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--- /dev/null
+\input lecnotes.tex
+
+\prednaska{8}{Pøevody problémù a NP-úplnost}{}
+
+\>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 {\csc ano}, nebo {\csc ne}.
+[Formálnì bychom se na~nìj mohli dívat jako na~mno¾inu $L$ vstupù, na~které je odpovìï {\csc ano},
+a místo $L(x)=\hbox{\csc ano}$ psát prostì $x\in L$.]
+
+Vstupy mìjme zakódované jen pomocí nul a jednièek (obecnì je jedno, jaký základ pro soustavu
+kódování zvolíme, pøevody mezi soustavami o nìjakém základu $\neq$ 1 jsou co do velikosti
+zápisu polynomiální). Rozhodovací problém je tedy
+$f:\ \{0,1\}^{\ast} \to \{0,1\}$, to jest funkce z mno¾iny v¹ech øetìzcù jednièek a nul
+do mno¾iny $\{1,0\}$, kde 1 na výstupu znamená {\csc ano}, 0 {\csc 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 {\it pøevést}) na $B$ (pí¹eme $A \rightarrow B$) právì tehdy,
+kdy¾ 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 $\O(a^\ell)$, kde $a=|q|$; 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.
+Nyní se ji¾ podíváme na nìjaké zajímavé problémy. Obecnì to budou problémy, na které
+polynomiální algoritmus není znám, a vzájemnými pøevody zjistíme ¾e jsou stejnì tì¾ké.
+
+\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 $\psi$.
+Pozdìji uká¾eme, ¾e lze podniknout pøevod na takovou formuli $\chi'$ v~CNF, která sice není
+ekvivalentní s $\psi$ (pøibydou nám promìnné, a ne ka¾dý roz¹íøený 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 (pøi ka¾dém dìlení klauzule {\it jiná}
+nová promìnná).
+%kterou nastavíme tak, abychom neovlivnili splnitelnost formule.
+
+\>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 tím pádem polynomiální vùèi pùvodní:
+v ka¾dém kroku se pøidají jen dva literály, tedy celkem {\it èas na pøevod}$\cdot
+2$ nových.
+
+\>Platí-li:
+\itemize\ibull
+\:$\alpha \Rightarrow$ zvolíme $x = 0$ (zajistí splnìní druhé poloviny nové formule),
+\:$\beta \Rightarrow$ zvolíme $x = 1$ (zajistí splnìní první poloviny nové formule),
+\:$\alpha ,\beta / \lnot\alpha ,\lnot\beta \Rightarrow$ zvolíme $x = 0/1$ (je nám to
+ jedno, celkové øe¹ení nám to neovlivní).
+\endlist
+
+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, jeho¾ nastavením se klauzuli
+rozhodneme splnit. Samozøejmì 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~ka¾dého z~$k$ 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 $k$ (v ka¾dém trojúhelníku zvolíme právì jednu z ohodnocených
+promìnných, nemù¾e se stát ¾e zvolíme vrcholy konfliktní hrany). Ptáme-li se tedy
+na nezávislou mno¾inu velikosti odpovídající
+poètu klauzulí, dostaneme odpovìï {\csc 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:}
+
+\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
+Tímto vynutíme NzMnu $\geq k$, co¾ jsme pøesnì chtìli. Takovýto pøevod je zøejmì polynomiální.
+
+\s{Pøíklad matice:} Jako pøíklad pou¾ijeme nezávislou mno¾inu z ukázky nezávislé mno¾iny.
+Nech» jsou vrcholy grafu oèíslované zleva a zeshora. 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 vrchol mno¾iny bude vybrán
+vrchol $v_4$, proto na druhém 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 na tento problém pøevést 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}
+
+\h{NP-úplné problémy}
+
+Dosud jsme zkoumali problémy, které se nás ptaly na to, jestli nìco existuje.
+Napøíklad jsme dostali formuli a problém splnitelnosti se nás ptal, zda
+existuje ohodnocení promìnných takové, ¾e formule platí. Nebo v~pøípadì
+nezávislých mno¾in jsme dostali graf a èíslo $k$ a ptali jsme se, jestli
+v~grafu existuje nezávislá mno¾ina, která obsahuje alespoò~$k$ vrcholù.
+Tyto otázky mìly spoleèné to, ¾e kdy¾ nám nìkdo napovìdìl nìjaký objekt, umìli
+jsme efektivnì øíci, zda je to ten, který hledáme. Napøíklad pokud dostaneme
+ohodnocení promìnných logické formule, staèí jen dosadit a spoèítat, kde
+formule dá \<true> nebo \<false>. Zjistit, ¾e nìjaký objekt je ten, který
+hledáme, umíme efektivnì. Tì¾ké na tom je takový objekt najít. Co¾ vede
+k~definici obecných vyhledávacích problémù, kterým se øíká tøída problémù NP.
+Nadefinujeme si ji poøádnì, ale nejdøíve zaèneme tro¹ièku jednodu¹¹í tøídou.
+
+\s{Definice:} P je {\I tøída rozhodovacích problémù}, které jsou øe¹itelné
+v~polynomiálním èase. Jinak øeèeno, problém
+$L \in {\rm P} \Leftrightarrow \exists $ polynom $f$ a~$\exists$ algoritmus~$A$
+takový, ¾e $\forall x: L(x)=A(x)$ a $A(x)$ dobìhne v~èase $\O(f(x))$.
+
+Tøída P tedy odpovídá tomu, o èem jsme se shodli, ¾e umíme efektivnì øe¹it.
+Nadefinujme nyní tøídu NP:
+
+\s{Definice:} NP je {\I tøída rozhodovacích problémù} takových, ¾e $L \in {\rm NP}$ právì tehdy, kdy¾ $\exists $ problém
+$K\in{\rm P}$ a $\exists$ polynom $g$ takový, ¾e pro
+$\forall x$ platí $L(x)=1 \Leftrightarrow \exists $ nápovìda $ y: \vert y \vert \leq g(\vert x \vert)$ a souèasnì $K(x,y)=1$.
+
+\s{Pozorování:} Splnitelnost logických formulí je v~NP. Staèí si toti¾ nechat napovìdìt, jak
+ohodnotit jednotlivé promìnné a pak ovìøit, jestli je formule splnìna. Nápovìda je polynomiálnì
+velká (dokonce lineárnì), splnìní zkontrolujeme také v~lineárním èase. Odpovíme tedy ano právì
+tehdy, existuje-li nápovìda, která nás pøesvìdèí, èili pokud je formule splnitelná.
+
+\s{Pozorování:} Tøída P le¾í uvnitø NP.
+V~podstatì øíkáme, ¾e kdy¾ máme problém, který umíme øe¹it v~polynomiálním èase
+bez nápovìdy, tak to zvládneme v~polynomiálním èase i s~nápovìdou.
+
+Problémy z minulé pøedná¹ky jsou v¹echny v NP (napø. pro nezávislou
+mno¾inu je onou nápovìdou pøímo mno¾ina vrcholù deklarující nezávislost),
+o jejich pøíslu¹nosti do P ale nevíme nic.
+Brzy uká¾eme, ¾e to jsou v jistém smyslu nejtì¾¹í problémy v~NP.
+Nadefinujme si:
+
+\s{Definice:} Problém $L$ je NP-{\I tì¾ký} právì tehdy, kdy¾ je na~nìj pøevoditelný
+ka¾dý problém z~NP (viz definici pøevodù z minulé pøedná¹ky).
