Abstract
We consider the Minimum Feasible Tileset problem: Given a set of symbols and subsets of these symbols (scenarios), find a smallest possible number of pairs of symbols (tiles) such that each scenario can be formed by selecting at most one symbol from each tile. We show that this problem is \(\mathsf {NP}\)-complete even if each scenario contains at most three symbols. Our main result is a 4/3-approximation algorithm for the general case. In addition, we show that the Minimum Feasible Tileset problem is fixed-parameter tractable both when parameterized with the number of scenarios and with the number of symbols.
Y. Disser—Supported by the Alexander von Humboldt-Foundation.
S. Kratsch—Supported by the German Research Foundation (DFG), KR 4286/1.
M. Sorge—Supported by the German Research Foundation (DFG), NI 369/12.
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Notes
- 1.
A compression to \(\mathcal {O}(|F|^d)\) size can be achieved by specifying one bit for each possible scenario in \(\mathcal {S} \) and setting it to one if the scenario is present and zero otherwise.
- 2.
Dell and Marx called this problem Perfect \(d\) -Set Matching.
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Disser, Y., Kratsch, S., Sorge, M. (2015). The Minimum Feasible Tileset Problem. In: Bampis, E., Svensson, O. (eds) Approximation and Online Algorithms. WAOA 2014. Lecture Notes in Computer Science(), vol 8952. Springer, Cham. https://doi.org/10.1007/978-3-319-18263-6_13
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