From 5b89c4dbe7d0b360e3ecf94f4499c6b3699b8171 Mon Sep 17 00:00:00 2001 From: Tobias Eidelpes Date: Sat, 30 May 2020 11:43:04 +0200 Subject: [PATCH] Explain parameter `epsilon` in approximation scheme --- paper/termpaper.tex | 7 +++++-- 1 file changed, 5 insertions(+), 2 deletions(-) diff --git a/paper/termpaper.tex b/paper/termpaper.tex index 54521e4..3135afb 100644 --- a/paper/termpaper.tex +++ b/paper/termpaper.tex @@ -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 pseudo-polynomial time. Finding a solution using the rule $\mathcal{R}_{sat_\#}^m$ 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 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 @@ -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 problems. \textcite{talmonFrameworkApprovalBasedBudgeting2019} show that the 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 algorithm with the same ratio exists not only for the case where the projects have unit cost but also for the general cost version.