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Berechenbarkeit: Bin Packing, Partitionen

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Thomas Ba. 14 years ago
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60
Berechenbarkeits-KomplexTh/Berechenbarkeits-KomplexTh.tex
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Da im vorderen Teil mit den Füll-Zahlen maximal 2 erreicht werden kann, muss die Summe der ausgewählten $y-$ und $z-$Zahlen $\ge 1$ je Klauselspalte sein.\\
\(\Rightarrow\) nach Konstruktion enthält also jede Klausel $\ge 1$ erfülltes Problem.
\underline{Knapsack (Rucksack)}
Gegeben: \( s_1,…,s_n, \ , w_1,…,w_n, \ , s, w \)\\
Gefragt: \( \exists I \subseteq \{ 1,…,n\} \)\\
\( \sum\limits_{i\in I} s_i \le S\) und \( \sum\limits_{i\in I} w_i \ge W \)\\
\( \in NP \)\\
Knapsack ist $NP$-Vollständig. Subset Sum $\le^P$ Knapsack.
Sei \(a_1,…,a_n,b\) Eingabe für Subset Sum. Konstruiere Eingabe für Knapsack:\\
\( S_i = a_i \ , S=b\)\\
\( w_i = a_i \ , W=b\)\\
Dann ist \( \sum\limits_{i\in I} a_i = b \Leftrightarrow \sum\limits_{i\in I} a_i \le b \) und \( \sum\limits_{i\in I} a_i \ge b\)
\underline{Partition}
Gegeben: \( a_1,…a_n\)\\
Gefragt: \( \exists I \subseteq \{ 1,…,n\} \)\\
\( \sum\limits_{i\in I} a_i = \sum\limits_{i\not\in I} a_i\)\\
\( \in NP\)-Vollständig
Subset Sum \( \le^P\) Partition.
Sei \(a_1,a_2, …, a_n,b\) Eingae für Subset Sum.\\
Sei \( A = \sum\limits_{i=1}^n a_i\)\\
Sei \( I \subseteq \{1,…,n\}i \) mit \( \sum\limits_{i\in I} a_i = b\)
\includegraphics{bilder/partition.eps}
Dann ist \( \sum\limits_{i\in I} a_i + A-b = \sum\limits_{i\not\in I} a_i + b\)
D.\,h. Eingabe \( a_i,…,a_n,b,A-b \) für Partition hat Lösung.\\
\underline{Aber}: die hat immer die Lösung: \( \underbrace{\sum\limits_{i=1}^n a_i}_{=A} = \underbrace{b+(A-b)}_{=A} \)
Stattdessen: \( a_i,…,a_n, b+1, A-b+1 \)\\
Dann ist \( \sum\limits_{i\in I} a_i + (A-b+1) = \sum\limits_{i\not\in I} a_i + (b+1) \)
Aber \( \sum\limits_{i=1}^n a_i = A \not= (b+1) + (A-b+1) = A+2 \)
\('\Leftarrow'\) Sei $I$ eine Lösung für das Partitionsproblem $a_1,…,a_n,\underbrace{b+1}_{=a_{n+1}}, \underbrace{A-b+1}_{=a_{n+2}} $\\
Da \( (b+1) + (A-b+1) = A+2 > A\) kann nur eine von beiden Zahlen zu $I$ gehören. Sei o.\,B.\,d.\,A.\footnote{o.\,B.\,d.\,A.: ohne Beschränkung der Allgemeinheit} sei $n+2 \in I$ (sonst betrachte $\overline{I}$).
Es gilt aber:\\
\( \sum\limits_{i\in I} a_i + A-b+1 = \sum\limits_{i\not\in I} a_i+b+1 \)\\
\(\Rightarrow \sum\limits_{i\in I} a_i + \underbrace{A - \sum\limits_{i\not\in I} a_i}_{ =\sum\limits_{i\in I} a_i } = 2b \)\\
\(\Rightarrow \sum\limits_{i\in I} a_i = b\)
\underline{Bin Packing}
Gegeben: \( a_1,…,a_n,B,k\)\\
Gefragt: Kann man \(a_1,…,a_n\) auf k Bins der Größe B verteilen?
\bsp \(1,3,3,7,5,4\ ,B=5\)\\
\includegraphics{bilder/bin-packing.eps}\\
\( \in NP\)
Partition $\le^P$ Bin Packing.\\
Sei \( a_1,…,a_n \) Eingabe für Partition.\\
Konstruiere Eingabe für Bin Packing:\\
\( a_1,…,a_n, B= \frac A2, k=2 \) mit \( A = \sum\limits_{i=1}^n a_i \)
\end{document}

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Berechenbarkeits-KomplexTh/Makefile
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Berechenbarkeits-KomplexTh.pdf: Berechenbarkeits-KomplexTh.tex bilder/*.xml
for FILE in $(find bilder/ -iname '*.xml') ; do ipetoipe -eps -runlatex $FILE ${FILE%.xml}.eps ; done
pdflatex Berechenbarkeits-KomplexTh.tex

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<gridsize name="4 pts" value="4"/>
<gridsize name="8 pts (~3 mm)" value="8"/>
<gridsize name="16 pts (~6 mm)" value="16"/>
<gridsize name="32 pts (~12 mm)" value="32"/>
<gridsize name="10 pts (~3.5 mm)" value="10"/>
<gridsize name="20 pts (~7 mm)" value="20"/>
<gridsize name="14 pts (~5 mm)" value="14"/>
<gridsize name="28 pts (~10 mm)" value="28"/>
<gridsize name="56 pts (~20 mm)" value="56"/>
<anglesize name="90 deg" value="90"/>
<anglesize name="60 deg" value="60"/>
<anglesize name="45 deg" value="45"/>
<anglesize name="30 deg" value="30"/>
<anglesize name="22.5 deg" value="22.5"/>
<tiling name="falling" angle="-60" step="4" width="1"/>
<tiling name="rising" angle="30" step="4" width="1"/>
</ipestyle>
<page>
<layer name="alpha"/>
<view layers="alpha" active="alpha"/>
<path layer="alpha" stroke="black" fill="orange" fillrule="eofill" tiling="rising">
80 704 m
80 752 l
96 752 l
96 704 l
</path>
<path stroke="black" fill="seagreen" tiling="rising">
112 704 m
112 752 l
128 752 l
128 704 l
</path>
<text matrix="1 0 0 1 0 4" transformations="translations" pos="80 688" stroke="black" type="label" width="22.504" height="7.463" depth="10.45" valign="baseline">$\sum\limits_{i\in I} a_i$</text>
<text matrix="1 0 0 1 0 4" transformations="translations" pos="112 688" stroke="black" type="label" width="4.276" height="6.918" depth="0" valign="baseline">$b$</text>
<path stroke="black" fill="seagreen" tiling="rising">
160 704 m
160 768 l
176 768 l
176 704 l
</path>
<path stroke="black">
64 704 m
224 704 l
</path>
<path stroke="black" fill="orange" tiling="rising">
192 704 m
192 768 l
208 768 l
208 704 l
</path>
<text matrix="1 0 0 1 0 4" transformations="translations" pos="160 688" stroke="black" type="label" width="22.504" height="7.458" depth="11.36" valign="baseline">$\sum\limits_{i\not\in I} a_i$</text>
<text matrix="1 0 0 1 0 4" transformations="translations" pos="192 688" stroke="black" type="label" width="23.924" height="6.919" depth="0.83" valign="baseline">$A-b$</text>
</page>
</ipe>
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