--- /dev/null
+2009-12-09
--- /dev/null
+\input lecnotes.tex
+
+\prednaska{10}{Pøevody problémù}{\vbox{\hbox{(zapsali Martin Chytil, Vladimír Kudelas,}\hbox{ Michal Kozák, Vojta Tùma)}}}
+
+\>Na této pøedná¹ce se budeme zabývat rozhodovacími problémy a jejich obtí¾ností.
+Za jednoduché budeme trochu zjednodu¹enì pova¾ovat ty problémy, na~nì¾ známe algoritmus
+pracující v~polynomiálním èase.
+
+\s{Definice:} {\I Rozhodovací problém} je takový problém, jeho¾ výstupem je v¾dy {\sc ano}, nebo {\sc ne}.
+[Formálnì bychom se na~nìj mohli dívat jako na~mno¾inu $L$ vstupù, na~které je odpovìï {\sc ano},
+a místo $L(x)=\hbox{\sc ano}$ psát prostì $x\in L$.]
+
+Vstupy mìjme zakódované jen pomocí nul a jednièek (obecnì je jedno, jaký základ pro soustavu
+kódování zvolíme, pøevody mezi poyladickými soustavami jsou co do velikosti polynomiální). Rozhodovací problém
+zadefinujeme jako $f:\ \{0,1\}^{\ast} \to \{0,1\}$, to jest funkci z mno¾iny v¹ech øetìzcù jednièek a nul
+do mno¾iny $\{1,0\}$, kde 1 na výstupu znamená {\sc ano}, 0 {\sc ne}.
+
+\s{Pøíklad:} Je dán bipartitní graf $G$ a $k \in {\bb N}$. Existuje v $G$
+párování, které obsahuje alespoò $k$ hran?
+
+To, co bychom ve~vìt¹inì pøípadù chtìli, je samozøejmì nejen zjistit, zda takové párování
+existuje, ale také nìjaké konkrétní najít. V¹imnìme si ale, ¾e kdy¾ umíme rozhodovat
+existenci párování v~polynomiálním èase, mù¾eme ho polynomiálnì rychle i najít:
+
+Mìjme èernou skøíòku (fungující v polynomiálním èase), která odpoví, zda daný
+graf má nebo nemá párování o~$k$ hranách. Odebereme z~grafu libovolnou hranu
+a zeptáme se, jestli i tento nový graf má párovaní velikosti~$k$. Kdy¾ má, pak tato
+hrana nebyla pro existenci párování potøebná, a~tak ji odstraníme. Kdy¾ naopak
+nemá (hrana patøí do ka¾dého párování po¾adované velikosti), tak si danou hranu
+poznamenáme a odebereme nejen ji a její vrcholy, ale také hrany, které do tìchto
+vrcholù vedly. Toto je korektní krok, proto¾e v pùvodním grafu tyto vrcholy
+byly navzájem spárované, a tedy nemohou být spárované s~¾ádnými jinými vrcholy.
+Na~nový graf aplikujeme znovu tentý¾ postup. Výsledkem je mno¾ina hran, které patøí
+do hledaného párování. Hran, a tedy i iterací na¹eho algoritmu, je polynomiálnì
+mnoho a skøíòka funguje v polynomiálním èase, tak¾e celý algoritmus je polynomiální.
+
+A~jak ná¹ rozhodovací problém øe¹it? Nejsnáze tak, ¾e ho pøevedeme
+na~{jiný\footnote{${}^{\dag}$}{vìrni matfyzáckým vtipùm}}, který
+u¾ vyøe¹it umíme. Tento postup jsme (právì u~hledání párování) u¾ pou¾ili
+v~kapitole o~Dinicovì algoritmu. Vytvoøili jsme vhodnou sí», pro kterou
+platilo, ¾e v~ní existuje tok velikosti~$k$ právì tehdy, kdy¾
+v~pùvodním grafu existuje párování velikosti~$k$.
+
+Takovéto pøevody mezi problémy mù¾eme definovat obecnì:
+
+\s{Definice:} Jsou-li $A$, $B$ rozhodovací problémy, pak øíkáme, ¾e $A$ lze {\I
+redukovat} (neboli pøevést) na $B$ (pí¹eme $A \rightarrow B$) $\equiv$
+existuje funkce $f$ spoèitatelná v polynomiálním èase taková, ¾e pro $\forall
+x: A(x) = B(f(x))$. V¹imnìme si, ¾e $f$ pracující v polynomiálním èase vstup
+zvìt¹í nejvíce polynomiálnì.
