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❈♦♠♣❧❡t❡ ❛①✐♦♠❛t✐③❛t✐♦♥s ♦❢ ❧❡①✐❝♦❣r❛♣❤✐❝ s✉♠s ❛♥❞ ♣r♦❞✉❝ts ♦❢ ♠♦❞❛❧ ❧♦❣✐❝s P❤✐❧✐♣♣❡ ❇❛❧❜✐❛♥✐ ✶ ■❧②❛ ❙❤❛♣✐r♦✈s❦② ✷ ✶ ■♥st✐t✉t ❞❡ ❘❡❝❤❡r❝❤❡ ❡♥ ■♥❢♦r♠❛t✐q✉❡ ❞❡ ❚♦✉❧♦✉s❡✱ ❈◆❘❙ ✖ ❚♦✉❧♦✉s❡ ❯♥✐✈❡rs✐t② ✷ ■♥st✐t✉t❡ ❢♦r ■♥❢♦r♠❛t✐♦♥ ❚r❛♥s♠✐ss✐♦♥ Pr♦❜❧❡♠s✱ ❘✉ss✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ❚♦♣♦❧♦❣②✱ ❆❧❣❡❜r❛✱ ❛♥❞ ❈❛t❡❣♦r✐❡s ✐♥ ▲♦❣✐❝ ✭❚❆❈▲ ✷✵✶✺✮ ■s❝❤✐❛ ✭■t❛❧②✮✱ ❏✉♥❡ ✷✷✱ ✷✵✶✺

▲✐❦❡ ✏✉s✉❛❧✑ ♣r♦❞✉❝t ♦❢ ♠♦❞❛❧ ❧♦❣✐❝s✱ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ s✉♠ ❛♥❞ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ♠♦❞❛❧ ❧♦❣✐❝s ❛r❡ ❞❡✜♥❡❞ s❡♠❛♥t✐❝❛❧❧② ✖ ✈✐❛ ❝♦rr❡s♣♦♥❞✐♥❣ ♦♣❡r❛t✐♦♥ ♦♥ t❤❡✐r ❢r❛♠❡s✳ ❲❡ ❝♦♥s✐❞❡r t✇♦ ♥❛t✉r❛❧ ♦♣❡r❛t✐♦♥s ♦♥ ♠♦❞❛❧ ❧♦❣✐❝s ✖ ❧❡①✐❝♦❣r❛♣❤✐❝ ✭♦r ♦r❞❡r❡❞ ✮ s✉♠s ❛♥❞ ♣r♦❞✉❝ts ✳ ❋♦r s✉❝❤ s②st❡♠s ✇❡ ♣r❡s❡♥t ❣❡♥❡r❛❧ ❝♦♠♣❧❡t❡♥❡ss r❡s✉❧ts✳

❲❡ ❝♦♥s✐❞❡r t✇♦ ♥❛t✉r❛❧ ♦♣❡r❛t✐♦♥s ♦♥ ♠♦❞❛❧ ❧♦❣✐❝s ✖ ❧❡①✐❝♦❣r❛♣❤✐❝ ✭♦r ♦r❞❡r❡❞ ✮ s✉♠s ❛♥❞ ♣r♦❞✉❝ts ✳ ❋♦r s✉❝❤ s②st❡♠s ✇❡ ♣r❡s❡♥t ❣❡♥❡r❛❧ ❝♦♠♣❧❡t❡♥❡ss r❡s✉❧ts✳ ▲✐❦❡ ✏✉s✉❛❧✑ ♣r♦❞✉❝t ♦❢ ♠♦❞❛❧ ❧♦❣✐❝s✱ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ s✉♠ ❛♥❞ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ♠♦❞❛❧ ❧♦❣✐❝s ❛r❡ ❞❡✜♥❡❞ s❡♠❛♥t✐❝❛❧❧② ✖ ✈✐❛ ❝♦rr❡s♣♦♥❞✐♥❣ ♦♣❡r❛t✐♦♥ ♦♥ t❤❡✐r ❢r❛♠❡s✳

