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Add example for greedy selection rules

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Tobias Eidelpes 2020-05-15 12:53:10 +02:00
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paper/termpaper.tex
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@ -27,6 +27,11 @@
\email{e1527193@student.tuwien.ac.at} \email{e1527193@student.tuwien.ac.at}
} }
% Numbered example environment
\newcounter{example}[section]
\newenvironment{example}[1][]{\refstepcounter{example}\par\medskip
\noindent \textbf{Example~\theexample. #1} \rmfamily}{\medskip}
\begin{document} \begin{document}
\maketitle \maketitle
@ -210,10 +215,10 @@ of this property.
\end{equation} \end{equation}
The third satisfaction function assumes that voters are content as long as there The third satisfaction function assumes that voters are content as long as there
is at least one of the projects they have approved is selected to be in the is at least one of the projects they have approved selected to be in the winning
winning set. Therefore, a voter achieves satisfaction 1 when at least one set. Therefore, a voter achieves satisfaction 1 when at least one approved
approved project ends up in the winning bundle, i.e. if $|A_v| > 0$ and 0 project ends up in the winning bundle, i.e. if $|A_v| > 0$ and 0 satisfaction
satisfaction otherwise (see equation~\ref{eq:5}). otherwise (see equation~\ref{eq:5}).
\begin{equation}\label{eq:5} \begin{equation}\label{eq:5}
sat_{0/1}(P_v,A) = sat_{0/1}(P_v,A) =
\begin{cases} \begin{cases}
@ -221,6 +226,26 @@ satisfaction otherwise (see equation~\ref{eq:5}).
0 & \mathsf{otherwise} 0 & \mathsf{otherwise}
\end{cases} \end{cases}
\end{equation} \end{equation}
The satisfaction functions from equations~\ref{eq:4} and \ref{eq:5} can also be
combined with the greedy rule, potentially giving slightly different outcomes
than $\mathcal{R}_{sat_\#}^g$.
\begin{example}
A set of projects $P = \{ p_2,p_3,p_4,p_5,p_6 \}$ and their associated cost
$p_i$ where project $p_i$ costs $i$ and a budget limit $B = 10$ is given.
Futhermore, five voters vote $v_1 = \{ p_2,p_5,p_6 \}$, $v_2 = \{ p_2,
p_3,p_4,p_5 \}$, $v_3 = \{ p_3,p_4,p_5 \}$, $v_4 = \{ p_4,p_5 \}$ and $v_5 =
\{ p_6 \}$. Under $\mathcal{R}_{sat_\#}^g$ the winning bundle is $\{ p_4,p_5
\}$, $\mathcal{R}_{sat_\$}^g$ gives $\{ p_4,p_5 \}$ and
$\mathcal{R}_{sat_{0/1}}^g$ $\{ p_2,p_3,p_5 \}$.
\end{example}
These three satisfaction functions cannot only be combined with a greedy
selection process. A different possibility is to always select a winning set of
projects that maximizes the sum of the voters' satisfaction:
\begin{equation}\label{eq:6}
\max_{A\subseteq P}\sum_{v\in V}sat(P_v,A)
\end{equation}
\printbibliography \printbibliography