Abstract
Data reduction techniques are widely applied to deal with computationally hard problems in real world applications. It has been a long-standing challenge to formally express the efficiency and accuracy of these “pre-processing” procedures. The framework of parameterized complexity turns out to be particularly suitable for a mathematical analysis of pre-processing heuristics. A kernelization algorithm is a pre-processing algorithm which simplifies the instances given as input in polynomial time, and the extent of simplification desired is quantified with the help of the additional parameter.
We give an overview of some of the early work in the area and also survey newer techniques that have emerged in the design and analysis of kernelization algorithms. We also outline the framework of Bodlaender et al. [9] and Fortnow and Santhanam [38] for showing kernelization lower bounds under reasonable assumptions from classical complexity theory, and highlight some of the recent results that strengthen and generalize this framework.
Keywords
- Planar Graph
- Vertex Cover
- Parameterized Problem
- Polynomial Kernel
- Truth Assignment
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
Abu-Khzam, F.N.: A kernelization algorithm for d-hitting set. J. Comput. Syst. Sci. 76(7), 524–531 (2010)
Alber, J., Fellows, M.R., Niedermeier, R.: Polynomial-time data reduction for dominating set. Journal of the ACM 51(3), 363–384 (2004)
Alon, N., Gutin, G., Kim, E.J., Szeider, S., Yeo, A.: Solving MAX-r-SAT above a tight lower bound. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), pp. 511–517. SIAM (2010)
Alon, N., Gutin, G., Krivelevich, M.: Algorithms with large domination ratio. J. Algorithms 50, 118–131 (2004)
Alon, N., Yuster, R., Zwick, U.: Color-coding. J. Assoc. Comput. Mach. 42(4), 844–856 (1995)
Arnborg, S., Courcelle, B., Proskurowski, A., Seese, D.: An algebraic theory of graph reduction. J. ACM 40(5), 1134–1164 (1993)
Bodlaender, H., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) Kernelization. In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009), pp. 629–638. IEEE (2009)
Bodlaender, H.L., de Fluiter, B.: Reduction Algorithms for Constructing Solutions in Graphs with Small Treewidth. In: Cai, J.-Y., Wong, C.K. (eds.) COCOON 1996. LNCS, vol. 1090, pp. 199–208. Springer, Heidelberg (1996)
Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)
Bodlaender, H.L., Hagerup, T.: Parallel algorithms with optimal speedup for bounded treewidth. SIAM J. Comput. 27, 1725–1746 (1998)
Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: A new technique for kernelization lower bounds. In: Proceedings of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). LIPIcs, vol. 9, pp. 165–176. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2011)
Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Preprocessing for Treewidth: A Combinatorial Analysis through Kernelization. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 437–448. Springer, Heidelberg (2011)
Bodlaender, H.L., Thomassé, S., Yeo, A.: Analysis of data reduction: Transformations give evidence for non-existence of polynomial kernels, Tech. Report CS-UU-2008-030, Department of Information and Computer Sciences, Utrecht University, Utrecht, The Netherlands (2008)
Bodlaender, H.L., van Antwerpen-de Fluiter, B.: Reduction algorithms for graphs of small treewidth. Inf. Comput. 167(2), 86–119 (2001)
Bourgain, J.: Walsh subspaces of lp-product space. Seminar on Functional Analysis, Exp. (4A), 9 (1980)
Burrage, K., Estivill-Castro, V., Fellows, M.R., Langston, M.A., Mac, S., Rosamond, F.A.: The Undirected Feedback Vertex Set Problem Has a Poly(k) Kernel. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 192–202. Springer, Heidelberg (2006)
Chen, J., Kanj, I.A., Jia, W.: Vertex cover: further observations and further improvements. Journal of Algorithms 41, 280–301 (2001)
Chor, B., Fellows, M., Juedes, D.W.: Linear Kernels in Linear Time, or How to Save k Colors in o(n2) Steps. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 257–269. Springer, Heidelberg (2004)
