Abstract
Given a graph G together with a capacity function c : V(G) →ℕ, we call S ⊆ V(G) a capacitated dominating set if there exists a mapping f: (V(G) ∖ S) →S which maps every vertex in (V(G) ∖ S) to one of its neighbors such that the total number of vertices mapped by f to any vertex v ∈ S does not exceed c(v). In the Planar Capacitated Dominating Set problem we are given a planar graph G, a capacity function c and a positive integer k and asked whether G has a capacitated dominating set of size at most k. In this paper we show that Planar Capacitated Dominating Set is W[1]-hard, resolving an open problem of Dom et al. [IWPEC, 2008 ]. This is the first bidimensional problem to be shown W[1]-hard. Thus Planar Capacitated Dominating Set can become a useful starting point for reductions showing parameterized intractablility of planar graph problems.
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References
Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs. Algorithmica 33(4), 46–493 (2002)
Alber, J., Fellows, M.R., Niedermeier, R.: Polynomial-time data reduction for dominating set. J. ACM 51(3), 363–384 (2004)
Demaine, E.D., Fomin, F.V., Hajiaghayi, M.T., Thilikos, D.M.: Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. J. ACM 52(6), 866–893 (2005)
Dom, M., Lokshtanov, D., Saurabh, S., Villanger, Y.: Capacitated Domination and Covering: A Parameterized Perspective. In: Grohe, M., Niedermeier, R. (eds.) IWPEC 2008. LNCS, vol. 5018, pp. 78–90. Springer, Heidelberg (2008)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Fellows, M.R., Hermelin, D., Rosamond, F.A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theoretical Computer Science 410(1), 53–61 (2009)
Fiala, J., Golovach, P.A., Kratochvíl, J.: Parameterized Complexity of Coloring Problems: Treewidth versus Vertex Cover. In: TAMC 2009. LNCS, vol. 5532, pp. 221–230. Springer, Heidelberg (2009)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Flum, J., Grohe, M., Weyer, M.: Bounded fixed-parameter tractability and log\(^{\mbox{2}}\) n nondeterministic bits. J. Comput. Syst. Sci. 72(1), 34–71 (2006)
Fomin, F.V., Golovach, P.A., Lokshtanov, D., Saurabh, S.: Clique-width: on the price of generality. In: The Proceedings of SODA, pp. 825–834 (2009)
Fomin, F.V., Thilikos, D.M.: Dominating Sets in Planar Graphs: Branch-Width and Exponential Speed-Up. SIAM Journal on Computing 36(2), 281–309 (2006)
Guo, J., Niedermeier, R.: Linear Problem Kernels for NP-Hard Problems on Planar Graphs. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 375–386. Springer, Heidelberg (2007)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)
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Bodlaender, H.L., Lokshtanov, D., Penninkx, E. (2009). Planar Capacitated Dominating Set Is W[1]-Hard. In: Chen, J., Fomin, F.V. (eds) Parameterized and Exact Computation. IWPEC 2009. Lecture Notes in Computer Science, vol 5917. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11269-0_4
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DOI: https://doi.org/10.1007/978-3-642-11269-0_4
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