SLIDE 1
s rt - - PowerPoint PPT Presentation
s rt - - PowerPoint PPT Presentation
s rt rts t t rr rts t rr r
SLIDE 2
SLIDE 3
▼♦t✐✈❛t✐♦♥
▸ ❚❤✐s t❛❧❦✿ st✉❞② ♦❢ ♠❛♥✐❢♦❧❞s ✉♣ t♦ ✭P▲✲✮❤♦♠❡♦♠♦r♣❤✐s♠
♠❛♥✐❢♦❧❞s ↔ tr✐❛♥❣✉❧❛t❡❞ ♠❛♥✐❢♦❧❞s ✭s✐♠♣❧✐❝✐❛❧❄✮
▸ ▸ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛r❣✉♠❡♥ts ✴ ❞✐s❝r❡t❡ ♠❡t❤♦❞s ♣r♦✈❡ ❣❡♦♠❡tr✐❝
❛♥❞ t♦♣♦❧♦❣✐❝❛❧ ♣r♦❜❧❡♠s
▸ ❋✉♥❞❛♠❡♥t❛❧ t❛s❦✿ ❞✐st✐♥❣✉✐s❤✐♥❣ ❜❡t✇❡❡♥ ♠❛♥✐❢♦❧❞s✱ ✐✳❡✳✱
❣✐✈❡♥ tr✐❛♥❣✉❧❛t✐♦♥s ▼ ❛♥❞ ◆✱ ✐s ▼ / ≅ ◆❄
SLIDE 4
▼♦t✐✈❛t✐♦♥
▸ ❚❤✐s t❛❧❦✿ st✉❞② ♦❢ ♠❛♥✐❢♦❧❞s ✉♣ t♦ ✭P▲✲✮❤♦♠❡♦♠♦r♣❤✐s♠
♠❛♥✐❢♦❧❞s ↔ tr✐❛♥❣✉❧❛t❡❞ ♠❛♥✐❢♦❧❞s ✭s✐♠♣❧✐❝✐❛❧❄✮
▸ ▸ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛r❣✉♠❡♥ts ✴ ❞✐s❝r❡t❡ ♠❡t❤♦❞s ♣r♦✈❡ ❣❡♦♠❡tr✐❝
❛♥❞ t♦♣♦❧♦❣✐❝❛❧ ♣r♦❜❧❡♠s
▸ ❋✉♥❞❛♠❡♥t❛❧ t❛s❦✿ ❞✐st✐♥❣✉✐s❤✐♥❣ ❜❡t✇❡❡♥ ♠❛♥✐❢♦❧❞s✱ ✐✳❡✳✱
❣✐✈❡♥ tr✐❛♥❣✉❧❛t✐♦♥s ▼ ❛♥❞ ◆✱ ✐s ▼ / ≅ ◆❄
SLIDE 5
▼♦t✐✈❛t✐♦♥
▸ ❈❛♥ ✇❡ ❞✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ ♠❛♥✐❢♦❧❞s❄
▸ ❉✐♠❡♥s✐♦♥
✶✿ ✓
▸ ❉✐♠❡♥s✐♦♥
✷✿ ✓
▸ ❉✐♠❡♥s✐♦♥
✸✿ ❨❡s ✐♥ t❤❡♦r②✳ ◆♦ ✐♥ ❣❡♥❡r❛❧ ✐♥ ♣r❛❝t✐❝❡✳
▸ ❉✐♠❡♥s✐♦♥ ≥ ✹✿ ◆♦✳
▸ ■✳❡✳✱ ✐ts tr✐✈✐❛❧✱ ❡①tr❡♠❡❧② ❞✐✣❝✉❧t✱ ♦r ✐♠♣♦ss✐❜❧❡ t♦ ❞✐st✐♥❣✉✐s❤
❜❡t✇❡❡♥ ♠❛♥✐❢♦❧❞s✳
▸ P❛rt✐❛❧ s♦❧✉t✐♦♥✿ t♦♣♦❧♦❣✐❝❛❧ ✐♥✈❛r✐❛♥ts✱ ♣r♦♣❡rt✐❡s ♦❢ ❛