❙✉♠ ♦❢ ❢r❛♠❡s ❉❡✜♥✐t✐♦♥ ▲❡t ■ = ( I , S ) ❜❡ ❛ ❢r❛♠❡✱ { ❋ i = ( W i , R i ) | i ∈ I } ❜❡ ❛ ❢❛♠✐❧② ♦❢ ❢r❛♠❡s✳ ❚❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ✭♦r ♦r❞❡r❡❞ ✮ s✉♠ � ❋ i ✐s t❤❡ ❢r❛♠❡ ■ ( W , R + , S + ) , ✇❤❡r❡ W ✐s t❤❡ ❞✐s❥♦✐♥ s✉♠ � W i = { ( w , i ) | i ∈ I , w ∈ W i } , ❛♥❞ I ( w , i ) R + ( u , j ) ⇐ ⇒ i = j & wR i u , ( w , i ) S + ( u , j ) ⇐ ⇒ iSj . ■ ✐s ✏✈❡rt✐❝❛❧✑✱ ❋ i ❛r❡ ✏❤♦r✐③♦♥t❛❧✑✳

❙✉♠ ♦❢ ❧♦❣✐❝s ❉❡✜♥✐t✐♦♥ � L ✶ ✐s t❤❡ ❧♦❣✐❝ ♦❢ s✉♠s ✇❤❡r❡ ✏❤♦r✐③♦♥t❛❧✑ ❢r❛♠❡s ❛r❡ L ✶ ✲❢r❛♠❡s✱ L ✷ ❛♥❞ t❤❡ ✏✈❡rt✐❝❛❧✑ ❢r❛♠❡ ✐s ❛♥ L ✷ ✲❢r❛♠❡✿ � � L ✶ = ▲♦❣ ( { ❋ i | ■ | = L ✷ , { ❋ i | i ✐♥ ■ } | = L ✶ } ) . L ✷ ■ Pr♦❜❧❡♠ ❚♦ ❝♦♥str✉❝t t❤❡ ❛①✐♦♠❛t✐③❛t✐♦♥ ♦❢ � L ✶ ✱ ❦♥♦✇✐♥❣ t❤❡ ❧♦❣✐❝s L ✶ , L ✷ ✳ L ✷

❬▲✳ ❇❡❦❧❡♠✐s❤❡✈✳ ❑r✐♣❦❡ s❡♠❛♥t✐❝s ❢♦r ♣r♦✈❛❜✐❧✐t② ❧♦❣✐❝ ✳ ✷✵✶✵❪ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❞❡❝✐❞❛❜✐❧✐t② ❛♥❞ ❝♦♠♣❧❡①✐t②✱ t❤❡ s✉♠ ♦♣❡r❛t✐♦♥ t✉r♥s ♦✉t t♦ ❜❡ ❛ ❣♦♦❞ ♦♣❡r❛t✐♦♥✦ ■♥ ♠❛♥② ❝❛s❡s t❤❡ s✉♠ ♦♣❡r❛t✐♦♥ ♣r❡s❡r✈❡s ❝♦♠♣❧❡①✐t② ♦❢ ❧♦❣✐❝s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛❧❧ t❤❡ ❛❜♦✈❡ ❧♦❣✐❝s ❛r❡ ✐♥ ✭❬❙❤✱ ✷✵✵✽❪✮❀ ✐t ❢♦❧❧♦✇s t❤❛t ✐s ✐♥ ✳ ❙✐♠✉❧t❛♥❡♦✉s❧②✱ ❙❡r❣❡② ❇❛❜❡♥②s❤❡✈ ❛♥❞ ❱❧❛❞✐♠✐r ❘②❜❛❦♦✈ ❞❡✈❡❧♦♣❡❞ ✜❧tr❛t✐♦♥s ❢♦r s✉♠s✱ ❛♥❞ ♣r♦✈❡❞ ❛ ♥✉♠❜❡r ♦❢ ❞❡❝✐❞❛❜✐❧✐t② r❡s✉❧ts✳ ❬❇❛❜❡♥②s❤❡✈✱ ❘②❜❛❦♦✈✳ ▲♦❣✐❝s ♦❢ ❑r✐♣❦❡ ♠❡t❛✲♠♦❞❡❧s✳ ✷✵✶✵❪ ❙♦♠❡ ❤✐st♦r② ■♥ ✷✵✵✼✱ ▲❡✈ ❇❡❦❧❡♠✐s❤❡✈ ❝♦♥str✉❝t❡❞ t❤❡ ❛①✐♦♠❛t✐③❛t✐♦♥ ♦❢ � GL GL