+
+\s{Pozorování:} $A\rightarrow B$ také znamená, ¾e problém~$B$ je alespoò tak tì¾ký
+jako problém~$A$ (tím myslíme, ¾e pokud lze $B$ øe¹it v~polynomiálním èase,
+lze tak øe¹it i~$A$): Nech» problém~$B$ umíme øe¹it v~èase $\O(b^k)$, kde
+$b$ je délka jeho vstupu. Nech» dále funkce $f$ pøevádìjící $A$ na $B$ pracuje
+v~èase $\O(a^\ell)$ pro vstup délky~$a$. Spustíme-li tedy $B(f(x))$ na~nìjaký
+vstup~$x$ problému~$A$, bude mít $f(x)$ délku $\O(a^\ell)$, tak¾e $B(f(x))$
+pobì¾í v~èase $\O(a^\ell + (a^\ell)^k) = \O(a^{k\ell})$, co¾ je polynomiální
+v~délce vstupu~$a$.
+
+
+\s{Pozorování:} Pøevoditelnost je
+\itemize\ibull
+\:reflexivní (úlohu mù¾eme pøevést na tu stejnou identickým zobrazením): $A \rightarrow A$,
+\:tranzitivní: Je-li $A \rightarrow B$ funkcí $f$, $B \rightarrow C$ funkcí $g$,
+pak $A \rightarrow C$ slo¾enou funkcí $g \circ f$
+(slo¾ení dvou polynomiálních funkcí je zase polynomiální funkce, jak u¾ jsme zpozorovali
+v~pøedchozím odstavci).
+\endlist
+\>Takovýmto relacím øíkáme kvaziuspoøádání -- nesplòují obecnì antisymetrii, tedy mù¾e nastat
+$A\rightarrow B$ a $B\rightarrow A$. Omezíme-li se v¹ak na tøídy navzájem pøevoditelných
+problémù, dostáváme ji¾ (èásteèné) uspoøádání. Existují i navzájem nepøevoditelné problémy --
+napøíklad problém v¾dy odpovídající 1 a problém v¾dy odpovídající 0.
+Podívejme se nyní na~pøevody mezi dal¹ími zajímavými problémy:
+
+\h{1. problém: SAT}
+\>Splnitelnost (satisfiability) logických formulí, tj. dosazení 1 èi
+0 za promìnné v logické formuli tak, aby formule dala výsledek 1.
+
+\>Zamìøíme se na speciální formu zadání formulí, {\I konjunktivní normální formu} (CNF),
+ které splòují následující podmínky:
+\itemize\ibull
+\:{\I formule} je zadána pomocí {\I klauzulí}\footnote{${}^{\dag}$}{bez politických konotací} oddìlených $\land$,
+\:ka¾dá {\I klauzule} je slo¾ená z {\I literálù} oddìlených $\lor$,
+\:ka¾dý {\I literál} je buïto promìnná nebo její negace.
+\endlist
+formule mají tedy tvar:
+$$\psi = (\ldots\lor\ldots\lor\ldots\lor\ldots) \land (\ldots\lor\ldots\lor\ldots\lor\ldots) \land \ldots $$
+
+\>{\I Vstup:} Formule $\psi$ v konjunktivní normální formì.
+
+\>{\I Výstup:} $\exists$ dosazení 1 a 0 za promìnné takové, ¾e hodnota formule $\psi(\ldots) = 1$.
+
+
+
+\>Pøevod nìjaké obecné formule $\psi$ na jí ekvivalentní $\chi$ v CNF mù¾e
+zpùsobit, ¾e $\chi$ je exponenciálnì velká vùèi $\psi$.
+Lze ov¹em podniknout pøevod na takovou formuli $\chi'$ v CNF, která sice není
+ekvivalentní s $\psi$ (tedy ne ka¾dý model
+$\psi$ je modelem $\chi'$), ale je splnitelná právì tehdy, kdy¾ je splnitelná $\psi$ --- co¾ nám
+pøesnì staèí --- a je lineárnì velká vùèi $\psi$.
+
+\h{2. problém: 3-SAT}
+\s{Definice:} 3-SAT je takový SAT, v nìm¾ ka¾dá klauzule obsahuje nejvý¹e tøi literály.