♠❛♥✐❢♦❧❞ ✇❤✐❝❤ ❞♦ ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ❝♦♥t✐♥✉♦✉s ❞❡❢♦r♠❛t✐♦♥
▸ ❚✉r❛❡✈✲❱✐r♦ ✐♥✈❛r✐❛♥ts✿ ♣❛rt✐❝✉❧❛r❧② ♣♦✇❡r❢✉❧ ❢❛♠✐❧② ♦❢
t♦♣♦❧♦❣✐❝❛❧ ✐♥✈❛r✐❛♥ts ❢♦r ✸✲♠❛♥✐❢♦❧❞s✶ ✷
▸ ▼❡t❤♦❞ ♦❢ ❝❤♦✐❝❡ ✇❤❡♥✱ ❢♦r ❡①❛♠♣❧❡✱ ❡♥✉♠❡r❛t✐♥❣ ✸✲♠❛♥✐❢♦❧❞s
✶▼❛t✈❡❡✈✱ ❆❧❣♦r✐t❤♠✐❝ ❚♦♣♦❧♦❣② ❛♥❞ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✸✲♠❛♥✐❢♦❧❞s✱ ✷✵✵✸ ✷❑❛✉✛♠❛♥♥ ❛♥❞ ▲✐♥s✱ ❈♦♠♣✉t✐♥❣ ❚✉r❛❡✈✲❱✐r♦ ✐♥✈✳ ❢♦r ✸✲♠❛♥✐❢♦❧❞s✱ ✶✾✾✶
SLIDE 6
❚✉r❛❡✈✲❱✐r♦ ✐♥✈❛r✐❛♥ts
❚❤❡ ❚✉r❛❡✈✲❱✐r♦ ✐♥✈❛r✐❛♥t ✇✐t❤ ♣❛r❛♠❡t❡rs r ❛♥❞ q ✐s ❛ ❢✉♥❝t✐♦♥ ❚❱r,q ∶ M → Q[ζ] ∩ R ✇❤❡r❡
▸ M ❂ s❡t ♦❢ tr✐❛♥❣✉❧❛t❡❞ ✸✲♠❛♥✐❢♦❧❞s ✭❝♦♥♥❡❝t❡❞✱ ❝❧♦s❡❞✮ ▸ ζ = ❡✐πq/r;
r,q ∈ Z ❝♦✲♣r✐♠❡❀ r ≥ ✸; ✵ < q < ✷r
▸ ❈❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✈✐❛ ♣✉r❡❧② ❝♦♠❜✐♥❛t♦r✐❛❧ ❢♦r♠✉❧❛❡✳
SLIDE 7
❚✉r❛❡✈✲❱✐r♦ ✐♥✈❛r✐❛♥ts ✕ st❛t❡✲s✉♠ ♠♦❞❡❧
▸ ▼ ∈ M tr✐❛♥❣✉❧❛t❡❞ ✸✲♠❛♥✐❢♦❧❞ ▸ ❱ ✱ ❊✱ ❋✱ ❚ ✐ts s❡t ♦❢ ✈❡rt✐❝❡s✱ ❡❞❣❡s✱ tr✐❛♥❣❧❡s✱ ❛♥❞ t❡tr❛❤❡❞r❛ ▸ ϕ ∶ ❊ → {✵,✶,...,r − ✷} ❡❞❣❡ ❝♦❧♦✉r✐♥❣ s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣
❝♦♥❞✐t✐♦♥s ❛t ❛❧❧ tr✐❛♥❣❧❡s t ♦❢ ▼✿
▸ e1 e2 e3
t
ϕ(❡✐) + ϕ(❡❥) ≥ ϕ(❡❦) ∀✐ ≠ ❥ ≠ ❦ ≠ ✐ ∑ϕ(❡✐) ≡ ✵ ♠♦❞ ✷ ❛♥❞ ≤ ✷r − ✹
▸ ❈❛❧❧ t❤❡ s❡t ♦❢ s✉❝❤ ❛❞♠✐ss✐❜❧❡ ❝♦❧♦✉r✐♥❣s ❆❞♠(▼,r) ▸ ❋♦r ❡❛❝❤ ϕ ∈ ❆❞♠(▼,r)✱ ❡❞❣❡ ❡ ∈ ❊✱ tr✐❛♥❣❧❡ t ∈ ❋✱ ❛♥❞
