The Dominating Set Problem in Geometric Intersection Graphs
We study the parameterized complexity of dominating sets in geometric intersection graphs. In one dimension, we investigate intersection graphs induced by translates of a fixed pattern Q that consists of a finite number of intervals and a finite number of isolated points. We prove that Dominating Set on such intersection graphs is polynomially solvable whenever Q contains at least one interval, and whenever Q contains no intervals and for any two point pairs in Q the distance ratio is rational. The remaining case where Q contains no intervals but does contain an irrational distance ratio is shown to be NP-complete and contained in FPT (when parameterized by the solution size). In two and higher dimensions, we prove that Dominating Set is contained in W[1] for intersection graphs of semi-algebraic sets with constant description complexity. This generalizes known results from the literature. Finally, we establish W[1]-hardness for a large class of intersection graphs.
dominating set
intersection graph
W-hierarchy
14:1-14:12
Regular Paper
Mark
de Berg
Mark de Berg
Sándor
Kisfaludi-Bak
Sándor Kisfaludi-Bak
Gerhard
Woeginger
Gerhard Woeginger
10.4230/LIPIcs.IPEC.2017.14
Dennis S. Arnon, George E. Collins, and Scott McCallum. Cylindrical algebraic decomposition I: the basic algorithm. SIAM J. Comput., 13(4):865-877, 1984. URL: http://dx.doi.org/10.1137/0213054.
http://dx.doi.org/10.1137/0213054
Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, 3rd edition, 2008.
Maw-Shang Chang. Efficient algorithms for the domination problems on interval and circular-arc graphs. SIAM J. Comput., 27(6):1671-1694, 1998. URL: http://dx.doi.org/10.1137/S0097539792238431.
http://dx.doi.org/10.1137/S0097539792238431
Mark de Berg, Sándor Kisfaludi-Bak, and Gerhard Woeginger. The dominating set problem in geometric intersection graphs. CoRR, abs/1709.05182, 2017. URL: http://arxiv.org/abs/1709.05182.
http://arxiv.org/abs/1709.05182
Rodney G. Downey and Michael R. Fellows. Fixed-parameter tractability and completeness I: basic results. SIAM J. Comput., 24(4):873-921, 1995. URL: http://dx.doi.org/10.1137/S0097539792228228.
http://dx.doi.org/10.1137/S0097539792228228
Michael R. Fellows, Danny Hermelin, Frances A. Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci., 410(1):53-61, 2009. URL: http://dx.doi.org/10.1016/j.tcs.2008.09.065.
http://dx.doi.org/10.1016/j.tcs.2008.09.065
Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. URL: http://dx.doi.org/10.1007/3-540-29953-X.
http://dx.doi.org/10.1007/3-540-29953-X
Sariel Har-Peled. Being fat and friendly is not enough. arXiv preprint arXiv:0908.2369, 2009.
Teresa W. Haynes, Stephen T. Hedetniemi, and Peter J. Slater. Domination in Graphs: Advanced Topics. Pure and Applied Mathematics. Marcel Dekker, Inc., 1998.
Dániel Marx. Parameterized complexity of independence and domination on geometric graphs. In Hans L. Bodlaender and Michael A. Langston, editors, Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, September 13-15, 2006, Proceedings, volume 4169 of Lecture Notes in Computer Science, pages 154-165. Springer, 2006. URL: http://dx.doi.org/10.1007/11847250_14.
http://dx.doi.org/10.1007/11847250_14
Dániel Marx and Michał Pilipczuk. Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams. arXiv preprint arXiv:1504.05476, 2015.
Venkatesh Raman and Saket Saurabh. 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. URL: http://dx.doi.org/10.1007/s00453-007-9148-9.
http://dx.doi.org/10.1007/s00453-007-9148-9
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