+
+Rozmyslete si, ¾e pokud umíme øe¹it nìjaký NP-tì¾ký problém v~polynomiálním èase,
+pak umíme vyøe¹it v~polynomiálním èase v¹e v~NP, a tedy ${\rm P}={\rm NP}$.
+
+My se budeme zabývat problémy, které jsou NP-tì¾ké a samotné jsou v~NP. Takovým problémùm se øíká NP-úplné.
+
+\s{Definice:} Problém $L$ je NP-{\I úplný} právì tehdy, kdy¾ $L$ je NP-tì¾ký a $L \in {\rm NP}$.
+
+NP-úplné problémy jsou tedy ve~své podstatì nejtì¾¹í problémy, které le¾í v~NP.
+Kdybychom umìli vyøe¹it nìjaký NP-úplný problém v~polynomiálním èase, pak
+v¹echno v~NP je øe¹itelné v~polynomiálním èase. Bohu¾el to, jestli nìjaký
+NP-úplný problém lze øe¹it v~polynomiálním èase, se neví. Otázka, jestli
+${\rm P}={\rm NP}$, je asi nejznámìj¹í otevøený problém v~celé teoretické
+informatice.
+
+Kde ale nìjaký NP-úplný problém vzít? K~tomu se nám bude velice hodit následující vìta:
+
+\s{Vìta (Cookova):} SAT je NP-úplný.
+
+\>Dùkaz je znaènì technický, pøibli¾nì ho naznaèíme pozdìji. Pøímým dùsledkem
+Cookovy vìty je, ¾e cokoli v~NP je pøevoditelné na SAT.
+K dokazování NP-úplnosti dal¹ích problémù pou¾ijeme následující vìtièku:
+
+\s{Vìtièka:} Pokud problém $L$ je NP-úplný a $L$ se dá pøevést na nìjaký problém $M\in{\rm NP}$, pak $M$ je také NP-úplný.
+
+\proof
+Tuto vìtièku staèí dokázat pro NP-tì¾kost, NP-úplnost plyne okam¾itì z~toho, ¾e
+problémy jsou NP-tì¾ké a le¾í v~NP (podle pøedpokladu).
+
+Víme, ¾e $L$ se dá pøevést na~$M$ nìjakou funkcí~$f$. Jeliko¾ $L$ je NP-úplný,
+pak pro ka¾dý problém $Q\in{\rm NP}$ existuje nìjaká funkce~$g$, která pøevede
+$Q$ na~$L$. Staèí tedy slo¾it funkci~$f$ s~funkcí~$g$, èím¾ získáme funkci pracující
+opìt v~polynomiálním èase, která pøevede~$Q$ na~$M$. Ka¾dý problém z~NP se tedy
+dá pøevést na problém~$M$.
+\qed
+
+\s{Dùsledek:} Cokoliv, na co jsme umìli pøevést SAT, je také NP-úplné.
+Napøíklad nezávislá mno¾ina, rùzné varianty SATu, klika v~grafu~\dots
+
+Jak taková tøída NP vypadá? Pøedstavme si v¹echny problémy tøídy NP, jakoby seøazené
+shora dolu podle obtí¾nosti problémù (tedy navzdor gravitaci), kde porovnání dvou
+problémù urèuje pøevoditelnost (viz obrázek).
+
+\figure{p-np.eps}{Struktura tøídy NP}{2.5cm}
+
+Obecnì mohou nastat dvì situace, proto¾e nevíme, jestli ${\rm P}={\rm NP}$.
+Jestli ano, pak v¹echno je jedna a ta samá tøída. To by bylo v nìkterých
+pøípadech nepraktické, napø. ka¾dá ¹ifra by byla jednodu¹e rozlu¹titelná.
+\foot{Poznámka o ¹ifrách -- libovolnou funkci vyèíslitelnou v polynomiálním
+èase bychom umìli v polynomiálním èase také invertovat.}
+Jestli ne, NP-úplné problémy urèitì nele¾í v P, tak¾e P a NP-úplné problémy
+jsou dvì disjunktní èásti NP. Také se dá dokázat (to dìlat nebudeme, ale je
+dobré to vìdìt), ¾e je¹tì nìco le¾í mezi nimi, tedy ¾e existuje problém, který
+je v~NP, není v~P a není NP-úplný (dokonce je takových problémù nekoneènì mnoho,
+v nekoneènì tøídách).
+
+\s{Katalog NP-úplných problémù}
+
+Uká¾eme si nìkolik základních NP-úplných problémù. O~nìkterých jsme to dokázali
+na~minulé pøedná¹ce, o~dal¹ích si to doká¾eme nyní, zbylým se na~zoubek podíváme
+na~cvièeních.
+
+\itemize\ibull
+\:{\I logické:}
+ \itemize\ibull
+ \:SAT (splnitelnost logických formulí v~CNF)
+ \:3-SAT (ka¾dá klauzule obsahuje max.~3 literály)
+ \:3,3-SAT (a navíc ka¾dá promìnná se vyskytuje nejvý¹e tøikrát)
+ \:SAT pro obecné formule (nejen CNF)
+ \:Obvodový SAT (není to formule, ale obvod)
+ \endlist
+\:{\I grafové:}
+ \itemize\ibull
+ \:Nezávislá mno¾ina (existuje mno¾ina alespoò~$k$ vrcholù taková, ¾e ¾ádné dva nejsou propojeny hranou?)
+ \:Klika (existuje úplný podgraf na~$k$ vrcholech?)
+ \:3D párování (tøi mno¾iny se zadanými trojicemi, existuje taková mno¾ina disjunktních trojic, ve~které jsou v¹echny prvky?)
+ \:Barvení grafu (lze obarvit vrcholy $k$~barvami tak, aby vrcholy stejné barvy nebyly nikdy spojeny hranou? NP-úplné u¾ pro~$k=3$)
+ \:Hamiltonovská cesta (cesta obsahující v¹echny vrcholy [právì jednou])
+ \:Hamiltonovská kru¾nice (kru¾nice, která nav¹tíví v¹echny vrcholy [právì jednou])
+ \endlist
+\:{\I èíselné:}
+ \itemize\ibull
+ \:Batoh (nejjednodu¹¹í verze: dána mno¾ina èísel, zjistit, zda existuje podmno¾ina se zadaným souètem)
+ \:Batoh -- optimalizace (podobnì jako u pøedchozího problému, ale místo mno¾iny èísel máme mno¾inu
+ pøedmìtù s vahami a cenami, chceme co nejdra¾¹í podmno¾inu její¾ váha nepøesáhne zadanou kapacitu
+ batohu)
+ \:Loupe¾níci (lze rozdìlit danou mno¾inu èísel na~dvì podmno¾iny se stejným souètem)
+ \:$Ax=b$ (soustava celoèíslených lineárních rovnic; $x_i$ mohou být pouze 0 nebo 1; NP-úplné i pokud $A_{ij}\in\{0,1\}$ a $b_i\in\{0,1\}$)
+ \:Celoèíselné lineární programování (existuje vektor nezáporných celoèísených $x$ takový, ¾e $Ax \leq b$ ?)
+ \endlist
+\endlist
+
+Nyní si uká¾eme, jak pøevést SAT na nìjaký problém. Kdy¾ chceme ukázat, ¾e na
+nìco se dá pøevést SAT, potøebujeme obvykle dvì vìci:
+konstrukci, která bude simulovat promìnné, tedy nìco, co nabývá dvou stavù
+\<true>/\<false>; a nìco, co bude reprezentovat klauzule a umí zaøídit, aby
+ka¾dá klauzule byla splnìna alespoò jednou promìnnou.