■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❞❡❝✐❞❛❜✐❧✐t② ❛♥❞ ❝♦♠♣❧❡①✐t②✱ t❤❡ s✉♠ ♦♣❡r❛t✐♦♥ t✉r♥s ♦✉t t♦ ❜❡ ❛ ❣♦♦❞ ♦♣❡r❛t✐♦♥✦ ■♥ ♠❛♥② ❝❛s❡s t❤❡ s✉♠ ♦♣❡r❛t✐♦♥ ♣r❡s❡r✈❡s ❝♦♠♣❧❡①✐t② ♦❢ ❧♦❣✐❝s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛❧❧ t❤❡ ❛❜♦✈❡ ❧♦❣✐❝s ❛r❡ ✐♥ ✭❬❙❤✱ ✷✵✵✽❪✮❀ ✐t ❢♦❧❧♦✇s t❤❛t ✐s ✐♥ ✳ ❙✐♠✉❧t❛♥❡♦✉s❧②✱ ❙❡r❣❡② ❇❛❜❡♥②s❤❡✈ ❛♥❞ ❱❧❛❞✐♠✐r ❘②❜❛❦♦✈ ❞❡✈❡❧♦♣❡❞ ✜❧tr❛t✐♦♥s ❢♦r s✉♠s✱ ❛♥❞ ♣r♦✈❡❞ ❛ ♥✉♠❜❡r ♦❢ ❞❡❝✐❞❛❜✐❧✐t② r❡s✉❧ts✳ ❬❇❛❜❡♥②s❤❡✈✱ ❘②❜❛❦♦✈✳ ▲♦❣✐❝s ♦❢ ❑r✐♣❦❡ ♠❡t❛✲♠♦❞❡❧s✳ ✷✵✶✵❪ ❙♦♠❡ ❤✐st♦r② ■♥ ✷✵✵✼✱ ▲❡✈ ❇❡❦❧❡♠✐s❤❡✈ ❝♦♥str✉❝t❡❞ t❤❡ ❛①✐♦♠❛t✐③❛t✐♦♥ ♦❢ �� � �� �� �� � � � GL , GL , GL , . . . GL GL GL GL GL GL ❬▲✳ ❇❡❦❧❡♠✐s❤❡✈✳ ❑r✐♣❦❡ s❡♠❛♥t✐❝s ❢♦r ♣r♦✈❛❜✐❧✐t② ❧♦❣✐❝ GLP ✳ ✷✵✶✵❪