+
+\s{Pøevod 3-SAT na SAT:}
+Vstup není potøeba nijak upravovat, 3-SAT splòuje vlastnosti SATu, proto 3-SAT
+$\rightarrow$ SAT (SAT je alespoò tak tì¾ký jako 3-SAT)
+
+\s {Pøevod SAT na 3-SAT:}
+Musíme formuli pøevést tak, abychom neporu¹ili splnitelnost.
+
+\>Trik pro dlouhé klauzule: Ka¾dou \uv{¹patnou} klauzuli
+$$(\alpha \lor \beta) \hbox{, t¾. } \vert\alpha\vert + \vert\beta\vert \ge 4
+,\ \vert\alpha\vert \geq 2,\ \vert\beta\vert\geq 2$$
+pøepí¹eme na: $$(\alpha \lor x) \land (\beta \lor \lnot x),$$
+kde $x$ je nová promìnná, kterou nastavíme tak, abychom neovlivnili splnitelnost formule.
+
+\>Platí-li:
+\itemize\ibull
+\:$\alpha \Rightarrow x = 0$ (zajistí splnìní druhé poloviny nové formule),
+\:$\beta \Rightarrow x = 1$ (zajistí splnìní první poloviny nové formule),
+\:$\alpha ,\beta / \lnot\alpha ,\lnot\beta \Rightarrow x = 0/1$ (je nám to jedno, celkové øe¹ení nám to neovlivní).
+\endlist
+
+\>Tento trik opakujeme tak dlouho, dokud je to tøeba -- formuli délky $k+l$
+roztrhneme na formule délky $k+1$ a $l+1$. Pokud klauzule pùlíme, dostaneme
+polynomiální èas (strom rekurze má logaritmicky pater -- formule délky alespoò 6
+se nám pøi rozdìlení zmen¹í na dvì instance velikosti maximálnì $2/3$ pùvodní, krat¹í
+formule nás netrápí; na ka¾dém patøe se vykoná tolik co na pøedchozím + $2^{hloubka}$
+za pøidané formule). Velikost výsledné formule je polynomiální vùèi pùvodní:
+¾ádný literál (výskyt promìnné) pùvodní klauzule neduplikujeme, a v nejhor¹ím
+pøípadì zùstane ka¾dý pùvodní literál v klauzuli sám s nìjakými dvìma pøidanými
+(klauzule o tøech pøidaných mù¾eme vypustit) -- tedy poèet literálu se
+maximálnì ztrojnásobí.
+
+Nabízí se otázka, proè mù¾eme pøidanou promìnnou $x$ nastavovat, jak se nám zlíbí.
+Vysvìtlení je prosté -- promìnná $x$ nám pùvodní formuli nijak neovlivní, proto¾e
+se v ní nevyskytuje, proto ji mù¾eme nastavit tak, jak chceme.
+
+\s{Poznámka:} U~3-SAT lze vynutit právì tøi literály, pro krátké klauzule
+pou¾ijeme stejný trik:
+$$(\alpha) \rightarrow (\alpha \vee \alpha) \rightarrow (\alpha \lor x) \land (\alpha \lor \lnot x).$$
+
+\h{3. problém: Hledání nezávislé mno¾iny v grafu}
+
+\>Existuje nezávislá mno¾ina vrcholù z~$G$ velikosti alespoò $k$?
+
+\s{Definice:} {\I Nezávislá mno¾ina} (NzMna) budeme øíkat ka¾dé mno¾inì vrcholù grafu
+takové, ¾e mezi nimi nevede ¾ádná hrana.
+
+\figure{nezmna.eps}{Pøíklad nezávislé mno¾iny}{1in}
+
+\>{\I Vstup:} Neorientovaný graf G, $k \in {\bb N}$.
+
+\>{\I Výstup:} $\exists A \subseteq V(G)$, $\vert A \vert \ge k$: $\forall u,v \in A \Rightarrow uv \not\in E(G)$?
+
+\s{Poznámka:} Ka¾dý graf má minimálnì jednu nezávislou mno¾inu, a tou je prázdná mno¾ina. Proto je potøeba zadat i minimální velikost hledané mno¾iny.
+
+\>Uká¾eme, jak na~tento probém pøevést 3-SAT.