+Roz¹iøme tedy ná¹ katalog problémù o následující:
+\h{3D párování (3D matching)}
+
+\>{\I Vstup:} Tøi mno¾iny, napø. $K$ (kluci), $H$ (holky), $Z$ (zvíøátka) a
+mno¾ina $T$ kompatibilních trojic (tìch, kteøí se spolu snesou),
+ tj. $T \subseteq K\times H\times Z$.
+
+\>{\I Výstup:} Perfektní podmno¾ina trojic $P\subseteq K\times H \times Z$ --
+ tj. taková podmno¾ina trojic, ¾e $(\forall k\in K\ \exists !p\in P: k\in p)
+ \wedge(\forall h\in H\ \exists !p\in P: h\in p)
+ \wedge(\forall z\in Z\ \exists !p\in P: z\in p)$ -- tedy ka¾dý byl vybrán
+ právì jednou.
+
+
+\h{ Uká¾eme, jak na 3D-párování pøevést 3,3-SAT }
+
+Uva¾ujme takovouto konfiguraci:
+
+\fig{3d.eps}{4cm}
+
+\>4 zvíøátka, 2 kluci, 2 dívky a 4 trojice, které oznaèíme $A, B, C, D$.
+Je¹tì pøedpokládáme, ¾e zvíøátka
+ se mohou úèastnit nìjakých jiných trojic, ale
+tito ètyøi lidé se vyskytují pouze v~tìchto ètyøech trojicích a~nikde jinde.
+V¹imneme si, ¾e existují právì dvì mo¾nosti, jak tento obrázek spárovat.
+Abychom spárovali kluka $k_1$, tak si musíme vybrat $A$ nebo $B$. Kdy¾ si
+vybereme $A$, $k_1$ i $d_2$ u¾ jsou spárovaní tak¾e si nesmíme vybrat $B$ ani
+$D$. Pak jediná mo¾nost, jak spárovat $d_1$ a~$k_2$ je $C$. Jedna mo¾nost je
+tedy vybrat si $A$ a $C$ a jeliko¾ je obrázek symetrický, tak kdy¾ vybereme
+místo $A$ trojici $B$, dostaneme $B$ a~$D$. V¾dy si tedy vybereme dvì protìj¹í
+trojice v~obrázku.
+
+Tyto konfigurace budeme pou¾ívat k~reprezentaci promìnných. Pro ka¾dou
+promìnnou si nakreslíme takový obrázek a~to, ¾e $A$ bude spárované s~$C$, bude
+odpovídat tomu, ¾e $x=1$, a~spárování $B$ a~$D$ bude odpovídat
+ $x=0$. Pokud jsme
+pou¾ili $A$ a~$C$, zvíøata se sudými èísly, tj. $z_2$ a~$z_4$, horní a~dolní,
+jsou nespárovaná a~pokud jsme pou¾ili $B$ a~$D$, zvíøátka $z_1$ a~$z_3$ zùstala
+nespárovaná. Pøes tato nespárovaná zvíøátka mù¾eme pøedávat informaci, jestli
+promìnná $x$ má hodnotu \<true> nebo \<false> do dal¹ích èástí vstupu.
+
+Zbývá vymyslet, jak reprezentovat klauzule. Klauzule jsou trojice popø. dvojice
+literálù, napø. $\kappa = (x \lor y \lor \lnot r)$, kde
+potøebujeme zajistit, aby $x$ bylo nastavené na $1$ nebo $y$ bylo nastavené
+na~$1$ nebo $r$ na $0$.
+
+\fig{klauzule.eps}{4cm}
+
+\>Pro takovouto klauzuli si poøídíme dvojici kluk-dívka, kteøí budou figurovat
+ve tøech trojicích se tøemi rùznými zvíøátky, co¾ mají být volná zvíøátka
+z~obrázkù pro pøíslu¹né promìnné (podle toho, má-li se promìnná vyskytnout
+s negací nebo ne). A~zaøídíme to tak, aby ka¾dé zvíøátko bylo
+pou¾ité maximálnì v~jedné takové trojici, co¾ jde proto, ¾e ka¾dý literál se
+vyskytuje maximálnì dvakrát a~pro ka¾dý literál máme dvì volná zvíøátka,
+z~èeho¾ plyne, ¾e zvíøátek je dost pro v¹echny klauzule. Pro dvojice se postupuje
+obdobnì.
+
+Je¹tì nám ale urèitì zbude $2p-k$ zvíøátek, kde $p$ je poèet promìnných, $k$
+poèet klauzulí -- ka¾dá promìnná vyrobí 4 zvíøátka, klauzule zba¹tí jedno
+a samotné ohodnocení 2 zvíøátka -- tak pøidáme je¹tì $2p-k$ párù
+kluk-dìvèe, kteøí milují
+v¹echna zvíøátka, a~ti vytvoøí zbývající trojice.
+
+Pokud formule byla splnitelná, pak ze splòujícího ohodnocení mù¾eme vyrobit
+párování s~na¹í konstrukcí. Obrázek pro ka¾dou promìnnou spárujeme podle
+ohodnocení, tj. promìnná je $0$ nebo $1$ a~pro ka¾dou klauzuli si vybereme
+nìkterou z~promìnných, kterými je ta klauzule splnìna. Funguje to ale
+i~opaènì: Kdy¾ nám nìkdo dá párovaní v~na¹í konstrukci, pak z nìho doká¾eme
+vyrobit splòující ohodnocení dané formule. Podíváme se, v~jakém stavu je
+promìnná, a~to je v¹echno. Z~toho, ¾e jsou správnì spárované klauzule, u¾
+okam¾itì víme, ¾e jsou v¹echny splnìné.
+
+Zbývá ovìøit, ¾e na¹e redukce funguje v~polynomiálním èase. Pro ka¾dou klauzuli
+spotøebujeme konstantnì mnoho èasu, $2p-k$ je také polynomiálnì mnoho a~kdy¾ v¹e
+seèteme, máme polynomiální èas vzhledem k~velikosti vstupní formule. Tím je
+pøevod hotový a~mù¾eme 3D-párování zaøadit mezi NP-úplné problémy.
+
+
+%RK
+
+
+\h{Náznak dùkazu Cookovy vìty}
+
+Abychom mohli budovat teorii NP-úplnosti, potøebujeme alespoò jeden problém,
+o kterém doká¾eme, ¾e je NP-úplný, z definice. Cookova vìta øíká o NP-úplnosti
+SAT-u, ale nám se to hodí dokázat o tro¹ku jiném problému -- {\I obvodovém SAT-u}.
+
+\>{\I Obvodový SAT} je splnitelnost, která nepracuje s~formulemi, ale s~booleovskými
+obvody (hradlovými sítìmi). Ka¾dá formule se dá pøepsat do booleovského obvodu,
+který ji poèítá, tak¾e dává smysl zavést splnitelnost i pro obvody. Na¹e obvody
+budou mít nìjaké vstupy a~jenom jeden výstup. Budeme se ptát, jestli se vstupy
+tohoto obvodu dají nastavit tak, abychom na výstupu dostali \<true>.
+
+\>Nejprve doká¾eme NP-úplnost {\I obvodového SAT-u} a~pak uká¾eme, ¾e se dá
+pøevést na obyèejný SAT v~CNF. Tím bude dùkaz Cookovy vìty hotový. Zaènìme
+s pomocným lemmatem.
