Abstract
Set Cover is one of the well-known classical NP-hard problems. Following some recent trends, we study the conflict-free version of the Set Cover problem. Here we have a universe \({\mathcal U}\), a family \({\mathcal F}\) of subsets of \({\mathcal U}\) and a graph \(G_{{\mathcal F}}\) on the vertex set \({\mathcal F}\) and we look for a subfamily \({\mathcal F}' \subseteq {\mathcal F}\) of minimum size that covers \({\mathcal U}\) and also forms an independent set in \(G_{{\mathcal F}}\). Here we initiate a systematic study of the problem in parameterized complexity by restricting the focus to the variants where Set Cover is fixed-parameter tractable (FPT). We give upper bounds and lower bounds for conflict-free version of the Set Cover with and without duplicate sets along with restrictions to the graph classes of \(G_{{\mathcal F}}\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
\({\mathcal O}^*\) notation ignores polynomial factors of input.
- 2.
Proof in full version.
- 3.
Proof in full version.
- 4.
Proof in full version.
References
Agrawal, A., Jain, P., Kanesh, L., Lokshtanov, D., Saurabh, S.: Conflict free feedback vertex set: a parameterized dichotomy. In: LIPIcs-Leibniz International Proceedings in Informatics, vol. 117. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2018)
Arkin, E.M., et al.: Conflict-free covering. In: Conference on Computational Geometry, p. 17 (2015)
Banik, A., Panolan, F., Raman, V., Sahlot, V., Saurabh, S.: Parameterized complexity of geometric covering problems having conflicts. Algorithms and Data Structures. LNCS, vol. 10389, pp. 61–72. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-62127-2_6
Cygan, M., et al.: On problems as hard as CNF-SAT. ACM Trans. Algorithms (TALG) 12(3), 41 (2016)
Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3
Darmann, A., Pferschy, U., Schauer, J., Woeginger, G.J.: Paths, trees and matchings under disjunctive constraints. Discrete Appl. Math. 159(16), 1726–1735 (2011)
Diestel, R.: Graph Theory. Springer, Heidelberg (2006)
Farber, M.: On diameters and radii of bridged graphs. Discrete Math. 73(3), 249–260 (1989)
Fomin, F.V., Lokshtanov, D., Panolan, F., Saurabh, S.: Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM (JACM) 63(4), 29 (2016)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, vol. 57. Elsevier, Amsterdam (2004)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)
Jain, P., Kanesh, L., Misra, P.: Conflict free version of covering problems on graphs: classical and parameterized. In: Fomin, F.V., Podolskii, V.V. (eds.) CSR 2018. LNCS, vol. 10846, pp. 194–206. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-90530-3_17
Kann, V.: Polynomially bounded minimization problems which are hard to approximate. In: Lingas, A., Karlsson, R., Carlsson, S. (eds.) ICALP 1993. LNCS, vol. 700, pp. 52–63. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-56939-1_61
Lin, B.: The parameterized complexity of the k-biclique problem. J. ACM (JACM) 65(5), 34 (2018)
Lokshtanov, D., Marx, D., Saurabh, S.: Slightly superexponential parameterized problems. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 760–776. Society for Industrial and Applied Mathematics (2011)
Lokshtanov, D., Misra, P., Panolan, F., Saurabh, S.: Deterministic truncation of linear matroids. ACM Trans. Algorithms (TALG) 14(2), 14 (2018)
Lokshtanov, D., Panolan, F., Saurabh, S., Sharma, R., Zehavi, M.: Covering small independent sets and separators with applications to parameterized algorithms. In: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 2785–2800. Society for Industrial and Applied Mathematics (2018)
Marx, D.: A parameterized view on matroid optimization problems. Theor. Comput. Sci. 410(44), 4471–4479 (2009)
Marx, D., Salmasi, A., Sidiropoulos, A.: Constant-factor approximations for asymmetric tsp on nearly-embeddable graphs. In: LIPIcs-Leibniz International Proceedings in Informatics, vol. 60. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2016)
Pferschy, U., Schauer, J.: The maximum flow problem with conflict and forcing conditions. In: Pahl, J., Reiners, T., Voß, S. (eds.) INOC 2011. LNCS, vol. 6701, pp. 289–294. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21527-8_34
Raman, V., Saurabh, S.: Short cycles make w-hard problems hard: FPT algorithms for w-hard problems in graphs with no short cycles. Algorithmica 52(2), 203–225 (2008)
van Bevern, R., Tsidulko, O.Y., Zschoche, P.: Fixed-parameter algorithms for maximum-profit facility location under matroid constraints. In: Proceedings of the 11th International Conference on Algorithms and Complexity (2019)
Acknowledgements
The authors thank Fahad Panolan and Saket Saurabh for several valuable discussions on the theme of the paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Jacob, A., Majumdar, D., Raman, V. (2019). Parameterized Complexity of Conflict-Free Set Cover. In: van Bevern, R., Kucherov, G. (eds) Computer Science – Theory and Applications. CSR 2019. Lecture Notes in Computer Science(), vol 11532. Springer, Cham. https://doi.org/10.1007/978-3-030-19955-5_17
Download citation
DOI: https://doi.org/10.1007/978-3-030-19955-5_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-19954-8
Online ISBN: 978-3-030-19955-5
eBook Packages: Computer ScienceComputer Science (R0)