From 5fcfe1a176352dc85a4bc25e0321b1d900b07717 Mon Sep 17 00:00:00 2001 From: Martin Mares Date: Mon, 7 Jan 2008 09:06:18 +0100 Subject: [PATCH] Pridana kapitola o NP-uplnosti. --- 11-np/11-np.tex | 153 +++++++++++ 11-np/3d.eps | 641 +++++++++++++++++++++++++++++++++++++++++++++ 11-np/3d.svg | 314 ++++++++++++++++++++++ 11-np/Makefile | 3 + 11-np/and.eps | 266 +++++++++++++++++++ 11-np/and.svg | 141 ++++++++++ 11-np/klauzule.eps | 483 ++++++++++++++++++++++++++++++++++ 11-np/klauzule.svg | 250 ++++++++++++++++++ 11-np/kobvod.eps | 538 +++++++++++++++++++++++++++++++++++++ 11-np/kobvod.svg | 162 ++++++++++++ 11-np/not.eps | 170 ++++++++++++ 11-np/not.svg | 121 +++++++++ 11-np/p-np.eps | 350 +++++++++++++++++++++++++ 11-np/p-np.svg | 213 +++++++++++++++ all/Makefile | 2 +- 15 files changed, 3806 insertions(+), 1 deletion(-) create mode 100644 11-np/11-np.tex create mode 100644 11-np/3d.eps create mode 100644 11-np/3d.svg create mode 100644 11-np/Makefile create mode 100644 11-np/and.eps create mode 100644 11-np/and.svg create mode 100644 11-np/klauzule.eps create mode 100644 11-np/klauzule.svg create mode 100644 11-np/kobvod.eps create mode 100644 11-np/kobvod.svg create mode 100644 11-np/not.eps create mode 100644 11-np/not.svg create mode 100644 11-np/p-np.eps create mode 100644 11-np/p-np.svg diff --git a/11-np/11-np.tex b/11-np/11-np.tex new file mode 100644 index 0000000..346be51 --- /dev/null +++ b/11-np/11-np.tex @@ -0,0 +1,153 @@ +\input lecnotes.tex + +\prednaska{11}{NP-úplné problémy}{(zapsali F. Kaèmarik, R. Krivák, D. Remi¹)} + +Dosud jsme zkoumali problémy, které se nás ptali na to, jestli nìco existuje. Napøíklad: dostali jsme 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 je veliká alespoò $k$? Tyto otázky mìli spoleèné to, ¾e kdy¾ nám nìkdo zadal nìjaký objekt, pak jsme umìli 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. Definujme 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 $\leq f(x)$ + +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. + +Máme-li lineárnì velké nápovìdy, co¾ jsou ohodnocení promìnných zadané formule, odpovíme ano, formule je splnitelná, pokud nám nìkdo mù¾e odpovìdìt na splòující ohodnocení. Tak¾e $K$ nám ovìøí, èi dosazením ohodnocení je formule splnìna a ptáme se tedy, èi existuje nápovìda taková, ¾e existuje ohodnocení takové, ¾e formule je splnìna. Splnitelnost logických formulí je urèitì v~tøídì NP. +V¹imneme si, ¾e celá 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. + +Nasbírali jsme problémy, které jsou v~NP, ale nevíme, jestli jsou v~P. Dokonce by se dalo øíct, ¾e jsou to nejtì¾¹í problémy v~NP. +Nadefinujme si: + +\s{Definice:} Problém $L$ je NP-tì¾ký právì tehdy, kdy¾ je pøevoditelný: +$\forall M \in \rm{NP}: M~\rightarrow~L$. +Také platí, ¾e pokud umíme øe¹it nìjaký NP problém v~polynomiálním èase, pak umíme vyøe¹it v~polynomiálním èase v¹e v~NP, +pokud nìjaké takové~$\in~\rm{P} \Rightarrow \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 úplný právì tehdy, kdy¾ $L$ je NP-tì¾ký a $L \in \rm{NP}$. + +NP-úplné problémy jsou ve své podstatì nejtì¾¹í problémy, které le¾í v~NP. Proto¾e le¾í v~NP platí, ¾e libovolný problém z NP umíme pøevést na nì. Kdybychom umìli pøevést nìjaký NP-úplný problém, který je øe¹itelný v~polynomiálním èase, pak v¹echno v~NP je øe¹itelné v~polynomiálním èase. Bohu¾el, to se neví. Otázka jestli $\rm{P}=\rm{NP}$ je asi nejznámìj¹í otevøený problém v~celé teoretické informatice. + +Uká¾eme si nìjaký NP-úplný problém. Velmi se nám bude hodit následující vìta: + +\s{Vìta (Cookova):} SAT je NP-úplný. + +\>Dùkaz je pøíli¹ technický, jenom ho pøibli¾nì naznaèíme pozdìji. Pøímím dùsledkem je, ¾e cokoli v~NP je pøevoditelné na SAT. +Dá se dokázat vìtièka: + +\s{Vìtièka:} $L$ je NP-úplný a $L$ se dá pøevést na $M$ ($L \rightarrow M$), $M \in \rm{NP} \Rightarrow M$ 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 $M$ a také z toho, ¾e $L$ je NP-úplný plyne, ¾e ka¾dý problém $Q$ z $NP$ se $Q$ dá pøevést na $L$. $Q$ se dá pøevést na $L$, $L$ se dá pøevést na $M$. Z toho plyne, ¾e $Q$ se dá pøevést na $M$. Staèí tedy slo¾it funkci $f(Q \rightarrow L)$ s~funkcí $g(L \rightarrow M)$. To urèitì také spoèteme v~polynomiálním èase. Tak nahlédneme, ¾e v¹echny problémy z $NP$ se dají pøevést na problém $M$. +\qed + +\s{Dùsledek:} Cokoliv, na co jsme umìli pøevést SAT je také NP-úplné. Napøíklad nezávislá mno¾ina, v¹echny varianty SATu, klika v~grafu... + +\figure{p-np.eps}{Obrázek 11.1}{2.5cm} + +\s{Katalog NP-úplných