+
+\s{Lemma:} Nech» $L$ je problém v P. Potom existuje polynom $p$ a algoritmus,
+který pro $\forall n \ge 0$ spoète v èase $p(n)$ hradlovou sí» $Bn$ s $n$ vstupy
+a 1 výstupem takovou, ¾e $\forall x \in \{ 0, 1 \}^n : Bn(x) = L(x)$.
+
+\proof
+Náznakem. Na základì zku¹eností z Principù poèítaèù intuitivnì chápeme poèítaèe
+jako nìjaké slo¾ité booleovské obvody, jejich¾ stav se mìní v~èase. Uva¾me tedy
+nìjaký problém $L \in {\rm P}$ a polynomiální algoritmus, který ho øe¹í. Pro vstup
+velikosti~$n$ dobìhne v~èase~$T$ polynomiálním v~$n$ a spotøebuje $\O(T)$ bunìk pamìti.
+Staèí nám tedy \uv{poèítaè s~pamìtí velkou $\O(T)$}, co¾ je nìjaký booleovský obvod
+velikosti polynomiální v~$T$, a~tedy i v~$n$. Vývoj v~èase o¹etøíme tak, ¾e sestrojíme~$T$
+kopií tohoto obvodu, ka¾dá z~nich bude odpovídat jednomu kroku výpoètu a bude
+propojena s~\uv{minulou} a \uv{budoucí} kopií. Tím sestrojíme booleovský obvod,
+který bude øe¹it problém~$L$ pro vstupy velikosti~$n$ a bude polynomiálnì velký
+vzhledem k~$n$.
+
+\s{Poznámka:}
+Pro dùkaz následující vìty si dovolíme drobnou úpravu v~definici tøídy NP.
+Budeme chtít, aby nápovìda
+mìla pevnou velikost, závislou pouze na~velikosti vstupu (tedy: $\vert y \vert
+= g(\vert x \vert)$ namísto $\vert y \vert \le g(\vert x \vert)$). Proè je taková
+úprava BÚNO? Jistì si dovedete pøedstavit,
+¾e pùvodní nápovìdu doplníme na po¾adovanou délku nìjakými \uv{mezerami}, které
+program ignoruje. (Tedy upravíme program tak, aby mu nevadilo, ¾e dostane na
+konci nápovìdy nìjak kódované mezery.)
+
+\s{Vìta:} Obvodový SAT je NP-úplný.
+
+\proof
+Máme tedy nìjaký problém $L$ z~NP a~chceme dokázat, ¾e $L$ se dá pøevést
+na~obvodový SAT (tj. NP-tì¾kost). Kdy¾ nám nìkdo pøedlo¾í nìjaký vstup $x$
+(chápeme jako posloupnost $x_1, x_2, \ldots, x_n$),
+spoèítáme velikost nápovìdy $g(n)$. Víme, ¾e kontrolní
+algoritmus~$K$ (který kontroluje, zda nápovìda je správnì) je v~P. Vyu¾ijeme
+pøedchozí lemma, abychom získali obvod, který pro konkrétní velikost vstupu
+$x$ poèítá to, co kontrolní algoritmus $K$. Na vstupu tohoto obvodu bude $x$
+(vstup problému $L$) a~nápovìda~$y$. Na výstupu nám øekne, jestli je nápovìda
+správná. Velikost vstupu tohoto obvodu bude tedy $p(g(n))$, co¾ je také polynom.
+
+\fig{kobvod.eps}{2.3cm}
+
+\>V tomto obvodu zafixujeme vstup $x$ (na místa vstupu dosadíme konkrétní hodnoty z $x$). Tím získáme obvod, jeho¾ vstup je jen $y$ a~ptáme se, zda za $y$ mù¾eme dosadit nìjaké hodnoty tak, aby na výstupu bylo \<true>. Jinými slovy, ptáme se, zda je tento obvod splnitelný.
+
+\>Pro libovolný problém z~NP tak doká¾eme sestrojit funkci, která pro ka¾dý vstup~$x$ v~polynomiálním èase vytvoøí obvod, který je splnitelný pravì tehdy, kdy¾ odpovìï tohoto problému na vstup $x$ má být \<true>. Tedy libovolný problém z~NP se dá
+v~polynomiálním èase pøevést na obvodový SAT.
+
+\>Obvodový SAT je v NP triviálnì -- staèí si nechat poradit vstup, sí»
+topologicky setøídit a v~tomto poøadí poèítat hodnoty hradel.
+\qed
+
+\s{Lemma:} Obvodový SAT se dá pøevést na 3-SAT.
+
+\proof
+Budeme postupnì budovat formuli v~konjunktivní normální formì. Ka¾dý booleovský obvod se dá pøevést v~polynomiálním èase na~ekvivalentní obvod, ve~kterém se vyskytují jen hradla {\csc and} a {\csc not}, tak¾e staèí najít klauzule odpovídající tìmto hradlùm. Pro ka¾dé hradlo v~obvodu zavedeme novou promìnnou popisující jeho výstup. Pøidáme klauzule, které nám kontrolují, ¾e toto hradlo máme ohodnocené konzistentnì.
+
+\>{\I Pøevod hradla \csc not}: na vstupu hradla budeme mít nìjakou promìnnou $x$ (která pøi¹la buïto pøímo ze~vstupu toho celého obvodu nebo je to promìnná, která vznikla na výstupu nìjakého hradla) a na výstupu promìnnou $y$. Pøidáme klauzule, které nám zaruèí, ¾e jedna promìnná bude negací té druhé:
+$$\matrix{ (x \lor y), \cr
+ (\neg{x} \lor \neg{y}). \cr }
+ \hskip 0.2\hsize
+\vcenter{\hbox{\epsfxsize=0.7cm\epsfbox{not.eps}}}
+$$
+
+\>{\I Pøevod hradla \csc and}: Hradlo má vstupy $x, y$ a~výstup $z$. Potøebujeme pøidat klauzule, které nám popisují, jak se má hradlo {\csc and} chovat. Tyto vztahy pøepí¹eme do~konjunktivní normální formy:
+$$
+\left. \matrix{
+ x\ \&\ y \Rightarrow z \cr
+ \neg{x} \Rightarrow \neg{z} \cr
+ \neg{y} \Rightarrow \neg{z} \cr
+}
+\ \quad
+ \right\}
+\quad
+\matrix{
+ (z \lor \neg{x} \lor \neg{y}) \cr
+ (\neg{z} \lor x) \cr
+ (\neg{z} \lor y) \cr
+ }
+ \hskip 0.1\hsize
+\vcenter{\hbox{\epsfxsize=0.7cm\epsfbox{and.eps}}}
+$$
+
+\>Kdy¾ chceme pøevádìt obvodový SAT na 3-SAT, obvod nejdøíve pøelo¾íme na takový, ve~kterém jsou jen hradla {\csc and} a~{\csc not}, a~pak hradla tohoto obvodu pøelo¾íme na klauzule. Formule vzniklá z~takovýchto klauzulí je splnitelná pravì tehdy, kdy¾ je splnitelný daný obvod. Pøevod pracuje v polynomiálním èase.
