Abstract
We show that a number of election-related problems with prices (such as, for example, bribery) are fixed-parameter tractable (in \({\mathsf {FPT}}\)) when parameterized by the number of candidates. For bribery, this resolves a nearly 10-year old family of open problems. Our results follow by a general technique that formulates voting problems as covering problems and extends the classic approach of using integer linear programming and the algorithm of Lenstra [19]. In this context, our central result is that Weighted Set Multicover parameterized by the universe size is fixed-parameter tractable. Our approach is also applicable to weighted electoral control for Approval voting. We improve previously known \({\mathsf {XP}}\)-memberships to \({\mathsf {FPT}}\)-memberships. Our preliminary experiments on real-world-based data show the practical usefulness of our approach for instances with few candidates.
Robert Bredereck–Supported by DFG project PAWS (NI 369/10).
Piotr Faliszewski–Supported by a DFG Mercator fellowship within project PAWS (NI 369/10) while staying at TU Berlin, and by AGH University grant 11.11.230.124 afterward.
Nimrod Talmon–Supported by DFG Research Training Group MDS (GRK 1408).
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Notes
- 1.
One problem for which our technique does not apply is Swap Bribery [10]; even though Dorn and Schlotter [8] claim that it is in \({\mathsf {FPT}}\) when parameterized by the number of candidates, their proof applies only to a restricted setting. The complexity of Swap Bribery parameterized by the number of candidates remains open.
- 2.
There is a name clash between the literature on covering problems and that on elections. In the former, “weights” refer to what voting literature would call “prices.” Weights of the voters are modeled as multiplicities of the elements in the multisets. We kept the naming conventions from respective parts of the literature to make our results more accessible to researchers from both communities.
- 3.
Remarkably, under reasonable complexity-theoretic assumptions, Dom et al. [7] have shown that no polynomial-size kernels exist for Set Cover (which is a special case of Weighted Set Multicover and Uniform Multiset Multicover), parameterized by the universe size and the solution size.
- 4.
While this result does not, as of yet, have direct application to voting, we believe it is quite interesting in itself.
- 5.
By removing candidates during instance generation one also removes voters only approving removed candidates.
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Bredereck, R., Faliszewski, P., Niedermeier, R., Skowron, P., Talmon, N. (2015). Elections with Few Candidates: Prices, Weights, and Covering Problems. In: Walsh, T. (eds) Algorithmic Decision Theory. ADT 2015. Lecture Notes in Computer Science(), vol 9346. Springer, Cham. https://doi.org/10.1007/978-3-319-23114-3_25
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