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Explain parameter epsilon in approximation scheme

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Tobias Eidelpes 2020-05-30 11:43:04 +02:00
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paper/termpaper.tex
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@ -277,7 +277,8 @@ problem is solvable by a dynamic programming algorithm in $O(B\cdot |P|)$ where
$P$ is the set of projects, $\mathcal{R}_{sat_\$}^m$ is solvable in $P$ is the set of projects, $\mathcal{R}_{sat_\$}^m$ is solvable in
pseudo-polynomial time. Finding a solution using the rule pseudo-polynomial time. Finding a solution using the rule
$\mathcal{R}_{sat_\#}^m$ however, is doable in polynomial time due to the $\mathcal{R}_{sat_\#}^m$ however, is doable in polynomial time due to the
problem's relation to the knapsack problem. If the input is represented in problem's relation to the knapsack problem. If the input (either projects or
voters) is represented in
unary, a dynamic programming algorithm is bounded by a polynomial in the length unary, a dynamic programming algorithm is bounded by a polynomial in the length
of the input. For $\mathcal{R}_{sat_{0/1}}^m$, finding a set of projects that of the input. For $\mathcal{R}_{sat_{0/1}}^m$, finding a set of projects that
gives at least a certain amount of satisfaction is \textsf{NP}-hard. Assuming gives at least a certain amount of satisfaction is \textsf{NP}-hard. Assuming
@ -336,7 +337,9 @@ using integer linear programming (ILP). Although integer programming is
\textsf{NP}-complete, efficient solvers are readily available for these types of \textsf{NP}-complete, efficient solvers are readily available for these types of
problems. \textcite{talmonFrameworkApprovalBasedBudgeting2019} show that the problems. \textcite{talmonFrameworkApprovalBasedBudgeting2019} show that the
rule $\mathcal{R}_{sat_{0/1}}^m$ is similar to the max cover problem which can rule $\mathcal{R}_{sat_{0/1}}^m$ is similar to the max cover problem which can
be approximated with a $(1-\frac{1}{e})$-approximation algorithm. In fact, be approximated with a $(1-\frac{1}{\epsilon})$-approximation algorithm, where
$\epsilon > 0$ is a fixed parameter that is chosen depending on the error of the
approximation. In fact,
\textcite{khullerBudgetedMaximumCoverage1999} show that an approximation \textcite{khullerBudgetedMaximumCoverage1999} show that an approximation
algorithm with the same ratio exists not only for the case where the projects algorithm with the same ratio exists not only for the case where the projects
have unit cost but also for the general cost version. have unit cost but also for the general cost version.