t❡tr❛❤❡❞r♦♥ ∆ ∈ ❚ ✇❡ ❞❡✜♥❡ ✇❡✐❣❤ts ∣❡∣ϕ✱ ∣t∣ϕ✱ ❛♥❞ ∣∆∣ϕ ✐♥ Q[ζ] ♦♥❧② ❞❡♣❡♥❞✐♥❣ ♦♥ ϕ ✭❛♥❞ r ❛♥❞ q✮
▸ ❚❱r,q(▼)
=
∑
ϕ∈❆❞♠(▼,r)
( ∏
❡∈❊
∣❡∣ϕ ⋅ ∏
t∈❋
∣t∣ϕ ⋅ ∏
∆∈❚
∣∆∣ϕ)
SLIDE 8
❚✉r❛❡✈✲❱✐r♦ ✐♥✈❛r✐❛♥ts ✕ st❛t❡✲s✉♠ ♠♦❞❡❧
▸ ▼ ∈ M tr✐❛♥❣✉❧❛t❡❞ ✸✲♠❛♥✐❢♦❧❞ ▸ ❱ ✱ ❊✱ ❋✱ ❚ ✐ts s❡t ♦❢ ✈❡rt✐❝❡s✱ ❡❞❣❡s✱ tr✐❛♥❣❧❡s✱ ❛♥❞ t❡tr❛❤❡❞r❛ ▸ ϕ ∶ ❊ → {✵,✶,...,r − ✷} ❡❞❣❡ ❝♦❧♦✉r✐♥❣ s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣
❝♦♥❞✐t✐♦♥s ❛t ❛❧❧ tr✐❛♥❣❧❡s t ♦❢ ▼✿
▸ e1 e2 e3
t
ϕ(❡✐) + ϕ(❡❥) ≥ ϕ(❡❦) ∀✐ ≠ ❥ ≠ ❦ ≠ ✐ ∑ϕ(❡✐) ≡ ✵ ♠♦❞ ✷ ❛♥❞ ≤ ✷r − ✹
▸ ❈❛❧❧ t❤❡ s❡t ♦❢ s✉❝❤ ❛❞♠✐ss✐❜❧❡ ❝♦❧♦✉r✐♥❣s ❆❞♠(▼,r) ▸ ❋♦r ❡❛❝❤ ϕ ∈ ❆❞♠(▼,r)✱ ❡❞❣❡ ❡ ∈ ❊✱ tr✐❛♥❣❧❡ t ∈ ❋✱ ❛♥❞
t❡tr❛❤❡❞r♦♥ ∆ ∈ ❚ ✇❡ ❞❡✜♥❡ ✇❡✐❣❤ts ∣❡∣ϕ✱ ∣t∣ϕ✱ ❛♥❞ ∣∆∣ϕ ✐♥ Q[ζ] ♦♥❧② ❞❡♣❡♥❞✐♥❣ ♦♥ ϕ ✭❛♥❞ r ❛♥❞ q✮
▸ ❚❱r,q(▼)
=
∑
ϕ∈❆❞♠(▼,r)
( ∏
❡∈❊
∣❡∣ϕ ⋅ ∏
t∈❋
∣t∣ϕ ⋅ ∏
∆∈❚
∣∆∣ϕ)
SLIDE 9
❚✉r❛❡✈✲❱✐r♦ ✐♥✈❛r✐❛♥ts ✕ st❛t❡✲s✉♠ ♠♦❞❡❧
▸ ▼ ∈ M tr✐❛♥❣✉❧❛t❡❞ ✸✲♠❛♥✐❢♦❧❞ ▸ ❱ ✱ ❊✱ ❋✱ ❚ ✐ts s❡t ♦❢ ✈❡rt✐❝❡s✱ ❡❞❣❡s✱ tr✐❛♥❣❧❡s✱ ❛♥❞ t❡tr❛❤❡❞r❛ ▸ ϕ ∶ ❊ → {✵,✶,...,r − ✷} ❡❞❣❡ ❝♦❧♦✉r✐♥❣ s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣
❝♦♥❞✐t✐♦♥s ❛t ❛❧❧ tr✐❛♥❣❧❡s t ♦❢ ▼✿
▸ e1 e2 e3
t
ϕ(❡✐) + ϕ(❡❥) ≥ ϕ(❡❦) ∀✐ ≠ ❥ ≠ ❦ ≠ ✐ ∑ϕ(❡✐) ≡ ✵ ♠♦❞ ✷ ❛♥❞ ≤ ✷r − ✹
▸ ❈❛❧❧ t❤❡ s❡t ♦❢ s✉❝❤ ❛❞♠✐ss✐❜❧❡ ❝♦❧♦✉r✐♥❣s ❆❞♠(▼,r) ▸ ❋♦r ❡❛❝❤ ϕ ∈ ❆❞♠(▼,r)✱ ❡❞❣❡ ❡ ∈ ❊✱ tr✐❛♥❣❧❡ t ∈ ❋✱ ❛♥❞