+\qed
+
+\s{Poznámka:}
+Kdy¾ jsme zavádìli SAT, omezili jsme se jen na formule, které jsou
+v~konjunktivní normální formì (CNF). Teï u¾ víme, ¾e splnitelnost obecné
+booleovské formule doká¾eme pøevést na obvodovou splnitelnost a tu pak
+pøevést na 3-SAT. Opaèný pøevod je samozøejmì triviální, tak¾e obecný SAT
+je ve~skuteènosti ekvivalentní s~na¹ím \uv{standardním} SATem pro CNF.
+
+\bye
+
+
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-\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 soustavami o nìjakém základu $\neq$ 1 jsou co do velikosti
-zápisu polynomiální). Rozhodovací problém je tedy
-$f:\ \{0,1\}^{\ast} \to \{0,1\}$, to jest funkce 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 {\it pøevést}) na $B$ (pí¹eme $A \rightarrow B$) právì tehdy,
-kdy¾ 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 $\O(a^\ell)$, kde $a=|q|$; 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.
-Nyní se ji¾ podíváme na nìjaké zajímavé problémy. Obecnì to budou problémy, na které
-polynomiální algoritmus není znám, a vzájemnými pøevody zjistíme ¾e jsou stejnì tì¾ké.
-
-\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 $\psi$.
-Pozdìji uká¾eme, ¾e lze podniknout pøevod na takovou formuli $\chi'$ v~CNF, která sice není
-ekvivalentní s $\psi$ (pøibydou nám promìnné, a ne ka¾dý roz¹íøený 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 (pøi ka¾dém dìlení klauzule {\it jiná}
-nová promìnná).
-%kterou nastavíme tak, abychom neovlivnili splnitelnost formule.
-
-\>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 tím pádem polynomiální vùèi pùvodní:
-v ka¾dém kroku se pøidají jen dva literály, tedy celkem {\it èas na pøevod}$\cdot
-2$ nových.
-
-\>Platí-li:
-\itemize\ibull
-\:$\alpha \Rightarrow$ zvolíme $x = 0$ (zajistí splnìní druhé poloviny nové formule),
-\:$\beta \Rightarrow$ zvolíme $x = 1$ (zajistí splnìní první poloviny nové formule),
-\:$\alpha ,\beta / \lnot\alpha ,\lnot\beta \Rightarrow$ zvolíme $x = 0/1$ (je nám to
- jedno, celkové øe¹ení nám to neovlivní).
-\endlist
-
-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, jeho¾ nastavením se klauzuli
-rozhodneme splnit. Samozøejmì 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~ka¾dého z~$k$ 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 $k$ (v ka¾dém trojúhelníku zvolíme právì jednu z ohodnocených
-promìnných, nemù¾e se stát ¾e zvolíme vrcholy konfliktní hrany). 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:}
-
-\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
-Tímto vynutíme NzMnu $\geq k$, co¾ jsme pøesnì chtìli. Takovýto pøevod je zøejmì polynomiální.
-
-\s{Pøíklad matice:} Jako pøíklad pou¾ijeme nezávislou mno¾inu z ukázky nezávislé mno¾iny.
-Nech» jsou vrcholy grafu oèíslované zleva a zeshora. 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 vrchol mno¾iny bude vybrán
-vrchol $v_4$, proto na druhém 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 na tento problém pøevést 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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-\input lecnotes.tex
-
-\prednaska{11}{NP-úplné problémy}{\vbox{\hbox{(zapsali F. Kaèmarik, R. Krivák, D. Remi¹}
- \hbox{ Michal Kozák, Vojta Tùma)}}}
-
-Dosud jsme zkoumali problémy, které se nás ptaly na to, jestli nìco existuje.
-Napøíklad jsme dostali formuli a problém splnitelnosti se nás ptal, zda
-existuje ohodnocení promìnných takové, ¾e formule platí. Nebo v~pøípadì
-nezávislých mno¾in jsme dostali graf a èíslo $k$ a ptali jsme se, jestli
-v~grafu existuje nezávislá mno¾ina, která obsahuje alespoò~$k$ vrcholù.
-Tyto otázky mìly spoleèné to, ¾e kdy¾ nám nìkdo napovìdìl nìjaký objekt, umìli
-jsme efektivnì øíci, zda je to ten, který hledáme. Napøíklad pokud dostaneme
-ohodnocení promìnných logické formule, staèí jen dosadit a spoèítat, kde
-formule dá \<true> nebo \<false>. Zjistit, ¾e nìjaký objekt je ten, který
-hledáme, umíme efektivnì. Tì¾ké na tom je takový objekt najít. Co¾ vede
-k~definici obecných vyhledávacích problémù, kterým se øíká tøída problémù NP.
-Nadefinujeme si ji poøádnì, ale nejdøíve zaèneme tro¹ièku jednodu¹¹í tøídou.
-
-\s{Definice:} P je {\I tøída rozhodovacích problémù}, které jsou øe¹itelné
-v~polynomiálním èase. Jinak øeèeno, problém
-$L \in {\rm P} \Leftrightarrow \exists $ polynom $f$ a~$\exists$ algoritmus~$A$
-takový, ¾e $\forall x: L(x)=A(x)$ a $A(x)$ dobìhne v~èase $\O(f(x))$.
-
-Tøída P tedy odpovídá tomu, o èem jsme se shodli, ¾e umíme efektivnì øe¹it.
-Nadefinujme nyní tøídu NP:
-
-\s{Definice:} NP je {\I tøída rozhodovacích problémù} takových, ¾e $L \in {\rm NP}$ právì tehdy, kdy¾ $\exists $ problém
-$K\in{\rm P}$ a $\exists$ polynom $g$ takový, ¾e pro
-$\forall x$ platí $L(x)=1 \Leftrightarrow \exists $ nápovìda $ y: \vert y \vert \leq g(\vert x \vert)$ a souèasnì $K(x,y)=1$.
-
-\s{Pozorování:} Splnitelnost logických formulí je v~NP. Staèí si toti¾ nechat napovìdìt, jak
-ohodnotit jednotlivé promìnné a pak ovìøit, jestli je formule splnìna. Nápovìda je polynomiálnì
-velká (dokonce lineárnì), splnìní zkontrolujeme také v~lineárním èase. Odpovíme tedy ano právì
-tehdy, existuje-li nápovìda, která nás pøesvìdèí, èili pokud je formule splnitelná.
-
-\s{Pozorování:} Tøída P le¾í uvnitø NP.
-V~podstatì øíkáme, ¾e kdy¾ máme problém, který umíme øe¹it v~polynomiálním èase
-bez nápovìdy, tak to zvládneme v~polynomiálním èase i s~nápovìdou.
-
-Problémy z minulé pøedná¹ky jsou v¹echny v NP (napø. pro nezávislou
-mno¾inu je onou nápovìdou pøímo mno¾ina vrcholù deklarující nezávislost),
-o jejich pøíslu¹nosti do P ale nevíme nic.
-Brzy uká¾eme, ¾e to jsou v jistém smyslu nejtì¾¹í problémy v~NP.
-Nadefinujme si:
-
-\s{Definice:} Problém $L$ je NP-{\I tì¾ký} právì tehdy, kdy¾ je na~nìj pøevoditelný
-ka¾dý problém z~NP (viz definici pøevodù z minulé pøedná¹ky).
-
-Rozmyslete si, ¾e pokud umíme øe¹it nìjaký NP-tì¾ký problém v~polynomiálním èase,
-pak umíme vyøe¹it v~polynomiálním èase v¹e v~NP, a tedy ${\rm P}={\rm NP}$.
