20101021 proof complexity_hirsch_lecture05

Ñëîæíîñòü ïðîïîçèöèîíàëüíûõ äîêàçàòåëüñòâ
Ýäóàðä Àëåêñååâè÷ Ãèðø
http://logic.pdmi.ras.ru/~hirsch
ÏÎÌÈ ÐÀÍ
21 îêòÿáðÿ 2010 ã.
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Cutting Plane: íèæíÿÿ îöåíêà
Ìîíîòîííàÿ èíòåðïîëÿöèÿ.
Ýêñïîíåíöèàëüíàÿ íèæíÿÿ îöåíêà íà èíòåðïîëÿíò.
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Òåîðåìà (Ïóäëàê)
Ïóñòü â A(x, y) ⊃ B(x, z) âñå âõîæäåíèÿ xi â A è â B ïîëîæèòåëüíû.
Òîãäà èç äîê-âà â Res (èëè CP) ïîëó÷àåòñÿ ìîíîòîííûé èíòåðïîëÿíò
áóëåâà (èëè àðèôìåòè÷åñêàÿ) C(x) ïîëèíîìèàëüíîãî ðàçìåðà.
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Òåîðåìà (Ïóäëàê)
Ïóñòü â A(x, y) ⊃ B(x, z) âñå âõîæäåíèÿ xi â A è â B ïîëîæèòåëüíû.
Òîãäà èç äîê-âà â Res (èëè CP) ïîëó÷àåòñÿ ìîíîòîííûé èíòåðïîëÿíò
áóëåâà (èëè àðèôìåòè÷åñêàÿ) C(x) ïîëèíîìèàëüíîãî ðàçìåðà.
Òåîðåìà (Ðàçáîðîâ; Àëîí-Áîïïàíà; Ïóäëàê)
Äëÿ áóëåâûõ (èëè àðèôìåòè÷åñêèõ) ñõåì,
ðàçäåëÿþùèõ nêëèêè è (n − 1)ðàñêðàøèâàåìûå ãðàôû,
|C| = 2Ω(
√
n) ïðè n = 1
8(m/ log m)2/3 .
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Íèæíèå îöåíêè, îñíîâàííûå íà øèðèíå
Ñõåìà äîêàçàòåëüñòâà:
Ëèíåéíàÿ íèæíÿÿ îöåíêà íà êàêîé-òî ïàðàìåòð (øèðèíó).
Ëåììà: â êîðîòêîì äîê-âå π ìîæíî ïîíèçèòü øèðèíó äî ∼ ln |π|.
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Ðåçîëþöèÿ:
Øèðèíà äèçúþíêöèè W(l1 ∨ . . . ∨ lk) êîëè÷åñòâî ïåðåìåííûõ.
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Ðåçîëþöèÿ:
Øèðèíà ôîðìóëû/âûâîäà øèðèíà ñàìîé øèðîêîé äèçúþíêöèè.
G w H: åñòü âûâîä øèðèíû ≤ w.
W (F) ìèíèìàëüíî âîçìîæíàÿ øèðèíà äîê-âà.
S (F) ìèíèìàëüíî âîçìîæíîå êîë-âî äèçúþíêöèé äîê-âà.
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Ðåçîëþöèÿ:
Ëåììà (äëÿ ôîðìóëû F îò n ïåðåìåííûõ)
W (F) ≤ W(F) + O( n ln S (F))
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Ðåçîëþöèÿ:
W (F) ≤ W(F) + O( n ln S (F))
Ëåììà (ê ëåììå)
1. F|x=0 w C =⇒ F w+1 C ∨ x.
2. W (F|x=0) ≤ w−1, W (F|x=1) ≤ w =⇒
=⇒ W (F) ≤ max{w, W({C ∈ F|¬x ∈ C})}.3 / 5

Ðåçîëþöèÿ:
W (F) ≤ W(F) + O( n ln S (F))
Ëåììà (ê ëåììå)
1. F|x=1 w C =⇒ F w+1 C ∨ ¬x.
2. W (F|x=1) ≤ w−1, W (F|x=0) ≤ w =⇒
=⇒ W (F) ≤ max{w, W({C ∈ F| x ∈ C})}.3 / 5

Öåéòèíñêèå ôîðìóëû
G = (V, E) ãðàô.
Ïåðåìåííàÿ xe ∈ {0, 1} äëÿ êàæäîãî e ∈ E.
Óñëîâèå
e v
xe = 1.
|V|
... 2.
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Óñëîâèå
e v
xe = cv.
Êîíñòàíòà cv ∈ {0, 1} äëÿ êàæäîé v ∈ V.
v∈V
cv = 1.
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Óñëîâèå
e v
xe = cv.
Êîíñòàíòà cv ∈ {0, 1} äëÿ êàæäîé v ∈ V.
v∈V
cv = 1.
Îïðåäåëåíèå
Ðàñøèðèòåëüíàÿ ñïîñîáíîñòü
e(G) = min |cut(V , V V )| V ⊆ V, 1
3|V| ≤ |V | ≤ 2
3|V| .
G ðàñøèðèòåëü, åñëè e(G) = Ω(|V|).
Ôàêò
Ñóùåñòâóþò ñâÿçíûå ðåãóëÿðíûå ðàñøèðèòåëè ñòåïåíè 3.
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Íèæíÿÿ îöåíêà Ω(|V|) íà øèðèíó âûâîäà
Â âûâîäå åñòü äèçúþíêöèÿ C, ñëåäóþùàÿ â òî÷íîñòè èç
óðàâíåíèé
v∈U e v
xe = cv , (1)
ãäå |U| ∈ 1
3|V|..2
3|V| .
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óðàâíåíèé
v∈U e v
xe = cv , (1)
ãäå |U| ∈ 1
3|V|..2
3|V| .
Êàæäàÿ ïåðåìåííàÿ èç ñå÷åíèÿ (U, V U) ìåíÿåò èñòèííîñòü (1)
äëÿ êàêîãî-òî íàáîðà. . .
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óðàâíåíèé
v∈U e v
xe = cv , (1)
ãäå |U| ∈ 1
3|V|..2
3|V| .
Êàæäàÿ ïåðåìåííàÿ èç ñå÷åíèÿ (U, V U) ìåíÿåò èñòèííîñòü (1)
äëÿ êàêîãî-òî íàáîðà. . .
. . . è äîëæíà âõîäèòü â C.
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20101021 proof complexity_hirsch_lecture05

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