Lower Bounds for Dominating Set in Ball Graphs and for Weighted Dominating Set in Unit-Ball Graphs

Mark T. de Berg; Sándor Kisfaludi-Bak

doi:10.1007/978-3-030-42071-0_5

Lower Bounds for Dominating Set in Ball Graphs and for Weighted Dominating Set in Unit-Ball Graphs

Mark T. de Berg, Sándor Kisfaludi-Bak

Algorithms, Geometry and Applications

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

1 Citation (Scopus)

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 2O(n1−1/d) time on unit-ball graphs in Rd , 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
there is no algorithm with running time 2o(n) for Dominating Set on (non-unit) ball graphs in R3 ;

there is no algorithm with running time 2o(n) for Weighted Dominating Set on unit-ball graphs in R3 ;

there is no algorithm with running time 2o(n) for Dominating Set, Connected Dominating Set, or Steiner Tree on intersections graphs of arbitrary convex (but non-constant-complexity) objects in the plane.

Original language	English
Title of host publication	Treewidth, Kernels, and Algorithms
Editors	F. Fomin, S. Kratsch, E. van Leeuwen
Pages	31-48
Number of pages	18
ISBN (Electronic)	978-3-030-42071-0
DOIs	https://doi.org/10.1007/978-3-030-42071-0_5
Publication status	Published - 2020

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	12160 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Access to Document

10.1007/978-3-030-42071-0_5

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de Berg, M. T., & Kisfaludi-Bak, S. (2020). Lower Bounds for Dominating Set in Ball Graphs and for Weighted Dominating Set in Unit-Ball Graphs. In F. Fomin, S. Kratsch, & E. van Leeuwen (Eds.), Treewidth, Kernels, and Algorithms (pp. 31-48). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12160 LNCS). https://doi.org/10.1007/978-3-030-42071-0_5

de Berg, Mark T. ; Kisfaludi-Bak, Sándor. / Lower Bounds for Dominating Set in Ball Graphs and for Weighted Dominating Set in Unit-Ball Graphs. Treewidth, Kernels, and Algorithms. editor / F. Fomin ; S. Kratsch ; E. van Leeuwen. 2020. pp. 31-48 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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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 2O(n1−1/d) time on unit-ball graphs in Rd , 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 thatthere is no algorithm with running time 2o(n) for Dominating Set on (non-unit) ball graphs in R3 ;there is no algorithm with running time 2o(n) for Weighted Dominating Set on unit-ball graphs in R3 ;there is no algorithm with running time 2o(n) for Dominating Set, Connected Dominating Set, or Steiner Tree on intersections graphs of arbitrary convex (but non-constant-complexity) objects in the plane.",

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de Berg, MT & Kisfaludi-Bak, S 2020, Lower Bounds for Dominating Set in Ball Graphs and for Weighted Dominating Set in Unit-Ball Graphs. in F Fomin, S Kratsch & E van Leeuwen (eds), Treewidth, Kernels, and Algorithms. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 12160 LNCS, pp. 31-48. https://doi.org/10.1007/978-3-030-42071-0_5

Lower Bounds for Dominating Set in Ball Graphs and for Weighted Dominating Set in Unit-Ball Graphs. / de Berg, Mark T.; Kisfaludi-Bak, Sándor.

Treewidth, Kernels, and Algorithms. ed. / F. Fomin; S. Kratsch; E. van Leeuwen. 2020. p. 31-48 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12160 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic › peer-review

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AU - de Berg, Mark T.

AU - Kisfaludi-Bak, Sándor

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N2 - 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 2O(n1−1/d) time on unit-ball graphs in Rd , 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 thatthere is no algorithm with running time 2o(n) for Dominating Set on (non-unit) ball graphs in R3 ;there is no algorithm with running time 2o(n) for Weighted Dominating Set on unit-ball graphs in R3 ;there is no algorithm with running time 2o(n) for Dominating Set, Connected Dominating Set, or Steiner Tree on intersections graphs of arbitrary convex (but non-constant-complexity) objects in the plane.

AB - 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 2O(n1−1/d) time on unit-ball graphs in Rd , 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 thatthere is no algorithm with running time 2o(n) for Dominating Set on (non-unit) ball graphs in R3 ;there is no algorithm with running time 2o(n) for Weighted Dominating Set on unit-ball graphs in R3 ;there is no algorithm with running time 2o(n) for Dominating Set, Connected Dominating Set, or Steiner Tree on intersections graphs of arbitrary convex (but non-constant-complexity) objects in the plane.

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de Berg MT, Kisfaludi-Bak S. Lower Bounds for Dominating Set in Ball Graphs and for Weighted Dominating Set in Unit-Ball Graphs. In Fomin F, Kratsch S, van Leeuwen E, editors, Treewidth, Kernels, and Algorithms. 2020. p. 31-48. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-030-42071-0_5