✮ x i i ∈ A k ( ❇◗P❆ ) Cx + Dy ≥ b y ij = x i x j ❢♦r ❛❧❧ ( i , j ) ∈ E ✭✷✮ ∈ { ✵ , ✶ } ❢♦r ❛❧❧ i ∈ N x i ◮ x i ✱ i ∈ N ✇❤❡r❡ N = { ✶ , . . . , n } ❢♦r ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n ✳ ◮ s❡t K ♦❢ ✐♥❞❡① s❡ts A k ✱ k ∈ K ✭❵❛ss✐❣♥♠❡♥t ❣r♦✉♣s✬✮ ◮ s✉❝❤ t❤❛t � k ∈ K A k = N ✳ ◮ ❜✐❧✐♥❡❛r t❡r♠s y ij = x i x j ✱ i , j ∈ N ✐♥ ♦❜❥❡❝t✐✈❡ ❛♥❞ ❝♦♥str❛✐♥ts✳

❇✐❧✐♥❡❛r ✵✲✶ Pr♦❜❧❡♠s ✇✐t❤ ❆ss✐❣♥♠❡♥t ❈♦♥str❛✐♥ts c T x + d T y ♠✐♥ � s✳t✳ = ✶ ❢♦r ❛❧❧ k ∈ K ✭✶✮ x i i ∈ A k ( ❇◗P❆ ) Cx + Dy ≥ b y ij = x i x j ❢♦r ❛❧❧ ( i , j ) ∈ E ✭✷✮ ∈ { ✵ , ✶ } ❢♦r ❛❧❧ i ∈ N x i ◮ x i ✱ i ∈ N ✇❤❡r❡ N = { ✶ , . . . , n } ❢♦r ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n ✳ ◮ s❡t K ♦❢ ✐♥❞❡① s❡ts A k ✱ k ∈ K ✭❵❛ss✐❣♥♠❡♥t ❣r♦✉♣s✬✮ ◮ s✉❝❤ t❤❛t � k ∈ K A k = N ✳ ◮ ❜✐❧✐♥❡❛r t❡r♠s y ij = x i x j ✱ i , j ∈ N ✐♥ ♦❜❥❡❝t✐✈❡ ❛♥❞ ❝♦♥str❛✐♥ts✳ ◮ ❝♦❧❧❡❝t❡❞ ✐♥ ❛♥ ♦r❞❡r❡❞ s❡t E ⊂ N × N ✱ s✳t✳ i ≤ j ❢♦r ( i , j ) ∈ E ✳

❋♦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✉❡ ✭❘▲❚✮✳ ❈♦♠♣❛❝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✳

✶ ❋✐♥❛❧❧②✱ r❡♣❧❛❝❡ ✇✐t❤ ❛ ❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r✐③❛t✐♦♥ ✈❛r✐❛❜❧❡ ✐❢ ♦r ✐❢ ✳ ❈❛♥ ❜❡ s❡❡♥ ❛s ❛ s♣❡❝✐❛❧ ✜rst✲❧❡✈❡❧ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❘❡❢♦r♠✉❧❛t✐♦♥✲▲✐♥❡❛r✐③❛t✐♦♥✲❚❡❝❤♥✐q✉❡ ✭❘▲❚✮✳ ❈♦♠♣❛❝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 A k ❜✉✐❧❞ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ s❡t B k ♦❢ ✈❛r✐❛❜❧❡ ✐♥❞✐❝❡s✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ j ∈ B k ✱ ❛❞❞ t❤❡ ❡q✉❛t✐♦♥ t❤❛t r❡s✉❧ts ✇❤❡♥ ♠✉❧t✐♣❧②✐♥❣ t❤❡ ❛ss✐❣♥♠❡♥t ❝♦♥str❛✐♥ts ✇✳r✳t✳ A k ✇✐t❤ x j ✿

❋✐♥❛❧❧②✱ r❡♣❧❛❝❡ ✇✐t❤ ❛ ❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r✐③❛t✐♦♥ ✈❛r✐❛❜❧❡ ✐❢ ♦r ✐❢ ✳ ❈❛♥ ❜❡ s❡❡♥ ❛s ❛ s♣❡❝✐❛❧ ✜rst✲❧❡✈❡❧ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡ ❘❡❢♦r♠✉❧❛t✐♦♥✲▲✐♥❡❛r✐③❛t✐♦♥✲❚❡❝❤♥✐q✉❡ ✭❘▲❚✮✳ ❈♦♠♣❛❝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 A k ❜✉✐❧❞ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ s❡t B k ♦❢ ✈❛r✐❛❜❧❡ ✐♥❞✐❝❡s✳ ❚❤❡♥✱ ❢♦r ❡❛❝❤ j ∈ B k ✱ ❛❞❞ t❤❡ ❡q✉❛t✐♦♥ t❤❛t r❡s✉❧ts ✇❤❡♥ ♠✉❧t✐♣❧②✐♥❣ t❤❡ ❛ss✐❣♥♠❡♥t ❝♦♥str❛✐♥ts ✇✳r✳t✳ A k ✇✐t❤ x j ✿ � x i = ✶ i ∈ A k