t❡tr❛❤❡❞r♦♥ ∆ ∈ ❚ ✇❡ ❞❡✜♥❡ ✇❡✐❣❤ts ∣❡∣ϕ✱ ∣t∣ϕ✱ ❛♥❞ ∣∆∣ϕ ✐♥ Q[ζ] ♦♥❧② ❞❡♣❡♥❞✐♥❣ ♦♥ ϕ ✭❛♥❞ r ❛♥❞ q✮
▸ ❚❱r,q(▼)
=
∑
ϕ∈❆❞♠(▼,r)
( ∏
❡∈❊
∣❡∣ϕ ⋅ ∏
t∈❋
∣t∣ϕ ⋅ ∏
∆∈❚
∣∆∣ϕ)
SLIDE 10
❚✉r❛❡✈✲❱✐r♦ ✐♥✈❛r✐❛♥ts ✕ st❛t❡✲s✉♠ ♠♦❞❡❧
▸ ▼ ∈ M tr✐❛♥❣✉❧❛t❡❞ ✸✲♠❛♥✐❢♦❧❞ ▸ ❱ ✱ ❊✱ ❋✱ ❚ ✐ts s❡t ♦❢ ✈❡rt✐❝❡s✱ ❡❞❣❡s✱ tr✐❛♥❣❧❡s✱ ❛♥❞ t❡tr❛❤❡❞r❛ ▸ ϕ ∶ ❊ → {✵,✶,...,r − ✷} ❡❞❣❡ ❝♦❧♦✉r✐♥❣ s❛t✐s❢②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣
❝♦♥❞✐t✐♦♥s ❛t ❛❧❧ tr✐❛♥❣❧❡s t ♦❢ ▼✿
▸ e1 e2 e3
t
ϕ(❡✐) + ϕ(❡❥) ≥ ϕ(❡❦) ∀✐ ≠ ❥ ≠ ❦ ≠ ✐ ∑ϕ(❡✐) ≡ ✵ ♠♦❞ ✷ ❛♥❞ ≤ ✷r − ✹
▸ ❈❛❧❧ t❤❡ s❡t ♦❢ s✉❝❤ ❛❞♠✐ss✐❜❧❡ ❝♦❧♦✉r✐♥❣s ❆❞♠(▼,r) ▸ ❋♦r ❡❛❝❤ ϕ ∈ ❆❞♠(▼,r)✱ ❡❞❣❡ ❡ ∈ ❊✱ tr✐❛♥❣❧❡ t ∈ ❋✱ ❛♥❞
t❡tr❛❤❡❞r♦♥ ∆ ∈ ❚ ✇❡ ❞❡✜♥❡ ✇❡✐❣❤ts ∣❡∣ϕ✱ ∣t∣ϕ✱ ❛♥❞ ∣∆∣ϕ ✐♥ Q[ζ] ♦♥❧② ❞❡♣❡♥❞✐♥❣ ♦♥ ϕ ✭❛♥❞ r ❛♥❞ q✮
▸ ❚❱r,q(▼)
=
∑
ϕ∈❆❞♠(▼,r)
( ∏
❡∈❊
∣❡∣ϕ ⋅ ∏
t∈❋
∣t∣ϕ ⋅ ∏
∆∈❚
∣∆∣ϕ)
SLIDE 11
❆♥ ❛❧t❡r♥❛t✐✈❡ ✈✐❡✇ ♦♥ ❛❞♠✐ss✐❜❧❡ ❝♦❧♦✉r✐♥❣s
r = ✸ ✭❝♦❧♦✉rs ✵✱ ✶✮ ∈ P✿
1 1
r = ✹ ✭❝♦❧♦✉rs ✵✱ ✶✱ ✷✮ ∈ ★P✲❤❛r❞✸✿
✸❑✐r❜②✱ ▼❡❧✈✐♥✱ ▲♦❝❛❧ s✉r❣❡r② ❢♦r♠✉❧❛s ❢♦r q✉❛♥t✉♠ ✐♥✈❛r✐❛♥ts ❛♥❞ t❤❡ ❆r❢
✐♥✈❛r✐❛♥t✱ ✷✵✵✹✳
SLIDE 12
❆♥ ❛❧t❡r♥❛t✐✈❡ ✈✐❡✇ ♦♥ ❛❞♠✐ss✐❜❧❡ ❝♦❧♦✉r✐♥❣s
r = ✸ ✭❝♦❧♦✉rs ✵✱ ✶✮ ∈ P✿
1 1
r = ✹ ✭❝♦❧♦✉rs ✵✱ ✶✱ ✷✮ ∈ ★P✲❤❛r❞✸✿
1 1 1 1 2 2 2
