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+%\usetikzlibrary{quotes} +%\usetikzlibrary{angles} +%\usetikzlibrary{babel} +%\usetikzlibrary{fit} +%\usepackage{datetime} +%\usepackage{xcolor} + +\DeclareMathOperator{\opt}{\mathrm{OPT}} +\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} + +\setlist[enumerate,1]{label={\arabic*.}} +\setlist[enumerate,2]{label={\alph*)}} + +\title{Approximationsalgorithmen} +\author{Marco Ammon} +\date{\today} + +\makeatletter +\g@addto@macro\bfseries{\boldmath} +\makeatother + +\begin{document} +\maketitle + +\tableofcontents +\clearpage +\section{Grundlegende Definitionen} +\subsection{Kombinatorisches Optimierungsproblem $\phi$} +\begin{align*} + \mathcal{D} &= \text{Menge der Eingaben $I$}\\ + \mathcal{S}(I \in \mathcal{D}) &= \text{Menge der zur Eingabe $I$ zulässigen Lösungen}\\ + f: \mathcal{S}(I) \mapsto \mathbb{N}^{\neq 0} &= \text{Bewertungs-/Kosten-/Maßfunction}\\ + \mathrm{ziel} \in \{\min, \max\} +\end{align*} +\begin{itemize} + \item Beschränkung auf natürliche Zahlen, weil Vergleich reeller Zahlen bislang nicht beweisbar schnell funktioniert. + \item Ausschluss der 0 für spätere Definitionen sinnvoll (lässt sich durch Modifikation von $f$ in der Regel trivial erreichen) +\end{itemize} + +Gesucht ist zu $I \in \mathcal{D}$ eine zulässige Lösung $\sigma_\mathrm{opt} \in \mathcal{S}(I)$, sodass +\begin{equation*} + \opt(I) = f(\sigma_\mathrm{opt}) = \mathrm{ziel}\{f(\sigma) \mid \sigma \in \mathcal{S}(I) \} +\end{equation*} + +\subsubsection{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 $A(I) = f(\sigma_I^A)$. + +\end{document}