-
-My se budeme zabývat problémy, které jsou NP-tì¾ké a samotné jsou v~NP. Takovým problémùm se øíká NP-úplné.
-
-\s{Definice:} Problém $L$ je NP-{\I úplný} právì tehdy, kdy¾ $L$ je NP-tì¾ký a $L \in {\rm NP}$.
-
-NP-úplné problémy jsou tedy ve~své podstatì nejtì¾¹í problémy, které le¾í v~NP.
-Kdybychom umìli vyøe¹it nìjaký NP-úplný problém v~polynomiálním èase, pak
-v¹echno v~NP je øe¹itelné v~polynomiálním èase. Bohu¾el to, jestli nìjaký
-NP-úplný problém lze øe¹it v~polynomiálním èase, se neví. Otázka, jestli
-${\rm P}={\rm NP}$, je asi nejznámìj¹í otevøený problém v~celé teoretické
-informatice.
-
-Kde ale nìjaký NP-úplný problém vzít? K~tomu se nám bude velice hodit následující vìta:
-
-\s{Vìta (Cookova):} SAT je NP-úplný.
-
-\>Dùkaz je znaènì technický, pøibli¾nì ho naznaèíme pozdìji. Pøímým dùsledkem
-Cookovy vìty je, ¾e cokoli v~NP je pøevoditelné na SAT.
-K dokazování NP-úplnosti dal¹ích problémù pou¾ijeme následující vìtièku:
-
-\s{Vìtièka:} Pokud problém $L$ je NP-úplný a $L$ se dá pøevést na nìjaký problém $M\in{\rm NP}$, pak $M$ je také NP-úplný.
-
-\proof
-Tuto vìtièku staèí dokázat pro NP-tì¾kost, NP-úplnost plyne okam¾itì z~toho, ¾e
-problémy jsou NP-tì¾ké a le¾í v~NP (podle pøedpokladu).
-
-Víme, ¾e $L$ se dá pøevést na~$M$ nìjakou funkcí~$f$. Jeliko¾ $L$ je NP-úplný,
-pak pro ka¾dý problém $Q\in{\rm NP}$ existuje nìjaká funkce~$g$, která pøevede
-$Q$ na~$L$. Staèí tedy slo¾it funkci~$f$ s~funkcí~$g$, èím¾ získáme funkci pracující
-opìt v~polynomiálním èase, která pøevede~$Q$ na~$M$. Ka¾dý problém z~NP se tedy
-dá pøevést na problém~$M$.
-\qed
-
-\s{Dùsledek:} Cokoliv, na co jsme umìli pøevést SAT, je také NP-úplné.
-Napøíklad nezávislá mno¾ina, rùzné varianty SATu, klika v~grafu~\dots
-
-Jak taková tøída NP vypadá? Pøedstavme si v¹echny problémy tøídy NP, jakoby seøazené
-shora dolu podle obtí¾nosti problémù (tedy navzdor gravitaci), kde porovnání dvou
-problémù urèuje pøevoditelnost (viz obrázek).
-
-\figure{p-np.eps}{Struktura tøídy NP}{2.5cm}
-
-Obecnì mohou nastat dvì situace, proto¾e nevíme, jestli ${\rm P}={\rm NP}$.
-Jestli ano, pak v¹echno je jedna a ta samá tøída. To by bylo v nìkterých
-pøípadech nepraktické, napø. ka¾dá ¹ifra by byla jednodu¹e rozlu¹titelná.
-\foot{Poznámka o ¹ifrách -- libovolnou funkci vyèíslitelnou v polynomiálním
-èase bychom umìli v polynomiálním èase také invertovat.}
-Jestli ne, NP-úplné problémy urèitì nele¾í v P, tak¾e P a NP-úplné problémy
-jsou dvì disjunktní èásti NP. Také se dá dokázat (to dìlat nebudeme, ale je
-dobré to vìdìt), ¾e je¹tì nìco le¾í mezi nimi, tedy ¾e existuje problém, který
-je v~NP, není v~P a není NP-úplný (dokonce je takových problémù nekoneènì mnoho,
-v nekoneènì tøídách).
-
-\s{Katalog NP-úplných problémù}
-
-Uká¾eme si nìkolik základních NP-úplných problémù. O~nìkterých jsme to dokázali
-na~minulé pøedná¹ce, o~dal¹ích si to doká¾eme nyní, zbylým se na~zoubek podíváme
-na~cvièeních.
-
-\itemize\ibull
-\:{\I logické:}
- \itemize\ibull
- \:SAT (splnitelnost logických formulí v~CNF)
- \:3-SAT (ka¾dá klauzule obsahuje max.~3 literály)
- \:3,3-SAT (a navíc ka¾dá promìnná se vyskytuje nejvý¹e tøikrát)
- \:SAT pro obecné formule (nejen CNF)
- \:Obvodový SAT (není to formule, ale obvod)
- \endlist
-\:{\I grafové:}
- \itemize\ibull
- \:Nezávislá mno¾ina (existuje mno¾ina alespoò~$k$ vrcholù taková, ¾e ¾ádné dva nejsou propojeny hranou?)
- \:Klika (existuje úplný podgraf na~$k$ vrcholech?)
- \:3D párování (tøi mno¾iny se zadanými trojicemi, existuje taková mno¾ina disjunktních trojic, ve~které jsou v¹echny prvky?)
- \:Barvení grafu (lze obarvit vrcholy $k$~barvami tak, aby vrcholy stejné barvy nebyly nikdy spojeny hranou? NP-úplné u¾ pro~$k=3$)
- \:Hamiltonovská cesta (cesta obsahující v¹echny vrcholy [právì jednou])
- \:Hamiltonovská kru¾nice (kru¾nice, která nav¹tíví v¹echny vrcholy [právì jednou])
- \endlist
-\:{\I èíselné:}
- \itemize\ibull
- \:Batoh (nejjednodu¹¹í verze: dána mno¾ina èísel, zjistit, zda existuje podmno¾ina se zadaným souètem)
- \:Batoh -- optimalizace (podobnì jako u pøedchozího problému, ale místo mno¾iny èísel máme mno¾inu
- pøedmìtù s vahami a cenami, chceme co nejdra¾¹í podmno¾inu její¾ váha nepøesáhne zadanou kapacitu
- batohu)
- \:Loupe¾níci (lze rozdìlit danou mno¾inu èísel na~dvì podmno¾iny se stejným souètem)
- \:$Ax=b$ (soustava celoèíslených lineárních rovnic; $x_i$ mohou být pouze 0 nebo 1; NP-úplné i pokud $A_{ij}\in\{0,1\}$ a $b_i\in\{0,1\}$)
- \:Celoèíselné lineární programování (existuje vektor nezáporných celoèísených $x$ takový, ¾e $Ax \leq b$ ?)
- \endlist
-\endlist
-
-Nyní si uká¾eme, jak pøevést SAT na nìjaký problém. Kdy¾ chceme ukázat, ¾e na
-nìco se dá pøevést SAT, potøebujeme obvykle dvì vìci:
-konstrukci, která bude simulovat promìnné, tedy nìco, co nabývá dvou stavù
-\<true>/\<false>; a nìco, co bude reprezentovat klauzule a umí zaøídit, aby
-ka¾dá klauzule byla splnìna alespoò jednou promìnnou.