+
+\s{Pøevod 3-SAT na NzMna:} Z ka¾dé klauzule vybereme jeden literál tak, abychom v rùzných klauzulích nevybírali konfliktnì, tj. $x$ a $\lnot x$.
+
+\s{Pøíklad:}
+$(x \lor y \lor z) \land (x \lor \lnot y \lor \lnot z) \land (\lnot x \lor \lnot y \lor p) $.
+
+\>Pro ka¾dou klauzuli sestrojíme graf (trojúhelník) a pøidáme \uv{konfliktní}
+hrany, tj. $x$ a $\lnot x$. Poèet vrcholù grafu odpovídá poètu literálù ve formuli,
+poèet hran je maximálnì kvadratický a pøevod je tedy polynomiální.
+
+Existuje-li v grafu nezávislá mno¾ina velikosti $k$, pak z právì $k$ trojúhelníkù
+vybere právì jeden vrchol, a pøitom ¾ádné dva vrcholy nebudou odpovídat literálu a
+jeho negaci -- tedy dostaneme ohodnocení promìnných splòujících alespoò $k$ klauzulí.
+Na druhou stranu, existuje-li ohodnocení $k$ klauzulí, pak pøímo odpovídá nezávislé
+mno¾inì velikosti $k$. Ptáme-li se tedy na nezávislou mno¾inu velikosti odpovídající
+poètu klauzulí, dostaneme odpovìï {\sc ano} právì tehdy, kdy¾ je formule splnitelná.
+
+Jsou-li ve formuli i klauzule krat¹í ne¾ 3, mù¾eme je buïto prodlou¾it metodou vý¹e
+popsanou; nebo si v grafu necháme dvoj- a jedno-úhelníky, které ¾ádné z na¹ich úvah
+vadit nebudou.
+
+\figure{nezmna_graf.eps}{Ukázka pøevodu 3-SAT na nezávislou mno¾inu}{3in}
+
+\s{Pøevod NzMna na SAT:}
+Máme promìnné $v_1, \ldots , v_n$ pro vrcholy.
+
+\>Nyní uká¾eme, jak pøevést problém hledání nezávislé mno¾iny, na SAT.
+
+\itemize\ibull
+\:Poøídíme si promìnné $v_1, \ldots, v_n$ odpovídající vrcholùm grafu. Promìnná $v_i$ bude
+ indikovat, zda se $i$-tý vrchol vyskytuje v~nezávislé mno¾inì (tedy pøíslu¹né ohodnocení
+ promìnných bude vlastnì charakteristická funkce nezávislé mno¾iny).
+\:Pro ka¾dou hranu $ij \in E(G)$ pøidáme klauzuli $(\lnot v_i \lor \lnot v_j)$. Tyto klauzule
+ nám ohlídají, ¾e vybraná mno¾ina je vskutku nezávislá..
+\:Je¹tì potøebujeme zkontrolovat, ¾e je mno¾ina dostateènì velká, tak¾e si její prvky
+ oèíslujeme èísly od~1 do~$k$. Oèíslování popí¹eme maticí promìnných $x_{ij}$, pøièem¾
+ $x_{ij}$ bude pravdivá právì tehdy, kdy¾ v~poøadí $i$-tý prvek nezávislé mno¾iny je vrchol~$v_j$
+-- pøidáme tedy klauzule, které nám øeknou, ¾e vybrané do nezávislé mno¾iny jsou právì
+ ty vrcholy, které jsou touto maticí oèíslované: $\forall i,j$, $x_{ij} \Rightarrow v_j$
+ (jen dodejme, ¾e $a\Rightarrow b$ je definované jako $\neg a\vee b$).
+\:Je¹tì potøebujeme zajistit, aby byla v~ka¾dém øádku i sloupci nejvý¹e jedna jednièka:
+ $\forall j,i,i^{'}, i\ne i^{'} : x_{ij} \Rightarrow \lnot x_{i^{'}j}$ a
+ $\forall i,j,j^{'}, j\ne j^{'} : x_{ij} \Rightarrow \lnot x_{ij^{'}}$..
+\:A~nakonec si ohlídáme, aby v~ka¾dém øádku byla alespoò jedna jednièka, klauzulí $\forall i :
+ x_{i1} \lor x_{i2} \lor \ldots \lor x_{in}$.
+\endlist
+Takovýto pøevod je zøejmì polynomiální.
