Abstract
Many graph problems such as maximum cut, chromatic number, hamiltonian cycle, and edge dominating set are known to be fixed-parameter tractable (FPT) when parameterized by the treewidth of the input graphs, but become W-hard with respect to the clique-width parameter. Recently, Gajarský et al. proposed a new parameter called modular-width using the notion of modular decomposition of graphs. They showed that the chromatic number problem and the partitioning into paths problem, and hence hamiltonian path and hamiltonian cycle, are FPT when parameterized by this parameter. In this paper, we study modular-width in parameterized parallel complexity and show that the weighted maximum clique problem and the maximum matching problem are fixed-parameter parallel-tractable (FPPT) when parameterized by this parameter.
This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Center “On-The-Fly Computing” (SFB 901) and the International Graduate School “Dynamic Intelligent Systems”.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abu-Khzam, F.N., Li, S., Markarian, C., Meyer auf der Heide, F., Podlipyan, P.: On the parameterized parallel complexity and the vertex cover problem. In: Chan, T.-H.H., Li, M., Wang, L. (eds.) COCOA 2016. LNCS, vol. 10043, pp. 477–488. Springer, Cham (2016). doi:10.1007/978-3-319-48749-6_35
Bodlaender, H., Downey, R., Fellows, M.: Applications of parameterized complexity to problems of parallel and distributed computation (1994, unpublished extended abstract)
Bodlaender, H.L.: Treewidth: characterizations, applications, and computations. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 1–14. Springer, Heidelberg (2006). doi:10.1007/11917496_1
Bodlaender, H.L., Koster, A.M.: Combinatorial optimization on graphs of bounded treewidth. Comput. J. 51(3), 255–269 (2008)
Bui-Xuan, B.-M., Telle, J.A., Vatshelle, M.: Boolean-width of graphs. Theoret. Comput. Sci. 412(39), 5187–5204 (2011)
Cesati, M., Di Ianni, M.: Parameterized parallel complexity. In: Pritchard, D., Reeve, J. (eds.) Euro-Par 1998. LNCS, vol. 1470, pp. 892–896. Springer, Heidelberg (1998). doi:10.1007/BFb0057945
Chein, M., Habib, M., Maurer, M.-C.: Partitive hypergraphs. Discret. Math. 37(1), 35–50 (1981)
Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)
Dahlhaus, E.: Efficient parallel modular decomposition (extended abstract). In: Nagl, M. (ed.) WG 1995. LNCS, vol. 1017, pp. 290–302. Springer, Heidelberg (1995). doi:10.1007/3-540-60618-1_83
Fomin, F.V., Golovach, P.A., Lokshtanov, D., Saurabh, S.: Clique-width: on the price of generality. In: Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 825–834. Society for Industrial and Applied Mathematics (2009)
Fomin, F.V., Golovach, P.A., Lokshtanov, D., Saurabh, S.: Algorithmic lower bounds for problems parameterized by clique-width. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 493–502. Society for Industrial and Applied Mathematics (2010)
Fomin, F.V., Golovach, P.A., Lokshtanov, D., Saurabh, S.: Intractability of clique-width parameterizations. SIAM J. Comput. 39(5), 1941–1956 (2010)
Gajarský, J., Lampis, M., Ordyniak, S.: Parameterized algorithms for modular-width. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 163–176. Springer, Cham (2013). doi:10.1007/978-3-319-03898-8_15
Ganian, R., Hliněný, P., Nešetřil, J., Obdržálek, J., Ossona de Mendez, P., Ramadurai, R.: When trees grow low: shrubs and fast MSO\(_1\). In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 419–430. Springer, Heidelberg (2012). doi:10.1007/978-3-642-32589-2_38
Habib, M., Paul, C.: A survey of the algorithmic aspects of modular decomposition. Comput. Sci. Rev. 4(1), 41–59 (2010)
Hagerup, T., Katajainen, J., Nishimura, N., Ragde, P.: Characterizing multiterminal flow networks and computing flows in networks of small treewidth. J. Comput. Syst. Sci. 57(3), 366–375 (1998)
Lagergren, J.: Efficient parallel algorithms for graphs of bounded tree-width. J. Algorithms 20(1), 20–44 (1996)
Miller, G.L., Reif, J.H.: Parallel tree contraction-part I: fundamentals (1989)
Miyano, S.: The lexicographically first maximal subgraph problems: P-completeness and NC algorithms. Math. Syst. Theory 22(1), 47–73 (1989)
Oum, S.-I.: Rank-width and vertex-minors. J. Comb. Theory Ser. B 95(1), 79–100 (2005)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Abu-Khzam, F.N., Li, S., Markarian, C., Meyer auf der Heide, F., Podlipyan, P. (2017). Modular-Width: An Auxiliary Parameter for Parameterized Parallel Complexity. In: Xiao, M., Rosamond, F. (eds) Frontiers in Algorithmics. FAW 2017. Lecture Notes in Computer Science(), vol 10336. Springer, Cham. https://doi.org/10.1007/978-3-319-59605-1_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-59605-1_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-59604-4
Online ISBN: 978-3-319-59605-1
eBook Packages: Computer ScienceComputer Science (R0)