Abstract
In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A well-developed example is the investigation of how the parameter treewidth influences the complexity of (other) graph problems. The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce the problem instances (not necessarily in obvious, or well-understood ways). The max leaf number of a connected graph G is the maximum number of leaves in a spanning tree for G. Exploring questions analogous to the well-studied case of treewidth, we can ask: how hard is it to solve 3-Coloring or Hamilton Path or Minimum Dominating Set for graphs of bounded max leaf number? We do two things:
(1) We describe much improved FPT algorithms for a large number of graph problems, for input of bounded max leaf number, based on the polynomial-time extremal structure theory associated to the parameter max leaf number.
(2) The way that we obtain these concrete algorithmic results is general and systematic. We describe the approach.
This research has been supported by the Australian Research Council through the Australian Centre for Bioinformatics, by the University of Newcastle Parameterized Complexity Research Unit under the auspices of the Deputy Vice-Chancellor for Research, and by a Fellowship to the Durham University Institute for Advanced Studies. The authors also gratefully acknowledge the support and kind hospitality provided by a William Best Fellowship at Grey College while the paper was in preparation.
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References
Abu-Khzam, F.N., Collins, R.L., Fellows, M.R., Langston, M.A., Suters, W.H., Symons, C.T.: Kernelization algorithms for the vertex cover problem: theory and experiments. In: Arge, L., Italiano, G., Sedgewick, R. (eds.) Proceedings of the 6th Workshop on Algorithm Engineering and Experiments (ALENEX), New Orleans, ACM/SIAM, Proc. Applied Mathematics 115 (January 2004)
Alber, J., Fellows, M., Niedermeier, R.: Polynomial time data reduction for dominating set. Journal of the ACM 51, 363–384 (2004)
Alber, J., Niedermeier, R.: Improved Tree Decomposition Based Algorithms for Domination-Like Problems. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 613–627. Springer, Heidelberg (2002)
Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12, 308–340 (1991)
Bodlaender, H.L., Koster, A.M.: Combinatorial optimisation on graphs of bounded treewidth. The Computer Journal (to appear 2007)
Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small width. SIAM J. Computing 25, 1305–1317 (1996)
Chor, B., Fellows, M., Juedes, D.: Linear kernels in linear time, or how to save k colors in O(n 2) steps. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 257–269. Springer, Heidelberg (2004)
Chen, J., Kanj, I., Xia, G.: Simplicity is beauty: improved upper bounds for vertex cover. Manuscript communicated by email, April (2005)
Courcelle, B.: The monadic second order logic of graphs I: Recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs. Journal of the ACM 52, 866–893 (2005)
Downey, R., Fellows, M., Stege, U.: Parameterized complexity: a framework for systematically confronting computational intractability. In: Graham, R., Kratochvil, J., Nesetril, J., Roberts, F. (eds.) Contemporary Trends in Discrete Mathematics, Proceedings of the DIMACS-DIMATIA Workshop on the Future of Discrete Mathematics, Prague, 1997. AMS-DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 49, pp. 49–99 (1999)
Demaine, E.D., Hajiaghayi, M.: Bidimensionality: New connections between FPT algorithms and PTASs. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), Vancouver, pp. 590–601. ACM Press, New York (2005)
Demaine, E.D., Hajiaghayi, M.: The bidimensionality theory and its algorithmic applications. The Computer Journal (to appear)
Estivill-Castro, V., Fellows, M., Langston, M., Rosamond, F.: Fixed-parameter tractability is P-time extremal structure theory I: The case of max leaf. In: Proceedings of ACiD 2005: Algorithms and Complexity in Durham, pp. 1–41 (2005)
Fellows, M.: Parameterized complexity: the main ideas and connections to practical computing. In: Fleischer, R., Moret, B.M.E., Schmidt, E.M. (eds.) Experimental Algorithmics. LNCS, vol. 2547, pp. 51–77. Springer, Heidelberg (2002)
Flum, J., Frick, M., Grohe, M.: Query evaluation via tree-decompositions. In: Van den Bussche, J., Vianu, V. (eds.) ICDT 2001. LNCS, vol. 1973, pp. 22–32. Springer, Heidelberg (2001)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Fomin, F., Kratsch, D., Woeginger, G.: Exact (exponential) algorithms for the dominating set problem. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 245–256. Springer, Heidelberg (2004)
Grohe, M., Marino, J.: Definability and descriptive complexity on databases with bounded treewidth. In: Beeri, C., Bruneman, P. (eds.) ICDT 1999. LNCS, vol. 1540, pp. 70–82. Springer, Heidelberg (1998)
Grohe, M.: The parameterized complexity of database queries. In: Proc. PODS 2001, pp. 82–92. ACM Press, New York (2001)
Gurevich, Y.: The Challenger-Solver Game: Variations on the Theme of P=? NP, Bulletin EATCS 39 pp. 112–121 (1989)
Guo, J., Gramm, J., Hueffner, F., Niedermeier, R., Wernicke, S.: Improved fixed-parameter algorithms for two feedback set problems. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 158–169. Springer, Heidelberg (2005)
Niedermeier, R.: Ubiquitous parameterization — invitation to fixed-parameter algorithms. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 84–103. Springer, Heidelberg (2004)
Niedermeier, R.: Invitation to Fixed Parameter Algorithms (forthcoming). Oxford University Press, Oxford (2005)
Nemhauser, G.L., Trotter, L.E.: Vertex packings: structural properties and algorithms. Mathematical Programming 8, 232–248 (1975)
Prieto-Rodriguez, E.: Systematic kernelization in FPT algorithm design. Ph.D. Thesis, School of EE&CS, University of Newcastle, Australia (2005)
Raman, V.: Parameterized Complexity. In: Proceedings of the 7th National Seminar on Theoretical Computer Science, Chennai, India, pp. 1–18 (1997)
Robertson, N., Seymour, P.: Graph minors: a survey. In: Anderson, J. (ed.) Surveys in Combinatorics, pp. 153–171. Cambridge University Press, Cambridge (1985)
Telle, J.A., Proskurowski, A.: Practical Algorithms on Partial k-Trees with an Application to Domination-Like Problems. In: Dehne, F., Sack, J.-R., Santoro, N. (eds.) WADS 1993. LNCS, vol. 709, pp. 610–621. Springer, Heidelberg (1993)
Weihe, K.: Covering trains by stations, or the power of data reduction. In: Proc. ALEX’98 pp. 1–8 (1998)
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Fellows, M., Rosamond, F. (2007). The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_28
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DOI: https://doi.org/10.1007/978-3-540-73001-9_28
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