+
+\s{Pøíklad matice:} Jako pøíklad pou¾ijeme nezávislou mno¾inu z ukázky nezávislé mno¾iny.
+Nech» jsou vrcholy grafu oèíslované zleva a ze zhora. Hledáme nezávislou mno¾inu velikosti $2$.
+Matice pak bude vypadat následovnì:
+$$ \pmatrix{1&0&0&0&0 \cr 0&0&0&1&0}$$
+\s{Vysvìtlení:} Jako první vrchol mno¾iny bude vybrán vrchol $v_1$, proto v prvním
+øádku a v prvním sloupci bude $1$. Jako druhý ($k$-tý) vrchol mno¾iny bude vybrán
+vrchol $v_4$, proto na druhém ($k$-tém) øádku a ve ètvrtém sloupci bude $1$. Na ostatních místech bude $0$.
+
+\h{4. problém: Klika}
+
+\>{\I Vstup:} Graf $G, k \in N$.
+
+\>{\I Výstup:} $\exists$ úplný podgraf grafu $G$ na $k$ vrcholech?
+\figure{klika.eps}{Pøíklad kliky}{2in}
+
+\s{Pøevod:} Prohodíme v grafu $G$ hrany a nehrany $\Rightarrow$ (hledání nezávislé mno¾iny $\leftrightarrow$ hledání kliky).
+
+\s{Dùvod:} Pokud existuje úplný graf na $k$ vrcholech, tak v~komplementárním grafu tyto vrcholy nejsou spojeny hranou, tj. tvoøí nezávislou mno¾inu,
+ a naopak.
+
+\figure{doplnek_nm.eps}{Prohození hran a nehran}{2in}
+
+
+\h{5. problém: 3,3-SAT}
+\s{Definice:} 3,3-SAT je speciální pøípad 3-SATu, kde ka¾dá promìnná se vyskytuje v~maximálnì tøech literálech.
+
+\s{Pøevod 3-SAT na 3,3-SAT:}
+Pokud se promìnná $x$ vyskytuje v~$k > 3$ literálech, tak nahradíme výskyty novými promìnnými $x_1, \ldots , x_k$ a pøidáme klauzule:
+$$
+(\lnot x_1 \lor x_2),
+(\lnot x_2 \lor x_3),
+(\lnot x_3 \lor x_4),
+\ldots,
+(\lnot x_{k-1} \lor x_k),
+(\lnot x_k \lor x_1),
+$$
+
+co¾ odpovídá:
+
+$$
+(x_1 \Rightarrow x_2),
+(x_2 \Rightarrow x_3),
+(x_3 \Rightarrow x_4),
+\ldots,
+(x_{k-1} \Rightarrow x_k),
+(x_k \Rightarrow x_1).
+$$
+
+Tímto zaruèíme, ¾e v¹echny nové promìnné budou mít stejnou hodnotu.
+
+Mimochodem, mù¾eme rovnou zaøídit, ¾e ka¾dý literál se vyskytuje nejvíce dvakrát (tedy ¾e
+ka¾dá promìnná se vyskytuje alespoò jednou pozitivnì a alespoò jednou negativnì). Pokud by
+se nìjaká promìnná objevila ve~tøech stejných literálech, mù¾eme na~ni
+také pou¾ít ná¹ trik a nahradit ji tøemi promìnnými. V~nových klauzulích se pak bude
+vyskytovat jak pozitivnì, tak negativnì.
+
+\h{6. problém: 3D párování (3D matching)}
+
+\>{\I Vstup:} Tøi mno¾iny, napø. $K$ (kluci), $H$ (holky), $Z$ (zvíøátka) a mno¾ina kompatibilních trojic (tìch, kteøí se spolu snesou).
+
+\>{\I Výstup:} Perfektní podmno¾ina trojic - tj. taková podmno¾ina trojic, která obsahuje v¹echna $K$, $H$ a $Z$.
+
+\>Uká¾eme, jak tento problém pøevést na 3,3-SAT (ov¹em to a¾ na dal¹í pøedná¹ce).
+
+\figure{3d_parovani.eps}{Ukázka 3D párování}{3in}
+
+\s{Závìr:} Obrázek ukazuje problémy, jimi¾ jsme se dnes zabývali, a vztahy mezi tìmito problémy.
+\figure{prevody.eps}{Pøevody mezi problémy}{3in}
+
+\bye
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projects / ads2.git / commitdiff