✸❑✐r❜②✱ ▼❡❧✈✐♥✱ ▲♦❝❛❧ s✉r❣❡r② ❢♦r♠✉❧❛s ❢♦r q✉❛♥t✉♠ ✐♥✈❛r✐❛♥ts ❛♥❞ t❤❡ ❆r❢
✐♥✈❛r✐❛♥t✱ ✷✵✵✹✳
SLIDE 13
❆❧❣♦r✐t❤♠ ■✿ tr❡❡✇✐❞t❤
▸ ❚❤❡ tr❡❡✇✐❞t❤ ♦❢ ❛ ❣r❛♣❤ ♠❡❛s✉r❡s ❤♦✇ ✏tr❡❡❧✐❦❡✑ ❛ ❣r❛♣❤ ✐s
✭tr❡❡s ❤❛✈❡ tr❡❡✇✐❞t❤ ✶✮
▸ ❚❤❡ tr❡❡✇✐❞t❤ ♦❢ ❛ tr✐❛♥❣✉❧❛t❡❞ ♠❛♥✐❢♦❧❞ ▼ ✐s t❤❡ tr❡❡✇✐❞t❤ ♦❢
✐ts ❞✉❛❧ ❣r❛♣❤
▸ ▲♦✇ tr❡❡✇✐❞t❤ ⇒ ❝❛♥ ❛rr❛♥❣❡ t❡tr❛❤❡❞r❛ ♦❢ ▼ ✐♥ ❛ tr❡❡ ✇✐t❤
❢❡✇ t❡tr❛❤❡❞r❛ ❣r♦✉♣❡❞ t♦❣❡t❤❡r ♣❡r ♥♦❞❡ ♦❢ t❤❡ tr❡❡ ✭⇒ t❤✐♥ tr❡❡ ❞❡❝♦♠♣♦s✐t✐♦♥✮
1 2 3 6 7 5 4 8 9
1, 2, 4 2, 3, 4 3, 4, 5 3, 5, 6 6, 7 2, 3, 8 8, 9 leaf nodes
▸ ❙✉✐t❛❜❧❡ ❢♦r ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣
SLIDE 14
❆❧❣♦r✐t❤♠ ■✿ tr❡❡✇✐❞t❤
■❞❡❛✿
▸ ●✐✈❡♥ ❛ tr✐❛♥❣✉❧❛t✐♦♥✱ ❝♦♠♣✉t❡ ❛ tr❡❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ ❢❡✇
t❡tr❛❤❡❞r❛ ♣❡r ♥♦❞❡ ✭✐❢ ♣♦ss✐❜❧❡✮
▸ ❊♥✉♠❡r❛t❡ ❛❞♠✐ss✐❜❧❡ ❝♦❧♦✉r✐♥❣s ❛♥❞ ✇❡✐❣❤ts ❢r♦♠ t❤❡ ❧❡❛✈❡
♥♦❞❡s ✉♣
▸ ●r♦✉♣✐♥❣ ♣❛rt✐❛❧ ❝♦❧♦✉r✐♥❣s t♦❣❡t❤❡r ✇❤❡r❡✈❡r t❤❡② ❧♦♦❦ t❤❡
s❛♠❡ ❛t t❤❡ ❝✉rr❡♥t ♥♦❞❡
SLIDE 15
❆❧❣♦r✐t❤♠ ■✿ tr❡❡✇✐❞t❤
❚❤❡♦r❡♠ ✭❇✉rt♦♥✱ ▼❛r✐❛✱ ❙✳ ✷✵✶✺✮
- ✐✈❡♥ ❛ tr✐❛♥❣✉❧❛t❡❞ ✸✲♠❛♥✐❢♦❧❞ ▼ ✇✐t❤ ♥ t❡tr❛❤❡❞r❛✱ ❛♥❞ ❛ tr❡❡
❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ▼ ✇✐t❤ ❧❛r❣❡st ♥♦❞❡ ♦❢ s✐③❡ ❦✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ ❚❱r,q ✐♥ ❖ (♥ ⋅ (r − ✶)✻❦ ⋅ ❦✷ ⋅ ❧♦❣ r).
