Abstract
Recently it was shown that many classic graph problems—Independent Set, Dominating Set, Hamiltonian Cycle, and more—can be solved in subexponential time on unit-ball graphs. More precisely, these problems can be solved in \(2^{O(n^{1-1/d})}\) time on unit-ball graphs in \(\mathbb {R}^d\), which is tight under ETH. The result can be generalized to intersection graphs of similarly-sized fat objects.
For Independent Set the same running time can be achieved for non-similarly-sized fat objects, and for the weighted version of the problem. We show that such generalizations most likely are not possible for Dominating Set: assuming ETH, we prove that
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there is no algorithm with running time \(2^{o(n)}\) for Dominating Set on (non-unit) ball graphs in \(\mathbb {R}^3\);
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there is no algorithm with running time \(2^{o(n)}\) for Weighted Dominating Set on unit-ball graphs in \(\mathbb {R}^3\);
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there is no algorithm with running time \(2^{o(n)}\) for Dominating Set, Connected Dominating Set, or Steiner Tree on intersections graphs of arbitrary convex (but non-constant-complexity) objects in the plane.
This research was supported by the Netherlands Organization for Scientific Research NWO under project no. 024.002.003 (Networks).
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Notes
- 1.
Actually, it turns out that one can also argue that the existence of these edges is not needed for the reduction to work. We prefer to work with the specific graph \(\mathcal {G}_\phi \) defined earlier, and therefore need to show that our geometric representation includes all edges in the clique.
- 2.
Alternatively, one could give the dummies a weight of 1, and add a unique neighbor to each of them with weight \(\infty \), which ensures that they are contained in all finite weight dominating sets.
References
de Berg, M., Bodlaender, H.L., Kisfaludi-Bak, S., Kolay, S.: An ETH-tight exact algorithm for Euclidean TSP. In: Proceedings of the 59th IEEE Symposium Foundations Computer Science (FOCS), pp. 450–461 (2018)
de Berg, M., Bodlaender, H.L., Kisfaludi-Bak, S., Marx, D., van der Zanden, T.C.: A framework for ETH-tight algorithms and lower bounds in geometric intersection graphs. In: Proceedings of the 50th ACM Symposium Theory Computer (STOC), pp. 574–586 (2018)
de Berg, M., Kisfaludi-Bak, S., Woeginger, G.: The complexity of dominating set in geometric intersection graphs. Theor. Comput. Sci. 769, 18–31 (2019)
Bodlaender, H.L., Cygan, M., Kratsch, S., Nederlof, J.: Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput. 243, 86–111 (2015)
Bodlaender, H.L., Drange, P.G., Dregi, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M.: A \(c^k n\) 5-approximation algorithm for treewidth. SIAM J. Comput. 45(2), 317–378 (2016)
van den Eijkhof, F., Bodlaender, H.L., Koster, A.M.C.A.: Safe reduction rules for weighted treewidth. Algorithmica 47(2), 139–158 (2007)
Breu, H., Kirkpatrick, D.G.: Unit disk graph recognition is NP-hard. Comput. Geom. Theory Appl. 9, 3–24 (1998)
Bringmann, K., Kisfaludi-Bak, S., Pilipczuk, M., van Leeuwen, E.J.: On geometric set cover for orthants. In: Proceedings of the 27th European Symposium on Algorithms (ESA), pp. 26:1–26:18 (2019)
Cygan, M., et al.: Parameterized Algorithms. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-319-21275-3
Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)
Kang, R.J., Müller, T.: Sphere and dot product representations of graphs. Discret. Comput. Geom. 47, 548–568 (2012)
Kisfaludi-Bak, S.: ETH-tight algorithms for geometric network problems. Ph.D. thesis, Technische Universiteit Eidnhoven (2019)
Kisfaludi-Bak, S., Marx, D., van der Zanden, T.C.: How does object fatness impact the complexity of packing in d dimensions? In: Proceedings of the 30th International Symposium on Algorithms and Computation (ISAAC) (2019, to appear)
Koebe, P.: Kontaktprobleme der konformen Abbildung (1936). Hirzel
Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. App. Math. 36(2), 177–189 (1979)
Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comput. 9(3), 615–627 (1980)
Marx, D.: The square root phenomenon in planar graphs. In: Proceedings of the 40th International Colloquium on Automata, Languages, and Programming (ICALP), part II, p. 28 (2013)
Marx, D., Pilipczuk, M.: Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 865–877. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48350-3_72
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de Berg, M., Kisfaludi-Bak, S. (2020). Lower Bounds for Dominating Set in Ball Graphs and for Weighted Dominating Set in Unit-Ball Graphs. In: Fomin, F.V., Kratsch, S., van Leeuwen, E.J. (eds) Treewidth, Kernels, and Algorithms. Lecture Notes in Computer Science(), vol 12160. Springer, Cham. https://doi.org/10.1007/978-3-030-42071-0_5
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