Exact and heuristic algorithms for the domination problem
•A good exact enumeration algorithm for a hard problem is fast in practice.•An integer linear programming (ILP) formulation.•A good exact enumeration algorithm for a hard problem is cable to deal with enough large instances.•A good approximation is important for hard problems.•Graph domination probl...
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| Vydáno v: | European journal of operational research Ročník 313; číslo 3; s. 926 - 936 |
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| Jazyk: | angličtina |
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Elsevier B.V
16.03.2024
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| ISSN: | 0377-2217, 1872-6860 |
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| Abstract | •A good exact enumeration algorithm for a hard problem is fast in practice.•An integer linear programming (ILP) formulation.•A good exact enumeration algorithm for a hard problem is cable to deal with enough large instances.•A good approximation is important for hard problems.•Graph domination problem is hard and has a number of important real-life applications.
In a simple connected graph G=(V,E), a subset of vertices S⊆V is a dominating set if any vertex v∈V∖S is adjacent to some vertex x from this subset. A number of real-life problems can be modeled using this problem which is known to be among the difficult NP-hard problems in its class. We formulate the problem as an integer liner program (ILP) and compare the performance with the two earlier existing exact state-of-the-art algorithms and exact implicit enumeration and heuristic algorithms that we propose here. Our exact algorithm was able to find optimal solutions much faster than ILP and the above two exact algorithms for middle-dense instances. For graphs with a considerable size, our heuristic algorithm was much faster than both, ILP and our exact algorithm. It found an optimal solution for more than half of the tested instances, whereas it improved the earlier known state-of-the-art solutions for almost all the tested benchmark instances. Among the instances where the optimum was not found, it gave an average approximation error of 1.18. |
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| AbstractList | •A good exact enumeration algorithm for a hard problem is fast in practice.•An integer linear programming (ILP) formulation.•A good exact enumeration algorithm for a hard problem is cable to deal with enough large instances.•A good approximation is important for hard problems.•Graph domination problem is hard and has a number of important real-life applications.
In a simple connected graph G=(V,E), a subset of vertices S⊆V is a dominating set if any vertex v∈V∖S is adjacent to some vertex x from this subset. A number of real-life problems can be modeled using this problem which is known to be among the difficult NP-hard problems in its class. We formulate the problem as an integer liner program (ILP) and compare the performance with the two earlier existing exact state-of-the-art algorithms and exact implicit enumeration and heuristic algorithms that we propose here. Our exact algorithm was able to find optimal solutions much faster than ILP and the above two exact algorithms for middle-dense instances. For graphs with a considerable size, our heuristic algorithm was much faster than both, ILP and our exact algorithm. It found an optimal solution for more than half of the tested instances, whereas it improved the earlier known state-of-the-art solutions for almost all the tested benchmark instances. Among the instances where the optimum was not found, it gave an average approximation error of 1.18. |
| Author | Parra Inza, Ernesto Hernández Mira, Frank Angel Sigarreta Almira, José María Vakhania, Nodari |
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| Keywords | Enumeration algorithm Graph theory Dominating set Combination optimization Time complexity Heuristic |
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coloring and cliques in telecommunications – volume: 8 start-page: 194 issue: 2 year: 2020 ident: 10.1016/j.ejor.2023.08.033_bib0003 article-title: On the total outer k-independent domination number of graphs publication-title: Mathematics doi: 10.3390/math8020194 – start-page: 41 year: 2012 ident: 10.1016/j.ejor.2023.08.033_bib0015 article-title: A faster algorithm for dominating set analyzed by the potential method – volume: 4 start-page: 233 issue: 3 year: 1979 ident: 10.1016/j.ejor.2023.08.033_bib0006 article-title: A greedy heuristic for the set-covering problem publication-title: Mathematics of operations research doi: 10.1287/moor.4.3.233 – volume: 49 start-page: 772 issue: 4 year: 2020 ident: 10.1016/j.ejor.2023.08.033_bib0005 article-title: From gap-exponential time hypothesis to fixed parameter tractable inapproximability: Clique, dominating set, and more publication-title: SIAM Journal on Computing doi: 10.1137/18M1166869 – start-page: 818 year: 2005 ident: 10.1016/j.ejor.2023.08.033_bib0018 article-title: Power domination problem in graphs – volume: 3 start-page: 425 issue: 4 year: 2003 ident: 10.1016/j.ejor.2023.08.033_bib0029 article-title: Power-aware broadcasting and activity scheduling in ad hoc wireless networks using connected dominating sets publication-title: Wireless Communications and Mobile Computing doi: 10.1002/wcm.125 |
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