▸ ❘✉♥♥✐♥❣ t✐♠❡ ✐s ♦❢ t②♣❡ ❣(❦) × ♣♦❧②(♥)✳ ■♥ t❤❡ ❧✐t❡r❛t✉r❡ s✉❝❤
❛♥ ❛❧❣♦r✐t❤♠ ✐s r❡❢❡rr❡❞ t♦ ❛s ✜①❡❞ ♣❛r❛♠❡t❡r tr❛❝t❛❜❧❡ ✭❋P❚✮✹ ✐♥ ❦ ✭✏tr❡❡✇✐❞t❤✑✮
▸ ❈♦♠♠♦♥ ❢♦r ❋P❚ ❛❧❣♦r✐t❤♠s ✐s ❛ ✈❡r② ❜❛❞ ♣❛r❛♠❡t❡r ❢✉♥❝t✐♦♥
❣ ∶ N → N ✭t♦✇❡r ♦❢ ❡①♣♦♥❡♥t✐❛❧s✮
▸ ❍❡r❡✿
❣(❦) = (r − ✶)✻❦ ⋅ ❦✷ ⋅ ❧♦❣ r ✈s✳ (r − ✶)∣❊∣
▸ ❚❤✐s ✐s ✇❤② ✇❡ ✐♠♣❧❡♠❡♥t❡❞ t❤❡ ❛❧❣♦r✐t❤♠ ✭❛❧s♦ ✈❡r② r❛r❡ ❢♦r
❋P❚ ❛❧❣♦r✐t❤♠s✮
✹❉♦✇♥❡②✱ ❋❡❧❧♦✇s✱ P❛r❛♠❡t❡r✐③❡❞ ❝♦♠♣❧❡①✐t②✱ ❙♣r✐♥❣❡r
SLIDE 16
❆❧❣♦r✐t❤♠ ■✿ tr❡❡✇✐❞t❤
Backtracking (seconds) FPT (seconds) 0.01 0.1 1 10 100 0.01 0.1 1 10
- treewidth 1 (2143 points)
treewidth 2 (10902 points) treewidth 3 (14 points) treewidth 4 (337 points) treewidth 5 (1 point) equal times
❘✉♥♥✐♥❣ t✐♠❡s ❢♦r ❚❱✼,✶ ❢♦r t❤❡ ♠✐♥✐♠❛❧ ✶✶✲t❡tr❛❤❡❞r❛ tr✐❛♥❣✉❧❛t✐♦♥s ♦❢ ❝❧♦s❡❞ ♣r✐♠❡ ♦r✐❡♥t❛❜❧❡ ✸✲♠❛♥✐❢♦❧❞s✳
SLIDE 17
❖❜s❡r✈❛t✐♦♥s
- ❖❖❉✿
▸ ✇♦r❦s ❢♦r ❛❧❧ ♣❛r❛♠❡t❡rs r ❛♥❞ q ▸ ❢❛st❡r t❤❛♥ ♥❛✐✈❡ ❡♥✉♠❡r❛t✐♦♥
◆❖❚ ●❖❖❉✿
▸ ♣r♦♣❡rt✐❡s ♦❢ tr✐❛♥❣✉❧❛t✐♦♥✱ ♥♦t ♠❛♥✐❢♦❧❞✱ ❞❡t❡r♠✐♥❡ r✉♥♥✐♥❣
t✐♠❡✿ ✏❡✈❡r② ♠❛♥✐❢♦❧❞ ❛❞♠✐ts ❛ tr✐❛♥❣✉❧❛t✐♦♥ ✇✐t❤ ❛r❜✐tr❛r✐❧② ❤✐❣❤ tr❡❡✇✐❞t❤✑
▸ ❡①❛❝t tr❡❡✇✐❞t❤ ♠✐❣❤t ❜❡ ❞✐✣❝✉❧t t♦ ❞❡t❡r♠✐♥❡ ▸ ❛❧❣♦r✐t❤♠ r❡q✉✐r❡s ❧❛r❣❡ ❛♠♦✉♥ts ♦❢ ♠❡♠♦r②
❇❊❚❚❊❘✿
▸ ❯s❡ ♣❛r❛♠❡t❡r ✇❤✐❝❤ ✐s ❛❧s♦ ❛ t♦♣♦❧♦❣✐❝❛❧ ✐♥✈❛r✐❛♥t ▸ ❊❛s② t♦ ❝♦♠♣✉t❡✱ ❡✈❡♥ ✐❢ ❧❛r❣❡
SLIDE 18
❆♥ ❛❧t❡r♥❛t✐✈❡ ✈✐❡✇ ♦♥ ❛❞♠✐ss✐❜❧❡ ❝♦❧♦✉r✐♥❣s
