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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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commit 31065aba3c
1 changed files with 30 additions and 5 deletions
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paper/termpaper.tex
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@ -27,6 +27,11 @@
\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}
\maketitle
@ -210,10 +215,10 @@ of this property.
\end{equation}
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
winning set. Therefore, a voter achieves satisfaction 1 when at least one
approved project ends up in the winning bundle, i.e. if $|A_v| > 0$ and 0
satisfaction otherwise (see equation~\ref{eq:5}).
is at least one of the projects they have approved selected to be in the winning
set. Therefore, a voter achieves satisfaction 1 when at least one approved
project ends up in the winning bundle, i.e. if $|A_v| > 0$ and 0 satisfaction
otherwise (see equation~\ref{eq:5}).
\begin{equation}\label{eq:5}
sat_{0/1}(P_v,A) =
\begin{cases}
@ -221,6 +226,26 @@ satisfaction otherwise (see equation~\ref{eq:5}).
0 & \mathsf{otherwise}
\end{cases}
\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