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-273,6 +273,23 @@ Weiter lassen sich damit obere bzw. untere Schranken der Optimallösung aus eine Der Ansatz der Konstruktion von Probleminstanzen anderer $NP$-schwerer Probleme und der anschließenden Verwendung eines Scaling-Arguments kann auch für weitere Probleme verwendet werden. Ebenfalls können damit bestimmte Bereiche für mögliche konstante relative Gütegarantien ausgeschlossen werden, etwa $\rho < \frac{3}{2}$ bei \problem{BinPacking}. +\section{Pseudo-polynomielle Algorithmen} +Sei $\Pi$ ein kombinatorisches Optimierungsproblem, sodass in allen Instanzen $I$ alle vorkommenden Zahlen natürliche Zahlen sind. Sei $\mathrm{maxnr}(I)$ die größte in $I$ vorkommende Zahl. Ein Algorithmus wird als pseudo-polynomiell bezeichnet, falls es ein Polynom $\mathrm{poly}(\cdot, \cdot)$ gibt, sodass für alle Instanzen $I$ seine Laufzeit $\mathrm{poly}(\abs{I}, \mathrm{maxnr}(I))$ ist. + +Kann also das Problem so eingeschränkt werden, dass für alle Instanzen $I$ die größte vorkommende Zahl durch ein Polynom begrenz wird, also $\mathrm{maxnr}(I) \le q(\abs{I})$ mit Polynom $q$ gilt, so ist auch die Laufzeit des Algorithmus polynomiell. + +\section{Starke NP-Vollständigkeit} +\subsection{Definition} +Ein NP-vollständiges Entscheidungsproblem $L$ wird als stark NP-vollständig bezeichnet, wenn es ein Polynom $q$ gibt, sodass $L_q = \{x \mid x \in L, \mathrm{maxnr}(x) \le q(\abs{x})\}$ NP-vollständig ist. Gibt es kein solches Polynom, gilt $L$ als schwach NP-vollständig. + +Eine äquivalente Charakterisierung ist: Das NP-vollständige Entscheidungsproblem $L$ ist stark vollständig, falls es keinen pseudo-polynomiellen Algorithmus für $L$ gibt (unter der Annahme $P \neq NP$). +\subsection{Beispiele} +\begin{itemize} + \item \problem{Hamilton} und \problem{Clique} sind stark NP-vollständig, da bei ihnen das Polynom $q(n) = n$ ausreicht. + \item \problem{TSP} ist stark NP-vollständig, da sogar das $\Delta\problem{TSP}$ mit Gewichten 1 und 2 NP-vollständig ist. + \item \problem{Rucksack} ist schwach NP-vollständig, weil es einen pseudopolynomiellen Algorithmus für die Optimierungsvariante gibt, der auch für das Entscheidungsproblem verwendet werden kann. +\end{itemize} + \section{Approximationsschemata} Während die Ergebnisse von Approximationsalgorithmen mit absoluter oder relativer Gütegarantie nur durch eine Modifikation oder Wechsel des Algorithmus verbessert werden können, ist es manchmal gewünscht, im Gegenzug für eine verlängerte Laufzeit eine bessere Güte zu erreichen. Dafür sind sogenannte Approximationsschemata geeignet. @@ -298,10 +315,13 @@ Sei $\Pi$ ein Optimierungsproblem und $A$ ein Approximationsalgorithmus für $\P Da die Werte zulässiger Lösungen immer ganzzahlig sind, folgt $\abs{\opt(I) - A(I, \varepsilon^*)} = 0$, damit also die Optimalität von $A(I, \varepsilon^*)$. \end{proof} -\section{Pseudo-polynomielle Algorithmen} -Sei $\Pi$ ein kombinatorisches Optimierungsproblem, sodass in allen Instanzen $I$ alle vorkommenden Zahlen natürliche Zahlen sind. Sei $\mathrm{maxnr}(I)$ die größte in $I$ vorkommende Zahl. Ein Algorithmus wird als pseudo-polynomiell bezeichnet, falls es ein Polynom $\mathrm{poly}(\cdot, \cdot)$ gibt, sodass für alle Instanzen $I$ seine Laufzeit $\mathrm{poly}(\abs{I}, \mathrm{maxnr}(I))$ ist. - -Kann also das Problem so eingeschränkt werden, dass für alle Instanzen $I$ die größte vorkommende Zahl durch ein Polynom begrenz wird, also $\mathrm{maxnr}(I) \le q(\abs{I})$ mit Polynom $q$ gilt, so ist auch die Laufzeit des Algorithmus polynomiell. +\subsection{Unmöglichkeitsergebnisse für Approximationschemata} +\begin{satz} + Sei $\Pi$ ein Optimierungsproblem. Wenn es ein Polynom $q(\cdot, \cdot)$ gibt, sodass $\forall I \in \mathcal{D}: \opt(I) \le q(\abs{I}, \mathrm{maxnr}(I))$ gilt, dann folgt aus der Existenz eines FPAS für $\Pi$, dass ein pseudo-polynomieller exakter Algorithmus für $\Pi$ existiert. +\end{satz} +\begin{satz} + Wenn es für eine Optimierungsvariante eines stark NP-vollständigen Problems ein FPAS gibt, dann folgt $P = NP$. +\end{satz} \section{\algo{GreedyIS} für \problem{IS}} \begin{algorithmic}[]