Add text for coping with intractability

paper/references.bib

@@ -144,6 +144,19 @@
   number = {2}
 }
 
+@article{khullerBudgetedMaximumCoverage1999,
+  title = {The Budgeted Maximum Coverage Problem},
+  author = {Khuller, Samir and Moss, Anna and Naor, Joseph},
+  year = {1999},
+  month = apr,
+  volume = {70},
+  pages = {39--45},
+  doi = {10.1016/S0020-0190(99)00031-9},
+  abstract = {The budgeted maximum coverage problem is: given a collection S of sets with associated costs defined over a domain of weighted elements, and a budget L, find a subset of S{${'}\subseteqq$}S such that the total cost of sets in S{${'}$} does not exceed L, and the total weight of elements covered by S{${'}$} is maximized. This problem is NP-hard. For the special case of this problem, where each set has unit cost, a (1-1/e)-approximation is known. Yet, prior to this work, no approximation results were known for the general cost version. The contribution of this paper is a (1-1/e)-approximation algorithm for the budgeted maximum coverage problem. We also argue that this approximation factor is the best possible, unless NP{$\subseteqq$}DTIME(nO(loglogn)).},
+  journal = {Information Processing Letters},
+  number = {1}
+}
+
 @inproceedings{langPortioningUsingOrdinal2019,
   title = {Portioning {{Using Ordinal Preferences}}: {{Fairness}} and {{Efficiency}}},
   shorttitle = {Portioning {{Using Ordinal Preferences}}},

paper/termpaper.tex

@@ -333,6 +333,28 @@ constant time.
     however, gives $\{ p_2,p_3,p_4 \}$.
 \end{example}
 
+A benefit of the three discussed satisfaction functions is that they can be
+viewed as constraint satisfaction problems (CSPs) and can thus be formulated
+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,
+\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.
+
+Instead of sacrificing exactness to get a better running time,
+\textcite{talmonFrameworkApprovalBasedBudgeting2019} show that the
+$\mathcal{R}_{sat_{0/1}}^m$ rule is fixed parameter tractable for the number of
+voters $|V|$. A problem is fixed parameter tractable if there exists an
+algorithm that decides each instance of the problem in $O(f(k)\cdot p(n))$ where
+$p(n)$ is a polynomial function and $f(k)$ an arbitrary function in $k$. It is
+crucial to note that $f(k)$ does not admit functions of the form $n^k$. The
+algorithm for the maximum rule tries to guess the number of voters that are
+represented by the same project. The estimation is then used to pick a project
+which has the lowest cost and satisfies exactly the estimated amount of voters.
+
 \printbibliography
 
 \end{document}