From 18001dea1194eda81e26866947dee184dddb43ce Mon Sep 17 00:00:00 2001
From: Never Gude <n.gude@posteo.de>
Date: Mon, 18 May 2026 19:20:02 +0200
Subject: [PATCH] Aufgabe 1 korrektur und Overfull \hbox in 2 beheben

---
 übung_4/agt_übung_4.pdf          | Bin 55402 -> 64351 bytes
 übung_4/aufgabe_1.tex            |  26 +-
 übung_4/aufgabe_2.tex            |   6 +-
 übung_4/figures.pdf              | Bin 22397 -> 31881 bytes
 übung_4/figures.pdf.autosave.ipe | 877 +++++++++++++++++++++++++++++++
 5 files changed, 900 insertions(+), 9 deletions(-)
 create mode 100644 übung_4/figures.pdf.autosave.ipe

diff --git a/übung_4/agt_übung_4.pdf b/übung_4/agt_übung_4.pdf
index 555124b534f4d8b4f4b03803e62d4542bd5aa818..ad1eeed52dd7fb4a18f7578496ddae81504c2997 100644
GIT binary patch
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diff --git a/übung_4/aufgabe_1.tex b/übung_4/aufgabe_1.tex
index 62f2623..21957eb 100644
--- a/übung_4/aufgabe_1.tex
+++ b/übung_4/aufgabe_1.tex
@@ -11,23 +11,33 @@
 	\end{figure}
 \item
 	Das Problem kann als Maximalflussproblem modelliert werden. Dafür haben wir
-	eine Quelle $s$, eine Senke $t$ und Knoten $v_i$ mit $i \in \NN_{\leq n}$.
-	Die Eingangskapazität eines Knoten $v_i$ entspricht dem Eingangsgrad $e_i$.
-	Die Ausgangskapazität entspricht dem Ausgangsgrad $a_i$. Siehe \autoref{fig:task1b}.
+	eine Quelle $s$, eine Senke $t$ und Knoten $v_{i_{e/a}}$ mit $i \in \NN_{\leq n}$.
 
-	Wenn es einen vollständigen Fluss gibt, gibt es eine Lösung für das Problem,
+	Die Eingangskapazität eines Knoten $v_{i_a}$ entspricht dem Ausgangsgrad $a_i$.
+
+	Die Ausgangskapazität eines Knoten $v_{i_e}$ entspricht dem Eingangsgrad $e_i$.
+
+	Für die Kantenkapazität $c \in \NN$ einer Kante $\tup{v_{i_a}, v_{j_e}}$ mit $i, j \leq n$ gilt,
+	wenn $i = j$, sind Selbstkanten erlaubt, wenn $c > 0$.
+	Ausserdem sind Mehrfachkanten erlaubt, wenn $c > 1$.
+	Siehe \autoref{fig:task1b}.
+
+	Wenn es einen vollständigen Fluss gibt, also alle in $t$ eingehenden Kanten vollständig benutzt werden, gibt es eine Lösung für das Problem,
 	sonst nicht.
 	\points{4}
+
 	\begin{figure}
 		\centering
-		\includegraphics[page=2, width=0.5\textwidth]{figures.pdf}
+		\includegraphics[page=2, width=0.8\textwidth]{figures.pdf}
 		\caption{Das Problem als Maximalflussproblem.}
 		\label{fig:task1b}
 	\end{figure}
 \item
-	Die Modellierung ist korrekt, da der Eingangs- und Ausgangsgrad jedes Knotens $v_i$
-	von oben beschränkt wird von der Eingangskapazität $e_i$ bzw. Ausgangskapazität $a_i$.
+	Die Modellierung ist korrekt, da die Eingangs- und Ausgangsgrade der Knoten
+	$v_i$ im Graphen durch die Kantenkapazitäten $a_i$ und $e_i$ Fluss beschränkt werden.
 
-	Zudem erfüllt die Flusserhaltung den Zweck, dass die Summe der Eingangsgrade gleich der Summe der Ausgangsgrade sind, was in einem Graphen erfüllt sein muss.
+	Durch wählen der Kantenkapazitäten der Kanten $(v_{i_a}, v_{j_e})$ im Fluss
+	können Selbst- und Mehrfachkanten ein- oder ausgeschlossen werden.
+	Damit entsprechen diese Kanten im Fluss den Kanten im Graphen.
 	\points{2}
 \end{tasks}
diff --git a/übung_4/aufgabe_2.tex b/übung_4/aufgabe_2.tex
index 067063b..147143a 100644
--- a/übung_4/aufgabe_2.tex
+++ b/übung_4/aufgabe_2.tex
@@ -23,7 +23,11 @@ entspricht, da dann alle neuen von $s$ ausgehenden Kanten vollständig ausgelast
 	Also ist $f'$ ein zulässiger $s$-$t$-Fluss mit Wert $|f'|=\sum_{v:b(v)>0}b(v)$
 	\item Sei $f'$ ein $s$-$t$-Fluss in $G'$. Da sein Wert maximal ist, sind alle Kanten zu $s$ und $t$ voll ausgelastet.\\
 	Für die Einschränkung $f:=f'\vert_E$ betrachtet man die Flusserhaltung jedes Knotens $v\in V$ in $G'$:
-	$$\textnormal{Nettozufluss}_f(v)=\textnormal{Nettozufluss}_{f'}(v)-[\textnormal{Beitrag der neuen Kante]}=0-(-b(v))=b(v)$$
+	\begin{align*}
+		\textnormal{Nettozufluss}_f(v) &= \textnormal{Nettozufluss}_{f'}(v)-[\textnormal{Beitrag der neuen Kante]} \\
+		&= 0-(-b(v)) \\
+		&= b(v)
+	\end{align*}
 	Also ist $f$ ein zulässiger b-Fluss in $G$.
 \end{enumerate}
 \points{2}
diff --git a/übung_4/figures.pdf b/übung_4/figures.pdf
index 9236211972a91a785da05261a11e0cccf69e69f1..12bdfb090eb08d97355438f34660711bdbbe2b4d 100644
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+<text transformations="translations" pos="192 160" stroke="black" type="minipage" width="384" height="50.1936" depth="38.304" valign="top" size="script">\begin{description}
+\item[\color{dark purple}{Lila Kanten}] Kantenkapazität $1$ wenn keine Mehrfachkanten erlaubt sind.
+
+\item[\color{dark gray}{Graue Kanten}] Kantenkapazität $0$ wenn keine Selbstkanten erlaubt sind.
+\end{description}</text>
+</page>
+</ipe>