Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
169 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Algorithms for the Global Domination Problem (2312.04526v1)

Published 7 Dec 2023 in cs.DM

Abstract: A dominating set D in a graph G is a subset of its vertices such that every vertex of the graph which does not belong to set D is adjacent to at least one vertex from set D. A set of vertices of graph G is a global dominating set if it is a dominating set for both, graph G and its complement. The objective is to find a global dominating set with the minimum cardinality. The problem is known to be NP-hard. Neither exact nor approximation algorithm existed . We propose two exact solution methods, one of them being based on an integer linear program (ILP) formulation, three heuristic algorithms and a special purification procedure that further reduces the size of a global dominated set delivered by any of our heuristic algorithms. We show that the problem remains NP-hard for restricted types of graphs and specify some families of graphs for which the heuristics guarantee the optimality. The second exact algorithm turned out to be about twice faster than ILP for graphs with more than 230 vertices and up to 1080 vertices, which were the largest benchmark instances that were solved optimally. The heuristics were tested for the existing 2284 benchmark problem instances with up to 14000 vertices and delivered solutions for the largest instances in less than one minute. Remarkably, for about 52% of the 1000 instances with the obtained optimal solutions, at least one of the heuristics generated an optimal solution, where the average approximation error for the remaining instances was 1.07%.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (19)
  1. Efficient heuristic algorithms for positive-influence dominating set in social networks. In IEEE INFOCOM 2018-IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS) (pp. 610–615). IEEE. doi:https://doi.org/10.1109/INFCOMW.2018.8406851.
  2. Bertossi, A. A. (1984). Dominating sets for split and bipartite graphs. Information processing letters, 19, 37–40. doi:https://doi.org/10.1016/0020-0190(84)90126-1.
  3. Factor domination in graphs. Discrete Mathematics, 86, 127–136. doi:https://doi.org/10.1016/0012-365X(90)90355-L.
  4. The generalized independent set problem: Polyhedral analysis and solution approaches. European Journal of Operational Research, 260, 41–55. doi:https://doi.org/10.1016/j.ejor.2016.11.050.
  5. Bounds on the global domination number. Quaestiones Mathematicae, 38, 563–572. doi:https://doi.org/10.2989/16073606.2014.981728.
  6. Social context, spatial structure and social network structure. Social networks, 34, 32–46. doi:https://doi.org/10.1016/j.socnet.2010.09.002.
  7. Global domination in planar graphs. J. Combin. Math. Combin. Comput, 66, 273–278. URL: https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.535.9636.
  8. Split graphs having dilworth number two. Canadian Journal of Mathematics, 29, 666–672. doi:https://doi.org/10.4153/CJM-1977-069-1.
  9. Computers and intractability volume 174. freeman San Francisco.
  10. Haynes, T. (2017). Domination in Graphs: Volume 2: Advanced Topics. Routledge.
  11. Exact and heuristic algorithms for the domination problem. European Journal of Operational Research, . doi:https://doi.org/10.1016/j.ejor.2023.08.033.
  12. Fixed set search applied to the clique partitioning problem. European Journal of Operational Research, 309, 65–81. doi:https://doi.org/10.1016/j.ejor.2023.01.044.
  13. A graph approximation heuristic for the vertex cover problem on planar graphs. European Journal of Operational Research, 72, 588–597. doi:https://doi.org/10.1016/0377-2217(94)90425-1.
  14. Merris, R. (2003). Split graphs. European Journal of Combinatorics, 24, 413–430. doi:https://doi.org/10.1016/S0195-6698(03)00030-1.
  15. Parra Inza, E. (2023). Random graph (1). Mendeley Data, V4. doi:https://doi.org/10.17632/rr5bkj6dw5.4.
  16. Sampathkumar, E. (1989). The global domination number of a graph. J. Math. Phys. Sci, 23, 377–385.
  17. Canonical partition of a graph defined by the degrees of its vertices. Isv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk, 5, 14–26.
  18. Positive influence dominating set in online social networks. In Combinatorial Optimization and Applications: Third International Conference, COCOA 2009, Huangshan, China, June 10-12, 2009. Proceedings 3 (pp. 313–321). Springer.
  19. On positive influence dominating sets in social networks. Theoretical Computer Science, 412, 265–269. doi:https://doi.org/10.1016/j.tcs.2009.10.001.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now