--- /dev/null
+\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~pøípade 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 zadal nìjaký objekt, umìli jsme efektivnì øíci, zda je to objekt, který hledáme. Napøíklad pokud dostaneme ohodnocení promìnných logické formule, staèí jen dosadit a spoèítat, ¾e formule je \<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. Definujeme 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 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 odpovídá tomu, o èem jsme se shodli, ¾e umíme efektivnì øe¹it. Nadefinujme tedy 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, tedy 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).
+
+Také platí, 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 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 $M\in{\rm NP}$ ($L \rightarrow M$), 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
+zhora nadolu 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.
+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 (mno¾ina alespoò~$k$ vrcholù taková, ¾e ¾ádné dva nejsou propojeny hranou)
+ \:Klika (úplný podgraf na~$k$ vrcholech)
+ \:3D párování (tøi mno¾iny se zadanými trojicemi, najít takovou mno¾inu disjunktních trojic, ve~které jsou v¹echny prvky)
+ \:Barvení grafu (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 váhami a cenami, chceme co nejdra¾¹í podmno¾inu její¾ váha nepøesáhne zadanou kapacitu
+ batohu)
+ \:Loupe¾níci (rozdìlit mno¾inu 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.
+\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 { Pøevoditelnost 3,3-SAT na 3D-párování }
+
+Najdeme si takovouto konfiguraci:
+
+\fig{3d.eps}{4cm}
+
+\>4 zvíøátka, 2 kluci, 2 dívky a~takové 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.
+
+Takovýto obrázek 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~$D$ odpovídá $x=0$. 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 tyto nespárovaná zvíøátka mù¾eme pøedávat informaci, jestli
+promìnná $x$ má hodnotu \<true> nebo \<false> do dal¹ích èástí grafu.
+
+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 páry.
+
+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 také 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 to
+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. 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» $B_n$ s $n$ vstupy a 1 výstupem takovou, ¾e $\forall x \in \{ 0, 1 \}^{n} : B_n(x) = L(x).$
+
+\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:}
+Je¹tì si dovolíme drobnou úpravu v~definici tøídy NP. Budeme chtít, aby nápovìda
+mìla pevnou velikost, závislou pouze na~velikosti vstupu (tedy: $\vert y \vert
+= g(\vert x \vert)$). 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 vektor $(x_1, x_2, \ldots, x_n)$),
+spoèítáme velikost nápovìdy $g(\vert x\vert)$. Víme, ¾e kontrolní
+algoritmus~$K$ (který kontroluje, zda nápovìda je správnì) je v~P. Vyu¾ijeme
+intuice o~obvodech, 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 $\vert x\vert + g(\vert x\vert)$, co¾ je 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 -- za nápovìdu staèí vzít ohodnocení vstupù, hradla topologicky setøídit a postupnì vyhodnocovat.
+\qed
+
+\s{Lemma:} Obvodový SAT se dá pøevést na 3-SAT.
+
+\proof
+Budeme postupnì budovat formuli v~konjunktivní normální formì. Pro ka¾dé hradlo v~obvodu zavedeme novou promìnnou popisující jeho výstup. Pøidáme klauzule, které nám kontrolují, ¾e toto hradlo máme ohodnocené konzistentnì. Ka¾dý booleovský obvod se dá pøevést na ekvivalentní obvod, ve~kterém se vyskytují jen hradla {\sc and} a {\sc not}, tak¾e staèí najít klauzule odpovídající tìmto hradlùm.
+
+\>{\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
--- /dev/null
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projects / ads2.git / commitdiff