-Roz¹iøme tedy ná¹ katalog problémù o následující:
-\h{3D párování (3D matching)}
-
-\>{\I Vstup:} Tøi mno¾iny, napø. $K$ (kluci), $H$ (holky), $Z$ (zvíøátka) a
-mno¾ina $T$ kompatibilních trojic (tìch, kteøí se spolu snesou),
- tj. $T \subseteq K\times H\times Z$.
-
-\>{\I Výstup:} Perfektní podmno¾ina trojic $P\subseteq K\times H \times Z$ --
- tj. taková podmno¾ina trojic, ¾e $(\forall k\in K\ \exists !p\in P: k\in p)
- \wedge(\forall h\in H\ \exists !p\in P: h\in p)
- \wedge(\forall z\in Z\ \exists !p\in P: z\in p)$ -- tedy ka¾dý byl vybrán
- právì jednou.
-
-
-\h{ Uká¾eme, jak na 3D-párování pøevést 3,3-SAT }
-
-Uva¾ujme takovouto konfiguraci:
-
-\fig{3d.eps}{4cm}
-
-\>4 zvíøátka, 2 kluci, 2 dívky a 4 trojice, které oznaèíme $A, B, C, D$.
-Je¹tì pøedpokládáme, ¾e zvíøátka
- se mohou úèastnit nìjakých jiných trojic, ale
-tito ètyøi lidé se vyskytují pouze v~tìchto ètyøech trojicích a~nikde jinde.
-V¹imneme si, ¾e existují právì dvì mo¾nosti, jak tento obrázek spárovat.
-Abychom spárovali kluka $k_1$, tak si musíme vybrat $A$ nebo $B$. Kdy¾ si
-vybereme $A$, $k_1$ i $d_2$ u¾ jsou spárovaní tak¾e si nesmíme vybrat $B$ ani
-$D$. Pak jediná mo¾nost, jak spárovat $d_1$ a~$k_2$ je $C$. Jedna mo¾nost je
-tedy vybrat si $A$ a $C$ a jeliko¾ je obrázek symetrický, tak kdy¾ vybereme
-místo $A$ trojici $B$, dostaneme $B$ a~$D$. V¾dy si tedy vybereme dvì protìj¹í
-trojice v~obrázku.
-
-Tyto konfigurace budeme pou¾ívat k~reprezentaci promìnných. Pro ka¾dou
-promìnnou si nakreslíme takový obrázek a~to, ¾e $A$ bude spárované s~$C$, bude
-odpovídat tomu, ¾e $x=1$, a~spárování $B$ a~$D$ bude odpovídat
- $x=0$. Pokud jsme
-pou¾ili $A$ a~$C$, zvíøata se sudými èísly, tj. $z_2$ a~$z_4$, horní a~dolní,
-jsou nespárovaná a~pokud jsme pou¾ili $B$ a~$D$, zvíøátka $z_1$ a~$z_3$ zùstala
-nespárovaná. Pøes tato nespárovaná zvíøátka mù¾eme pøedávat informaci, jestli
-promìnná $x$ má hodnotu \<true> nebo \<false> do dal¹ích èástí vstupu.
-
-Zbývá vymyslet, jak reprezentovat klauzule. Klauzule jsou trojice popø. dvojice
-literálù, napø. $\kappa = (x \lor y \lor \lnot r)$, kde
-potøebujeme zajistit, aby $x$ bylo nastavené na $1$ nebo $y$ bylo nastavené
-na~$1$ nebo $r$ na $0$.
-
-\fig{klauzule.eps}{4cm}
-
-\>Pro takovouto klauzuli si poøídíme dvojici kluk-dívka, kteøí budou figurovat
-ve tøech trojicích se tøemi rùznými zvíøátky, co¾ mají být volná zvíøátka
-z~obrázkù pro pøíslu¹né promìnné (podle toho, má-li se promìnná vyskytnout
-s negací nebo ne). A~zaøídíme to tak, aby ka¾dé zvíøátko bylo
-pou¾ité maximálnì v~jedné takové trojici, co¾ jde proto, ¾e ka¾dý literál se
-vyskytuje maximálnì dvakrát a~pro ka¾dý literál máme dvì volná zvíøátka,
-z~èeho¾ plyne, ¾e zvíøátek je dost pro v¹echny klauzule. Pro dvojice se postupuje
-obdobnì.
-
-Je¹tì nám ale urèitì zbude $2p-k$ zvíøátek, kde $p$ je poèet promìnných, $k$
-poèet klauzulí -- ka¾dá promìnná vyrobí 4 zvíøátka, klauzule zba¹tí jedno
-a samotné ohodnocení 2 zvíøátka -- tak pøidáme je¹tì $2p-k$ párù
-kluk-dìvèe, kteøí milují
-v¹echna zvíøátka, a~ti vytvoøí zbývající trojice.
-
-Pokud formule byla splnitelná, pak ze splòujícího ohodnocení mù¾eme vyrobit
-párování s~na¹í konstrukcí. Obrázek pro ka¾dou promìnnou spárujeme podle
-ohodnocení, tj. promìnná je $0$ nebo $1$ a~pro ka¾dou klauzuli si vybereme
-nìkterou z~promìnných, kterými je ta klauzule splnìna. Funguje to ale
-i~opaènì: Kdy¾ nám nìkdo dá párovaní v~na¹í konstrukci, pak z nìho doká¾eme
-vyrobit splòující ohodnocení dané formule. Podíváme se, v~jakém stavu je
-promìnná, a~to je v¹echno. Z~toho, ¾e jsou správnì spárované klauzule, u¾
-okam¾itì víme, ¾e jsou v¹echny splnìné.
-
-Zbývá ovìøit, ¾e na¹e redukce funguje v~polynomiálním èase. Pro ka¾dou klauzuli
-spotøebujeme konstantnì mnoho èasu, $2p-k$ je také polynomiálnì mnoho a~kdy¾ v¹e
-seèteme, máme polynomiální èas vzhledem k~velikosti vstupní formule. Tím je
-pøevod hotový a~mù¾eme 3D-párování zaøadit mezi NP-úplné problémy.
-
-
-%RK
-
-
-\h{Náznak dùkazu Cookovy vìty}
-
-Abychom mohli budovat teorii NP-úplnosti, potøebujeme alespoò jeden problém,
-o kterém doká¾eme, ¾e je NP-úplný, z definice. Cookova vìta øíká o NP-úplnosti
-SAT-u, ale nám se to hodí dokázat o tro¹ku jiném problému -- {\I obvodovém SAT-u}.
-
-\>{\I Obvodový SAT} je splnitelnost, která nepracuje s~formulemi, ale s~booleovskými
-obvody (hradlovými sítìmi). Ka¾dá formule se dá pøepsat do booleovského obvodu,
-který ji poèítá, tak¾e dává smysl zavést splnitelnost i pro obvody. Na¹e obvody
-budou mít nìjaké vstupy a~jenom jeden výstup. Budeme se ptát, jestli se vstupy
-tohoto obvodu dají nastavit tak, abychom na výstupu dostali \<true>.
-
-\>Nejprve doká¾eme NP-úplnost {\I obvodového SAT-u} a~pak uká¾eme, ¾e se dá
-pøevést na obyèejný SAT v~CNF. Tím bude dùkaz Cookovy vìty hotový. Zaènìme
-s pomocným lemmatem.
-
-\s{Lemma:} Nech» $L$ je problém v P. Potom existuje polynom $p$ a algoritmus,
-který pro $\forall n \ge 0$ spoète v èase $p(n)$ hradlovou sí» $Bn$ s $n$ vstupy
-a 1 výstupem takovou, ¾e $\forall x \in \{ 0, 1 \}^n : Bn(x) = L(x)$.
