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zDkk<{u9j*1rxvv?W8aBSZbqicE?8PesPq?TDA$z4o&=`ka)fwwyc z$6k#rg+F{gB6e|8VQgMODd~2ZCJzj~>e}eqcyY_{TJ?e4F`iBn*w5uC5FxM-6Zssw zoNfSb+>^z4EYa7FmC~d&X85xrmNu{N1!3D^EOaqYumzez`FrF0%;{LU+%ng1#^{=$ zb*UT8&xE=WHfi}(oxDBo(!OALnQL%W=*H7eqHuOSDce5CI4X|5P5lD)jyvq|m4=%M z&ZZG6467Zf1Q5-bcQ9=zl@81*N z^uz?$QJ=n!TP=Wxq>+9(ZBbx@Mrg|Q3vAxyKs`_O{-XUIm*PrwIcFpi2Z&1p2)MY^ Ibkqs{1DGuQZU6uP diff --git a/übung_6/aufgabe_3.tex b/übung_6/aufgabe_3.tex index dd21680..f13f25d 100644 --- a/übung_6/aufgabe_3.tex +++ b/übung_6/aufgabe_3.tex @@ -24,13 +24,33 @@ \points{3} \item + \points{2} + \item + Sei M das vom Algorithmus berechnete maximale Matching und M* das optimale maximale Matching. \\ + Für jede Kante $e*=\{u,v\}\in$ M* gilt: mindestens einer der beiden Knoten $u$ oder $v$ muss über eine Kante aus M abgedeckt werden, sonst wäre der M nicht nicht-erweiterbar. \\ + Eine Kante aus M hat genau zwei Endknoten und kann daher höchstens zwei verschiedene Kanten aus M* ''blockieren''. Da alle Kanten in M* disjunt sind kann man daraus folgern: + $$ |M*|\leq 2\cdot |M|\quad \Rightarrow\quad |M|\geq \frac{1}{2}|M*|$$ + Somit ist der Algorithmus aus der Teilaufgabe a) eine 1/2 Approximation für ein optimales Matching. \points{3} \item - \points{2} - \item + Zielfunktion: + $$ \arg\min \sum_{e\in E} x_e \geq 1$$ + Entscheidungsvariablen: für jede Kante $e\in E$: + $$x_e\in\{0,1\}\quad \forall e\in E$$ + Die Variable nimmt den Wert 1 an, wenn die Kante im Matching M ist und 0 falls sie das nicht ist.\\ + Nebenbedingungen: + \begin{enumerate} + \item Matching: Jeder Knoten darf von maximal einer Matching-Kante berührt werden + $$\forall v\in V \colon \sum_{e\in \delta(v)}x_e\leq1$$ + $\delta(v)$ ist die Menge aller Kanten welche an $v$ anliegen + \item Nicht-Erweiterbarkeit: Für jede Kante $\{u,v\}$ muss die Summe der Matching-Kanten an $u$ und $v$ mindestens 1 sein + $$ \forall \{u,v\}\in E \colon \sum_{e\in\delta(u)}x_e+\sum_{e'\in\delta(v)}x_e'\geq 1$$ + $\delta(v)$ ist die Menge aller Kanten welche an $v$ anliegen. Äquivalent für $\delta(u)$. + \end{enumerate} + \points{3} \item