Abstract
Measure & Conquer (M&C) is a prominent technique for analyzing exact algorithms for computationally hard problems, in particular, graph problems. It tries to balance worse and better situations within the algorithm analysis. This has led, e.g., to algorithms for Minimum Vertex Cover with a running time of \(\mathcal{O}(c^{n})\) for some constant c≈1.2, where n is the number of vertices in the graph.
Several obstacles prevent the application of this technique in parameterized algorithmics, making it rarely applied in this area. However, these difficulties can be handled in some situations. We will exemplify this with two problems related to Vertex Cover, namely Connected Vertex Cover and Edge Dominating Set. For both problems, several parameterized algorithms have been published, all based on the idea of first enumerating minimal vertex covers. Using M&C in this context will allow us to improve on the hitherto published running times. In contrast to some of the earlier suggested algorithms, ours will use polynomial space.
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References
Abu-Khzam, F.N., Collins, R.L., Fellows, M.R., Langston, M.A., Sutters, W.H., Symons, C.T.: Kernelization algorithms for the vertex cover problem: theory and experiments. In: Arge, L., Italiano, G.F., Sedgewick, R. (eds.) Proceedings of the Sixth Workshop on Algorithm Engineering and Experiments and the First Workshop on Analytic Algorithmics and Combinatorics pp. 62–69. SIAM, Philadelphia (2004)
Binkele-Raible, D., Fernau, H.: A new upper bound for Max-2-SAT: a graph-theoretic approach. J. Discrete Algorithms 8, 388–401 (2010)
Binkele-Raible, D., Fernau, H.: Enumerate and measure, improving parameter budget management. In: Raman, V., Saurabh, S. (eds.) Parameterized and Exact Computation (IPEC 2010). Lecture Notes in Computer Science, vol. 6478, pp. 38–49. Springer, Berlin (2010)
Chen, J., Kanj, I.A., Xia, G.: Labeled search trees and amortized analysis: improved upper bounds for NP-hard problems. Algorithmica 43, 245–273 (2005)
Daligault, J., Gutin, G., Kim, E.J., Yeo, A.: FPT algorithms and kernels for the directed k-leaf problem. J. Comput. Syst. Sci. 76, 144–152 (2010)
Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through colors and IDs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S.E., Thomas, W. (eds.) International Colloquium on Automata, Languages and Programming (ICALP 2009), Part I. Lecture Notes in Computer Science, vol. 5555, pp. 378–389. Springer, Berlin (2009)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)
Fernau, H.: Edge dominating set: efficient enumeration-based exact algorithms. In: Bodlaender, H.L., Langston, M. (eds.) International Workshop on Parameterized and Exact Computation (IWPEC 2006). Lecture Notes in Computer Science, vol. 4169, pp. 142–153. Springer, Berlin (2006)
Fernau, H., Fomin, F.V., Philip, G., Saurabh, S.: The curse of connectivity: t-total vertex (edge) cover. In: Thai, M.T., Sahni, S. (eds.) International Computing and Combinatorics Conference (COCOON 2010). Lecture Notes in Computer Science, vol. 6196, pp. 34–43. Springer, Berlin (2010)
Fernau, H., Gaspers, S., Raible, D.: Exact and parameterized algorithms for Max Internal Spanning Tree. In: Paul, C., Habib, M. (eds.) Graph-Theoretic Concepts in Computer Science (WG 2009). Lecture Notes in Computer Science, vol. 5911, pp. 100–111. Springer, Berlin (2010). Long version accepted for publication with Algorithmica
Fernau, H., Manlove, D.F.: Vertex and edge covers with clustering properties: Complexity and algorithms. J. Discrete Algorithms 7, 149–167 (2009)
Fernau, H., Raible, D.: Exact algorithms for maximum acyclic subgraph on a superclass of cubic graphs. In: Nakano, S.-I., Rahman, Md.S. (eds.) Workshop on Algorithms and Computation (WALCOM 2008). Lecture Notes in Computer Science, vol. 4921, pp. 144–156. Springer, Berlin (2008)
Fomin, F.V., Gaspers, S., Saurabh, S., Stepanov, A.A.: On two techniques of combining branching and treewidth. Algorithmica 54, 181–207 (2009)
Fomin, F.V., Grandoni, F., Kratsch, D.: A Measure & Conquer approach for the analysis of exact algorithms. J. ACM 56(5) (2009)
Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Texts in Theoretical Computer Science. Springer, Berlin (2010)