r = ✸ ✭❝♦❧♦✉rs ✵✱ ✶✮ ∈ P✿
1 1
r = ✹ ✭❝♦❧♦✉rs ✵✱ ✶✱ ✷✮ ∈ ★P✲❤❛r❞✺✿
1 1 1 1 2 2 2
✺❑✐r❜②✱ ▼❡❧✈✐♥✱ ▲♦❝❛❧ s✉r❣❡r② ❢♦r♠✉❧❛s ❢♦r q✉❛♥t✉♠ ✐♥✈❛r✐❛♥ts ❛♥❞ t❤❡ ❆r❢
✐♥✈❛r✐❛♥t✱ ✷✵✵✹✳
SLIDE 19
❆♥ ❛❧t❡r♥❛t✐✈❡ ✈✐❡✇ ♦♥ ❛❞♠✐ss✐❜❧❡ ❝♦❧♦✉r✐♥❣s
r = ✸ ✭❝♦❧♦✉rs ✵✱ ✶✮ ∈ P✿ r = ✹ ✭❝♦❧♦✉rs ✵✱ ✶✱ ✷✮ ∈ ★P✲❤❛r❞✺✿
1 1 1 1 2 2 2
✺❑✐r❜②✱ ▼❡❧✈✐♥✱ ▲♦❝❛❧ s✉r❣❡r② ❢♦r♠✉❧❛s ❢♦r q✉❛♥t✉♠ ✐♥✈❛r✐❛♥ts ❛♥❞ t❤❡ ❆r❢
✐♥✈❛r✐❛♥t✱ ✷✵✵✹✳
SLIDE 20
❆♥ ❛❧t❡r♥❛t✐✈❡ ✈✐❡✇ ♦♥ ❛❞♠✐ss✐❜❧❡ ❝♦❧♦✉r✐♥❣s
r = ✸ ✭❝♦❧♦✉rs ✵✱ ✶✮ ∈ P✿ r = ✹ ✭❝♦❧♦✉rs ✵✱ ✶✱ ✷✮ ∈ ★P✲❤❛r❞✺✿
✺❑✐r❜②✱ ▼❡❧✈✐♥✱ ▲♦❝❛❧ s✉r❣❡r② ❢♦r♠✉❧❛s ❢♦r q✉❛♥t✉♠ ✐♥✈❛r✐❛♥ts ❛♥❞ t❤❡ ❆r❢
✐♥✈❛r✐❛♥t✱ ✷✵✵✹✳
SLIDE 21
❆♥ ❛❧t❡r♥❛t✐✈❡ ✈✐❡✇ ♦♥ ❛❞♠✐ss✐❜❧❡ ❝♦❧♦✉r✐♥❣s
SLIDE 22
❆♥ ❛❧t❡r♥❛t✐✈❡ ✈✐❡✇ ♦♥ ❛❞♠✐ss✐❜❧❡ ❝♦❧♦✉r✐♥❣s
SLIDE 23
❆❧❣♦r✐t❤♠ ■■✿ β✶(▼,Z✷)
▲❡♠♠❛ ✭▼❛r✐❛✱ ❙✳ ✷✵✶✻✮ ▲❡t ϕ ∈ ❆❞♠(▼,✹) ❛♥❞ ❧❡t ϕ✵ ❜❡ t❤❡ r❡❞✉❝t✐♦♥ ♦❢ ϕ ✭✐✳❡✳✱ ❛❧❧ ❝♦❧♦rs ♠♦❞ ✷✮✳ ❚❤❡♥ ∣▼∣ϕ = (−✶)α(± √ ✷)χ(❙ϕ✵), ✇❤❡r❡ α ❞❡♥♦t❡s t❤❡ ♥✉♠❜❡r ♦❢ ♦❝t❛❣♦♥s ✐♥ ❙ϕ✳ Pr♦♦❢ ✭s❦❡t❝❤✮✿
SLIDE 24
❆❧❣♦r✐t❤♠ ■■✿ β✶(▼,Z✷)
▲❡♠♠❛ ✭▼❛r✐❛✱ ❙✳ ✷✵✶✻✮ ▲❡t ϕ ∈ ❆❞♠(▼,✹) ❛♥❞ ❧❡t ϕ✵ ❜❡ t❤❡ r❡❞✉❝t✐♦♥ ♦❢ ϕ ✭✐✳❡✳✱ ❛❧❧ ❝♦❧♦rs ♠♦❞ ✷✮✳ ❚❤❡♥ ∣▼∣ϕ = (−✶)α(± √ ✷)χ(❙ϕ✵), ✇❤❡r❡ α ❞❡♥♦t❡s t❤❡ ♥✉♠❜❡r ♦❢ ♦❝t❛❣♦♥s ✐♥ ❙ϕ✳ Pr♦♦❢ ✭s❦❡t❝❤✮✿
SLIDE 25
❆❧❣♦r✐t❤♠ ■■✿ β✶(▼,Z✷)