❙♦♠❡ ❤✐st♦r② ■♥ ✷✵✵✼✱ ▲❡✈ ❇❡❦❧❡♠✐s❤❡✈ ❝♦♥str✉❝t❡❞ t❤❡ ❛①✐♦♠❛t✐③❛t✐♦♥ ♦❢ �� � �� �� �� � � � GL , GL , GL , . . . GL GL GL GL GL GL ❬▲✳ ❇❡❦❧❡♠✐s❤❡✈✳ ❑r✐♣❦❡ s❡♠❛♥t✐❝s ❢♦r ♣r♦✈❛❜✐❧✐t② ❧♦❣✐❝ GLP ✳ ✷✵✶✵❪ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❞❡❝✐❞❛❜✐❧✐t② ❛♥❞ ❝♦♠♣❧❡①✐t②✱ t❤❡ s✉♠ ♦♣❡r❛t✐♦♥ t✉r♥s ♦✉t t♦ ❜❡ ❛ ❣♦♦❞ ♦♣❡r❛t✐♦♥✦ ■♥ ♠❛♥② ❝❛s❡s t❤❡ s✉♠ ♦♣❡r❛t✐♦♥ ♣r❡s❡r✈❡s ❝♦♠♣❧❡①✐t② ♦❢ ❧♦❣✐❝s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛❧❧ t❤❡ ❛❜♦✈❡ ❧♦❣✐❝s ❛r❡ ✐♥ PSPACE ✭❬❙❤✱ ✷✵✵✽❪✮❀ ✐t ❢♦❧❧♦✇s t❤❛t GLP ✐s ✐♥ PSPACE ✳ ❙✐♠✉❧t❛♥❡♦✉s❧②✱ ❙❡r❣❡② ❇❛❜❡♥②s❤❡✈ ❛♥❞ ❱❧❛❞✐♠✐r ❘②❜❛❦♦✈ ❞❡✈❡❧♦♣❡❞ ✜❧tr❛t✐♦♥s ❢♦r s✉♠s✱ ❛♥❞ ♣r♦✈❡❞ ❛ ♥✉♠❜❡r ♦❢ ❞❡❝✐❞❛❜✐❧✐t② r❡s✉❧ts✳ ❬❇❛❜❡♥②s❤❡✈✱ ❘②❜❛❦♦✈✳ ▲♦❣✐❝s ♦❢ ❑r✐♣❦❡ ♠❡t❛✲♠♦❞❡❧s✳ ✷✵✶✵❪

❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❢♦r♠✉❧❛s ❄ ❛r❡ ❙❛❤❧q✈✐st ❢♦r♠✉❧❛s✳ ❋♦r ❋ ✷ ✱ ✇❡ ❤❛✈❡✿ ✶ ❋ ✶ ✷ ✷ ❋ ✷ ✶ ✷ ✶ ❋ ✷ ✳ ✷ ✶ ▲❡♠♠❛ ✭✷✵✶✹✮ ❈♦♥s✐❞❡r ❛ r♦♦t❡❞ ❢r❛♠❡ ❋ ✷ ✳ ✶ ❋ ✐✛ ❋ ✐s ❛ ✲♠♦r♣❤✐❝ ✐♠❛❣❡ ♦❢ ❛ s✉♠✳ ❈♦r♦❧❧❛r② α = � ✷ p → � ✶ � ✷ p , β = � ✷ p → � ✷ � ✶ p , γ = ♦ ✷ p → � ✶ ♦ ✷ p � GL = GL ∗ GL + { α, β, γ } GL

❛r❡ ❙❛❤❧q✈✐st ❢♦r♠✉❧❛s✳ ❋♦r ❋ ✷ ✱ ✇❡ ❤❛✈❡✿ ✶ ❋ ✶ ✷ ✷ ❋ ✷ ✶ ✷ ✶ ❋ ✷ ✳ ✷ ✶ ▲❡♠♠❛ ✭✷✵✶✹✮ ❈♦♥s✐❞❡r ❛ r♦♦t❡❞ ❢r❛♠❡ ❋ ✷ ✳ ✶ ❋ ✐✛ ❋ ✐s ❛ ✲♠♦r♣❤✐❝ ✐♠❛❣❡ ♦❢ ❛ s✉♠✳ ❈♦r♦❧❧❛r② α = � ✷ p → � ✶ � ✷ p , β = � ✷ p → � ✷ � ✶ p , γ = ♦ ✷ p → � ✶ ♦ ✷ p � GL = GL ∗ GL + { α, β, γ } GL ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❢♦r♠✉❧❛s α, β, γ ❄