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z&s|q!HPeQ*7st9c!EefiE3ZL6Qwu~t4CZu~_{K*ffpObMc`G4?+5 z=bsS=s=N3wu0=(3Mx}EGSUWq4b$+~^73LY^dy$x?DzCwiqH?fvv-1LIXk>u0fd2uB CYR2jS diff --git a/zusammenfassung.tex b/zusammenfassung.tex index 0a22f35..b209600 100644 --- a/zusammenfassung.tex +++ b/zusammenfassung.tex @@ -25,8 +25,8 @@ %\usepackage{datetime} %\usepackage{xcolor} -\DeclareMathOperator{\opt}{\mathrm{OPT}} -\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} +\DeclareMathOperator{\opt}{OPT} +\DeclarePairedDelimiter\abs{\lvert}{\rvert} \setlist[enumerate,1]{label={\arabic*.}} \setlist[enumerate,2]{label={\alph*)}} @@ -44,8 +44,8 @@ \tableofcontents \clearpage -\section{Grundlegende Definitionen} -\subsection{Kombinatorisches Optimierungsproblem $\phi$} +\section{Kombinatorisches Optimierungsproblem $\phi$} +\subsection{Definition} \begin{align*} \mathcal{D} &= \text{Menge der Eingaben $I$}\\ \mathcal{S}(I \in \mathcal{D}) &= \text{Menge der zur Eingabe $I$ zulässigen Lösungen}\\ @@ -62,10 +62,34 @@ Gesucht ist zu $I \in \mathcal{D}$ eine zulässige Lösung $\sigma_\mathrm{opt} \opt(I) = f(\sigma_\mathrm{opt}) = \mathrm{ziel}\{f(\sigma) \mid \sigma \in \mathcal{S}(I) \} \end{equation*} -\subsubsection{Beispiele} +\subsection{Beispiele} TODO: TSP, Rucksackproblem, etc. -\subsection{$t(n)$-Zeit-Approximationsalgorithmus $A$} -Für Eingabe $I \in \mathcal{D}$ berechnet $A$ in Zeit $t(\abs I \abs)$ eine Ausgabe $\sigma_I^A \in \mathcal{S}(I)$. Es gilt die Schreibweise $A(I) = f(\sigma_I^A)$. +\section{$t(n)$-Zeit-Approximationsalgorithmus $A$} +Für Eingabe $I \in \mathcal{D}$ berechnet $A$ in Zeit $t(\abs{I})$ eine Ausgabe $\sigma_I^A \in \mathcal{S}(I)$. Es gilt die Schreibweise $A(I) = f(\sigma_I^A)$. +\section{Konstante Gütegarantie} +\subsection{Definition} +\begin{itemize} + \item $A$ hat bei Eingabe $I$ absolute Güte von + \begin{equation*} + \kappa_A(I) = \abs{A(I) - \opt(I)} + \end{equation*} + \item Die absolute Worst-Case-Güte von $A$ abhängig von der Eingabelänge $n = \abs{I}$ ist die Funktion + \begin{equation*} + \kappa_A^{\mathrm{wc}}(n) = \max\{\kappa_A(I) \mid I \in \mathcal{D}, \abs{I} <= n\} + \end{equation*} + \item $A$ garantiert eine absolute Güte von $\kappa_A: \mathbb{N} \mapsto \mathbb{N}$, falls für alle $n \in \mathbb{N}$ gilt: + \begin{equation*} + \kappa_A^{\mathrm{wc}} \le \kappa_A(n) + \end{equation*} + \item $A$ hat eine absolute Abweichung von $\kappa'_A: \mathbb{N} \mapsto \mathbb{N}$, falls für unendlich viele $n$ gilt + \begin{equation*} + \kappa'_A(n) \le \kappa_A^{wc}(n) + \end{equation*} + Eine unendlich große Menge $\mathcal{D}' \subseteq \mathcal{D}$ heißt $\kappa'_A(n)$-Zeugenmenge gegen $A$, wenn für alle $I \in \mathcal{D}'$ gilt: + \begin{equation*} + \kappa_A(I) \ge \kappa'_A(\abs{I}) + \end{equation*} +\end{itemize} \end{document}