▲❡♠♠❛ ✭▼❛r✐❛✱ ❙✳ ✷✵✶✻✮ ▲❡t ϕ ∈ ❆❞♠(▼,✹) ❛♥❞ ❧❡t ϕ✵ ❜❡ t❤❡ r❡❞✉❝t✐♦♥ ♦❢ ϕ ✭✐✳❡✳✱ ❛❧❧ ❝♦❧♦rs ♠♦❞ ✷✮✳ ❚❤❡♥ ∣▼∣ϕ = (−✶)α(± √ ✷)χ(❙ϕ✵), ✇❤❡r❡ α ❞❡♥♦t❡s t❤❡ ♥✉♠❜❡r ♦❢ ♦❝t❛❣♦♥s ✐♥ ❙ϕ✳ Pr♦♦❢ ✭s❦❡t❝❤✮✿
SLIDE 26
❆❧❣♦r✐t❤♠ ■■✿ β✶(▼,Z✷)
❚❤❡♦r❡♠ ✭▼❛r✐❛✱ ❙✳ ✷✵✶✼✮ ▼ ✶✲✈❡rt❡①✱ ♥✲t❡tr❛❤❡❞r❛ tr✐❛♥❣✉❧❛t❡❞ ✸✲♠❛♥✐❢♦❧❞ ✇✐t❤ ✜rst ❇❡tt✐ ♥✉♠❜❡r β✶(▼,Z✷)✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛♥ ❛❧❣♦r✐t❤♠ t♦ ❝♦♠♣✉t❡ ❚❱✹,q(▼) ✇✐t❤ r✉♥♥✐♥❣ t✐♠❡ ❖(✷β✶(▼,Z✷)♥✸) ✐♥ ❖(♥✷) ♠❡♠♦r② ❛♥❞ ✇✐t❤ ❖(✷β✶(▼,Z✷)) ❝②❝❧♦t♦♠✐❝ ✜❡❧❞ ♦♣❡r❛t✐♦♥s✳ Pr❛❝t✐❝❛❧ ✐♠♣r♦✈❡♠❡♥ts✿ ◆❡✇ ❛❧❣♦✳ ❚r❡❡✇✐❞t❤ ❛❧❣♦✳✻ Z✲❤♦♠✳ ✐♥ ❘❡❣✐♥❛ ≤ ✶✶ t❡t✳ ❝❡♥s✉s ✶✵.✾✻ s❡❝✳ ✹✾✽ s❡❝✳ ✼.✼✷ s❡❝✳ ❚❤❡♦r❡t✐❝❛❧ ✐♠♣r♦✈❡♠❡♥ts✿ ❉✐st✐♥❣✉✐s❤❡s r♦✉❣❤❧② t✇✐❝❡ ❛s ♠❛♥② ♠❛♥✐❢♦❧❞s ❛s Z✲❤♦♠♦❧♦❣② ♦♥ ✐ts ♦✇♥
✻❇✉rt♦♥✱ ▼❛r✐❛✱ ❙✳✱ ❆❧❣♦r✐t❤♠s ❛♥❞ ❝♦♠♣❧❡①✐t② ❢♦r ❚✉r❛❡✈✲❱✐r♦ ✐♥✈✬s✳✱ ✷✵✶✺
SLIDE 27
❆❧❣♦r✐t❤♠ ■■✿ β✶(▼,Z✷)
Treewidth−FPT (seconds) β1−FPT (seconds) 0.001 0.003 0.01 0.03 0.1 0.3 0.00003 0.0001 0.0003 0.001 0.003 0.01
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- β1 = 1 (5632 points)
β1 = 2 (2043 points) β1 = 3 (334 points) β1 = 4 (19 points) tw = 1 (2143 points) tw = 2 (10902 points) tw = 3 (14 points) tw = 4 (337 points) tw = 5 (1 point) equal time
❘✉♥♥✐♥❣ t✐♠❡s ❢♦r ❚❱✹,✶ ❢♦r t❤❡ ♠✐♥✐♠❛❧ ✶✶✲t❡tr❛❤❡❞r❛ tr✐❛♥❣✉❧❛t✐♦♥s ♦❢ ❝❧♦s❡❞ ♣r✐♠❡ ♦r✐❡♥t❛❜❧❡ ✸✲♠❛♥✐❢♦❧❞s✳
SLIDE 28