problémù:} +\itemize\ibull +\:{\I logické:} +SAT, 3-SAT, 3,3-SAT, obvodový SAT, SAT pro obecné formule +\:{\I grafové:} +Nezávislá mno¾ina, klika v~grafu, 3D párování, 3-barvení, Hamiltonova cesta/kru¾nice +\:{\I èíselné:} +problém batohu, loupe¾níci, $Ax=b$, celoèíselné lineární programování +\endlist + +\h { Pøevoditelnost 3,3-SAT na 3D-párování } + +Kdy¾ chceme ukázat, ¾e na nìco se dá pøevést SAT, potøebujeme obvykle dvì vìci. Konstrukci, která bude simulovat promìnné, nìco co nabývá dvou stavù $true$/$false$ a nìco co bude reprezentovat klauzule a umí zaøídit, aby ka¾dá klauzule byla splnìna alespoò jednou promìnnou. +Jenom pro pøipomenutí, máme mno¾inu klukù, dìvèat, zvíøátek a nìjaké trojice, kdo se s~kým snese a chceme vybrat trojice tak, aby se v~nich ka¾dý kluk, holka, zvíøátko vyskytovalo právì jednou. +Najdeme si takovouto konfiguraci: + +\figure{3d.eps}{Obrázek 11.2}{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 $B$, tak dostaneme $B$ a~$D$. V¾dy si 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 budou vypadat jako trojice literálù: +$\kappa = (x \lor y \lor \lnot r) $ +Potøebujeme zajistit, aby $x$ bylo nastavené na $1$ nebo $y$ bylo nastavené na $1$ nebo $r$ na $0$. + +\figure{klauzule.eps}{Obrázek 11.3}{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~obrázku pro pøíslu¹né promìnné. 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. Pak nám ale urèitì zbude $2p-k$ zvíøátek, kde $p$ je poèet promìnných, 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, pak 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 vety} + +Abychom mohli budovat teorii NP-úplnosti, tak 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~boolovskými obvody. Ka¾dá formule se dá pøepsat do boolovského obvodu, který ji poèítá, tak ¾e dává smysl zavést splnitelnost 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 $1$. + +\>Nejprve doká¾eme NP-úplnost {\I obvodového SAT-u} a~pak uká¾eme, ¾e se dá pøevést na formulový SAT. Tím bude dùkaz Cookovy vìty hotový. + +\s{Vìta:} Obvodový SAT je NP-úplný. + +\proof +Náznak. Na základì zku¹eností principu poèítaèù intuitivnì chápeme, ¾e pokud nìjaký problém $L \in \rm{P}$, pak existuje polynomiálnì velký boolovský obvod, který poèítá $L$. + +\>Dovolíme si drobnou úpravu v~definici tøídy NP. Budeme chtít, aby nápovìda byla pravì tak velká jako velikost 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.) + +\>Máme nìjaký problém $L$ z~NP a~chceme dokázat, ¾e $L$ se dá pøevést na obvodový SAT. Kdy¾ nám nìkdo pøedlo¾í nìjaký vstup $x$ (chápeme jako vektor $(x_1, x_2, \dots)$), spoèítáme velikost nápovìdy $g(\vert x\vert)$. Víme, ¾e kontrolní algoritmus $K$ (který kontroluje, zda nápovìda je správnì) je v~P. Vyu¾ijeme intuice v~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 to nápovìda, která k~tomu patøí nebo nepatøí. Velikost vstupu tohoto obvodu bude tedy $\vert x\vert + g(\vert x\vert)$, co¾ je polynom. + +\figure{kobvod.eps}{Obrázek 11.4 - Obvod pro kontrolní algoritmus $K$}{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 byla 1. Jinými slovy, ptáme se, zda je tento obvod splnitelný. + +\>Pro libovolný problém z~NP tak doká¾eme vyrobit 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 jedna. Tedy libovolný problém z~NP se dá +v~polnomiálním èase pøevést na obvodový SAT. +\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 boolovský obvod se dá pøevést na ekvivalentní obvod, v~kterém se vyskytují jen hradla {\sc and} a {\sc not}, tak ¾e 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: +$$\matrix{ (x \lor y), \cr + (\neg{x} \lor \neg{y}). \cr }$$ +\figure{not.eps}{Obrázek 11.5 - Hradlo \sc not}{0.8cm} + +\>{\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: +$$ +\left. \matrix{ + x\ \&\ y \Rightarrow z \cr + \neg{x} \Rightarrow \neg{z} \cr + \neg{y} \Rightarrow \neg{z} \cr +} +\ \quad + \right\} +\quad +\matrix{ + (z \lor \neg{x} \lor \neg{y}) \cr + (\neg{z} \lor x) \cr + (\neg{z} \lor y) \cr + } $$ +\figure{and.eps}{Obrázek 11.6 - Hradlo \sc and}{0.9cm} + +\>Kdy¾ chceme pøevádìt obvodový SAT na 3-SAT, tak ten obvod nejdøíve pøelo¾íme na obvod, v~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. +\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 +boolovské formule doká¾eme pøevést na obvodovou splnitelnost a tu pak +pøevést na 3-SAT. 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+X:=$(shell for a in 1 2 3 4 5 6 7 8 9 10 11 12 ; do echo ../$$a-*/$$a-*.tex ; done) %universe: all ChangeLog -- 2.39.5