From e2f7cc0614f5c80a64cff78ceee8165af57f7cc4 Mon Sep 17 00:00:00 2001 From: Martin Mares Date: Mon, 14 Jan 2008 14:55:52 +0100 Subject: [PATCH 1/1] Drobna vylepseni prevodu nezavisle mnoziny na SAT. --- 10-prevody/10-prevody.tex | 20 +- 10-prevody/matice.eps | 8892 ------------------------------------- 10-prevody/matice.svg | 130 - 3 files changed, 12 insertions(+), 9030 deletions(-) delete mode 100644 10-prevody/matice.eps delete mode 100644 10-prevody/matice.svg diff --git a/10-prevody/10-prevody.tex b/10-prevody/10-prevody.tex index 9e638dc..9e55361 100644 --- a/10-prevody/10-prevody.tex +++ b/10-prevody/10-prevody.tex @@ -64,7 +64,7 @@ v~p \>Splnitelnost logických formulí, tj. dosazení \ èi \ za promìnné v logické formuli tak, aby formule dala výsledek \. \>Zamìøíme se na speciální formu zadání formulí, {\I konjunktivní normální formu} (CNF). -$$(\ldots\lor\ldots\lor\ldots\lor\ldots) \land (\ldots\lor\ldots\lor\ldots\lor\ldots) \land \ldots $$ +$$(\ldots\lor\ldots\lor\ldots\lor\ldots) \land (\ldots\lor\ldots\lor\ldots\lor\ldots) \land \ldots $$ \>{\I Vstup:} Formule $\varphi$ v konjunktivní normální formì. @@ -76,7 +76,7 @@ $$(\ldots\lor\ldots\lor\ldots\lor\ldots) \land (\ldots\lor\ldots\lor\ldots\lor\l \:{\I formule} je zadána pomocí {\I klauzulí} oddìlených $\land$, \:ka¾dá {\I klauzule} je slo¾ená z {\I literálù} oddìlených $\lor$, \:ka¾dý {\I literál} je buïto promìnná nebo její negace. -\endlist +\endlist \>Uká¾eme, ¾e staèí vyøe¹it jednodu¹¹í problém 3-SAT. @@ -124,10 +124,10 @@ takov \s{Poznámka:} Ka¾dý graf má minimálnì jednu nezávislou mno¾inu, a tou je prázdná mno¾ina. Proto je potøeba zadat i minimální velikost hledané mno¾iny. \>Uká¾eme, jak na~tento probém pøevést 3-SAT. - + \s{Pøevod:} Z ka¾dé klauzule vybereme jeden literál tak, abychom v rùzných klauzulích nevybírali konfliktnì, tj. $x$ a $\lnot x$. -\s{Pøíklad:} +\s{Pøíklad:} $(x \lor y \lor z) \land (x \lor \lnot y \lor \lnot z) \land (\lnot x \lor \lnot y \lor p) $. \>Pro ka¾dou klauzuli sestrojíme graf (trojúhelník) a pøidáme \uv{konfliktní} hrany, tj. $x$ a $\lnot x$. @@ -163,20 +163,24 @@ M x_{i1} \lor x_{i2} \lor \ldots \lor x_{in}$. \endlist -\figure{matice.eps}{Výsledná matice}{3in} +\s{Pøíklad matice:} Jako pøíklad pou¾ijeme nezávislou mno¾inu z ukázky nezávislé mno¾iny. +Nech» jsou vrcholy grafu oèíslované zleva a ze zhora. Hledáme nezávislou mno¾inu velikosti $2$. +Matice pak bude vypadat následovnì: +$$ \pmatrix{1&0&0&0&0 \cr 0&0&0&1&0}$$ +\s{Vysvìtlení:} Jako první vrchol mno¾iny bude vybrán vrchol $v_1$, proto v prvním øádku a v prvním sloupci bude $1$. Jako druhý ($k$-tý) vrchol mno¾iny bude vybrán vrchol $v_4$, proto na druhém ($k$-tém) øádku a ve ètvrtém sloupci bude $1$. Na ostatních místech bude $0$. \h{4. problém: Klika} \>{\I Vstup:} Graf $G, k \in N$. \>{\I Výstup:} $\exists$ úplný podgraf grafu $G$ na $k$ vrcholech? -\figure{klika.eps}{Pøíklad kliky}{3in} +\figure{klika.eps}{Pøíklad kliky}{2in} \s{Pøevod:} Prohodíme v grafu $G$ hrany a nehrany $\Rightarrow$ hledání nezávislé mno¾iny. \s{Dùvod:} Pokud existuje úplný graf na $k$ vrcholech, tak v~\uv{invertovaném} grafu tyto vrcholy nejsou spojeny hranou, tj. tvoøí nezávislou mno¾inu. -\figure{doplnek_nm.eps}{Prohození hran a nehran}{3in} +\figure{doplnek_nm.eps}{Prohození hran a nehran}{2in} \h{5. problém: 3D párování (3D matching)} @@ -214,7 +218,7 @@ $$ (x_k \Rightarrow x_1). $$ -Tímto zaruèíme, ¾e v¹echny promìnné budou mít stejnou hodnotu. Navíc si lze v¹imnout, ¾e ka¾dý literál se vyskytuje nejvíce dvakrát. +Tímto zaruèíme, ¾e v¹echny promìnné budou mít stejnou hodnotu. Navíc si lze v¹imnout, ¾e ka¾dý literál se vyskytuje nejvíce dvakrát. \s{Závìr:} Obrázek ukazuje problémy, jimi¾ jsme se dnes zabývali, a vztahy mezi tìmito problémy. \figure{prevody.eps}{Pøevody mezi problémy}{3in} diff --git a/10-prevody/matice.eps b/10-prevody/matice.eps deleted file mode 100644 index 9837ffb..0000000 --- a/10-prevody/matice.eps +++ /dev/null @@ -1,8892 +0,0 @@ -%!PS-Adobe-3.0 EPSF-3.0 -%%Creator: 0.45pre1 -%%Pages: 1 -%%Orientation: Portrait -%%BoundingBox: 0 0 208 185 -%%HiResBoundingBox: -3e-007 5.278125e-006 207.3569 184.78833 -%%EndComments -%%Page: 1 1 -0 185 translate -0.8 -0.8 scale -0 0 0 setrgbcolor -[] 0 setdash -1 setlinewidth -0 setlinejoin -0 setlinecap -gsave [1 0 0 1 0 0] concat -gsave [1 0 0 1 -77.384766 -40.448204] concat -gsave -0 0 0 setrgbcolor -newpath -83.15918 74.729454 moveto -83.15918 84.476524 lineto -83.159174 85.208948 83.1914 85.676232 83.255859 85.87838 curveto -83.320306 86.080529 83.455072 86.246056 83.660156 86.374962 curveto 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