-
-\proof
-Náznakem. Na základì zku¹eností z Principù poèítaèù intuitivnì chápeme poèítaèe
-jako nìjaké slo¾ité booleovské obvody, jejich¾ stav se mìní v~èase. Uva¾me tedy
-nìjaký problém $L \in {\rm P}$ a polynomiální algoritmus, který ho øe¹í. Pro vstup
-velikosti~$n$ dobìhne v~èase~$T$ polynomiálním v~$n$ a spotøebuje $\O(T)$ bunìk pamìti.
-Staèí nám tedy \uv{poèítaè s~pamìtí velkou $\O(T)$}, co¾ je nìjaký booleovský obvod
-velikosti polynomiální v~$T$, a~tedy i v~$n$. Vývoj v~èase o¹etøíme tak, ¾e sestrojíme~$T$
-kopií tohoto obvodu, ka¾dá z~nich bude odpovídat jednomu kroku výpoètu a bude
-propojena s~\uv{minulou} a \uv{budoucí} kopií. Tím sestrojíme booleovský obvod,
-který bude øe¹it problém~$L$ pro vstupy velikosti~$n$ a bude polynomiálnì velký
-vzhledem k~$n$.
-
-\s{Poznámka:}
-Pro dùkaz následující vìty si dovolíme drobnou úpravu v~definici tøídy NP.
-Budeme chtít, aby nápovìda
-mìla pevnou velikost, závislou pouze na~velikosti vstupu (tedy: $\vert y \vert
-= g(\vert x \vert)$ namísto $\vert y \vert \le g(\vert x \vert)$). Proè je taková
-úprava BÚNO? Jistì si dovedete pøedstavit,
-¾e pùvodní nápovìdu doplníme na po¾adovanou délku nìjakými \uv{mezerami}, které
-program ignoruje. (Tedy upravíme program tak, aby mu nevadilo, ¾e dostane na
-konci nápovìdy nìjak kódované mezery.)
-
-\s{Vìta:} Obvodový SAT je NP-úplný.
-
-\proof
-Máme tedy nìjaký problém $L$ z~NP a~chceme dokázat, ¾e $L$ se dá pøevést
-na~obvodový SAT (tj. NP-tì¾kost). Kdy¾ nám nìkdo pøedlo¾í nìjaký vstup $x$
-(chápeme jako posloupnost $x_1, x_2, \ldots, x_n$),
-spoèítáme velikost nápovìdy $g(n)$. Víme, ¾e kontrolní
-algoritmus~$K$ (který kontroluje, zda nápovìda je správnì) je v~P. Vyu¾ijeme
-pøedchozí lemma, abychom získali obvod, který pro konkrétní velikost vstupu
-$x$ poèítá to, co kontrolní algoritmus $K$. Na vstupu tohoto obvodu bude $x$
-(vstup problému $L$) a~nápovìda~$y$. Na výstupu nám øekne, jestli je nápovìda
-správná. Velikost vstupu tohoto obvodu bude tedy $p(g(n))$, co¾ je také polynom.
-
-\fig{kobvod.eps}{2.3cm}
-
-\>V tomto obvodu zafixujeme vstup $x$ (na místa vstupu dosadíme konkrétní hodnoty z $x$). Tím získáme obvod, jeho¾ vstup je jen $y$ a~ptáme se, zda za $y$ mù¾eme dosadit nìjaké hodnoty tak, aby na výstupu bylo \<true>. Jinými slovy, ptáme se, zda je tento obvod splnitelný.
-
-\>Pro libovolný problém z~NP tak doká¾eme sestrojit funkci, která pro ka¾dý vstup~$x$ v~polynomiálním èase vytvoøí obvod, který je splnitelný pravì tehdy, kdy¾ odpovìï tohoto problému na vstup $x$ má být \<true>. Tedy libovolný problém z~NP se dá
-v~polynomiálním èase pøevést na obvodový SAT.
-
-\>Obvodový SAT je v NP triviálnì -- staèí si nechat poradit vstup, sí»
-topologicky setøídit a v~tomto poøadí poèítat hodnoty hradel.
-\qed
-
-\s{Lemma:} Obvodový SAT se dá pøevést na 3-SAT.
-
-\proof
-Budeme postupnì budovat formuli v~konjunktivní normální formì. Ka¾dý booleovský obvod se dá pøevést v~polynomiálním èase na~ekvivalentní obvod, ve~kterém se vyskytují jen hradla {\sc and} a {\sc not}, tak¾e staèí najít klauzule odpovídající tìmto hradlùm. Pro ka¾dé hradlo v~obvodu zavedeme novou promìnnou popisující jeho výstup. Pøidáme klauzule, které nám kontrolují, ¾e toto hradlo máme ohodnocené konzistentnì.
-
-\>{\I Pøevod hradla \sc not}: na vstupu hradla budeme mít nìjakou promìnnou $x$ (která pøi¹la buïto pøímo ze~vstupu toho celého obvodu nebo je to promìnná, která vznikla na výstupu nìjakého hradla) a na výstupu promìnnou $y$. Pøidáme klauzule, které nám zaruèí, ¾e jedna promìnná bude negací té druhé:
-$$\matrix{ (x \lor y), \cr
- (\neg{x} \lor \neg{y}). \cr }
- \hskip 0.2\hsize
-\vcenter{\hbox{\epsfxsize=0.7cm\epsfbox{not.eps}}}
-$$
-
-\>{\I Pøevod hradla \sc and}: Hradlo má vstupy $x, y$ a~výstup $z$. Potøebujeme pøidat klauzule, které nám popisují, jak se má hradlo {\sc and} chovat. Tyto vztahy pøepí¹eme do~konjunktivní normální formy:
-$$
-\left. \matrix{
- x\ \&\ y \Rightarrow z \cr
- \neg{x} \Rightarrow \neg{z} \cr
- \neg{y} \Rightarrow \neg{z} \cr
-}
-\ \quad
- \right\}
-\quad
-\matrix{
- (z \lor \neg{x} \lor \neg{y}) \cr
- (\neg{z} \lor x) \cr
- (\neg{z} \lor y) \cr
- }
- \hskip 0.1\hsize
-\vcenter{\hbox{\epsfxsize=0.7cm\epsfbox{and.eps}}}
-$$
-
-\>Kdy¾ chceme pøevádìt obvodový SAT na 3-SAT, obvod nejdøíve pøelo¾íme na takový, ve~kterém jsou jen hradla {\sc and} a~{\sc not}, a~pak hradla tohoto obvodu pøelo¾íme na klauzule. Formule vzniklá z~takovýchto klauzulí je splnitelná pravì tehdy, kdy¾ je splnitelný daný obvod. Pøevod pracuje v polynomiálním èase.
-\qed
-
-\s{Poznámka:}
-Kdy¾ jsme zavádìli SAT, omezili jsme se jen na formule, které jsou
-v~konjunktivní normální formì (CNF). Teï u¾ víme, ¾e splnitelnost obecné
-booleovské formule doká¾eme pøevést na obvodovou splnitelnost a tu pak
-pøevést na 3-SAT. Opaèný pøevod je samozøejmì triviální, tak¾e obecný SAT
-je ve~skuteènosti ekvivalentní s~na¹ím \uv{standardním} SATem pro CNF.
-
-\bye
-
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projects / ads2.git / commitdiff