Fujito, T.: On approximability of the independent/connected edge dominating set problems. Inf. Process. Lett. 79, 261–266 (2001)
Fujito, T., Doi, T.: A 2-approximation NC algorithm for connected vertex cover and tree cover. Inf. Process. Lett. 90, 59–63 (2004)
Gaspers, S.: Measure & Conquer for parameterized branching algorithms. Parameterized Complexity News from September 2009 (pp. 5–6)
Gaspers, S., Sorkin, G.B.: A universally fastest algorithms for Max 2-Sat, Max 2-CSP, and everything in between. In: Symposium on Discrete Algorithms (SODA 2009), pp. 606–615. ACM Press, New York (2009)
Guo, J., Niedermeier, R., Wernicke, S.: Parameterized complexity of vertex cover variants. Theory Comput. Syst. 41, 501–520 (2007)
Kneis, J., Langer, A., Rossmanith, P.: A new algorithm for finding trees with many leaves. In: International Symposium on Algorithms and Computation (ISAAC 2008). Lecture Notes in Computer Science, vol. 5369, pp. 270–281. Springer, Berlin (2008)
Mölle, D., Richter, S., Rossmanith, P.: Enumerate and expand: improved algorithms for connected vertex cover and tree cover. Theory Comput. Syst. 43, 234–253 (2008)
Nederlof, J.: Fast polynomial-space algorithms using Möbius inversion: improving on Steiner tree and related problems. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S.E., Thomas, W. (eds.) International Colloquium on Automata, Languages and Programming (ICALP 2009), Part I. Lecture Notes in Computer Science, vol. 5555, pp. 713–725. Springer, Berlin (2009)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, London (2006)
Plesník, J.: Equivalence between the minimum covering problem and the maximum matching problem. Discrete Math. 49, 315–317 (1984)
Plesník, J.: Constrained weighted matchings and edge coverings in graphs. Discrete Appl. Math. 92, 229–241 (1999)
Prieto, E.: Systematic kernelization in FPT algorithm design. PhD thesis, The University of Newcastle, Australia (2005)
Raible, D., Fernau, H.: An amortized search tree analysis for k-Leaf Spanning Tree. In: van Leeuwen, J., Muscholl, A., Peleg, D., Pokorný, J., Rumpe, B. (eds.) Theory and Practice of Computer Science, Software Seminar (SOFSEM 2010). Lecture Notes in Computer Science, vol. 5901, pp. 672–684. Springer, Berlin (2010)
Raman, V., Saurabh, S., Sikdar, S.: Improved exact exponential algorithms for vertex bipartization and other problems. In: Coppo, M., Lodi, E., Pinna, G.M. (eds.) Italian Conference on Theoretical Computer Science (ICTCS 2005). Lecture Notes in Computer Science, vol. 3701, pp. 375–389. Springer, Berlin (2005)
Razgon, I.: Faster computation of maximum independent set and parameterized vertex cover for graphs with maximum degree 3. J. Discrete Algorithms 7(2), 191–212 (2009)
van Rooij, J.M.M., Bodlaender, H.L.: Design by Measure and Conquer, a faster exact algorithm for dominating set. In: Albers, S., Weil, P. (eds.) International Symposium on Theoretical Aspects of Computer Science (STACS 2008). LIPIcs, vol. 1, pp. 657–668. Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Hamburg (2008)
van Rooij, J.M.M., Bodlaender, H.L.: Exact algorithms for edge domination. In: Grohe, M., Niedermeier, R. (eds.) Parameterized and Exact Computation (IWPEC 2008). Lecture Notes in Computer Science, vol. 5018, pp. 214–225. Springer, Berlin (2008). Details on the parameterized algorithm can be found in the TR version. The long version was accepted for publication with Algorithmica
van Rooij, J.M.M., Nederlof, J., van Dijk, T.C.: Inclusion/exclusion meets Measure and Conquer. In: Fiat, A., Sanders, P. (eds.) Algorithms—ESA 2009, 17th Annual European Symposium. Lecture Notes in Computer Science, vol. 5757, pp. 554–565. Springer, Berlin (2009)
Truss, A., Weller, M.: Lessons in magic: AGAPE Spring School (May 24–29, Corsica). Parameterized Complexity News from September 2009 (pp. 2–3; supplemented by Fig. 1 on p. 1)
Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM J. Appl. Math. 38, 364–372 (1980)
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Binkele-Raible, D., Fernau, H. Parameterized Measure & Conquer for Problems with No Small Kernels. Algorithmica 64, 189–212 (2012). https://doi.org/10.1007/s00453-011-9566-6
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DOI: https://doi.org/10.1007/s00453-011-9566-6