Crowston, R., Gutin, G., Jones, M., Kim, E.J., Ruzsa, I.Z.: Systems of Linear Equations over \(\mathbb{F}_2\) and Problems Parameterized above Average. In: Kaplan, H. (ed.) SWAT 2010. LNCS, vol. 6139, pp. 164–175. Springer, Heidelberg (2010)
Cygan, M., Kratsch, S., Pilipczuk, M., Pilipczuk, M., Wahlström, M.: Clique cover and graph separation: New incompressibility results. CoRR, abs/1111.0570 (2011)
de Fluiter, B.: Algorithms for Graphs of Small Treewidth. PhD thesis, Utrecht University (1997)
Dell, H., Marx, D.: Kernelization of packing problems. In: SODA, pp. 68–81 (2012)
Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: STOC, pp. 251–260 (2010)
Diestel, R.: Graph theory, 3rd edn. Graduate Texts in Mathematics, vol. 173. Springer, Berlin (2005)
Dom, M., Guo, J., Hüffner, F., Niedermeier, R., Truß, A.: Fixed-Parameter Tractability Results for Feedback Set Problems in Tournaments. In: Calamoneri, T., Finocchi, I., Italiano, G.F. (eds.) CIAC 2006. LNCS, vol. 3998, pp. 320–331. Springer, Heidelberg (2006)
Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, Heidelberg (1999)
Downey, R.G., Fellows, M.R., Stege, U.: Computational tractability: the view from Mars. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS (69), 73–97 (1999)
Drucker, A.: On the hardness of compressing an AND of SAT instances, Theory Lunch, February 17, Center for Computational Intractability (2012), http://intractability.princeton.edu/blog/2012/03/theory-lunch-february-17/
Erdős, P., Rado, R.: Intersection theorems for systems of sets. J. London Math. Soc. 35, 85–90 (1960)
Fellows, M.R.: The Lost Continent of Polynomial Time: Preprocessing and Kernelization. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 276–277. Springer, Heidelberg (2006)
Fellows, M.R., Langston, M.A.: An analogue of the myhill-nerode theorem and its use in computing finite-basis characterizations (extended abstract). In: FOCS, pp. 520–525 (1989)
Fellows, M.R., Lokshtanov, D., Misra, N., Mnich, M., Rosamond, F.A., Saurabh, S.: The complexity ecology of parameters: An illustration using bounded max leaf number. Theory Comput. Syst. 45(4), 822–848 (2009)
Fernau, H., Fomin, F.V., Lokshtanov, D., Raible, D., Saurabh, S., Villanger, Y.: Kernel(s) for problems with no kernel: On out-trees with many leaves. In: STACS 2009, pp. 421–432. Schloss Dagstuhl—Leibniz-Zentrum fuer Informatik (2009)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science, An EATCS Series, Springer, Berlin (2006)
Fomin, F., Lokshtanov, D., Misra, N., Saurabh, S.: Planar-\({\cal F}\) Deletion: Approximation, Kernelization and Optimal FPT algorithms (2012) (unpublished manuscript)
Fomin, F.V., Lokshtanov, D., Misra, N., Philip, G., Saurabh, S.: Hitting forbidden minors: Approximation and kernelization. In: Proceedings of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). LIPIcs, vol. 9, pp. 189–200. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2011)
Fomin, F.V., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Bidimensionality and kernels. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), pp. 503–510. SIAM (2010)
Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: STOC 2008: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 133–142. ACM (2008)
Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1990)
Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Data reduction and exact algorithms for clique cover. ACM Journal of Experimental Algorithmics 13 (2008)
Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38, 31–45 (2007)
Gutin, G., Kim, E.J., Szeider, S., Yeo, A.: A probabilistic approach to problems parameterized above or below tight bounds. J. Comput. Syst. Sci. 77, 422–429 (2011)
Gutin, G., van Iersel, L., Mnich, M., Yeo, A.: All Ternary Permutation Constraint Satisfaction Problems Parameterized above Average Have Kernels with Quadratic Numbers of Variables. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part I. LNCS, vol. 6346, pp. 326–337. Springer, Heidelberg (2010)
Hall, P.: On representatives of subsets. J. London Math. Soc. 10, 26–30 (1935)
Håstad, J., Venkatesh, S.: On the advantage over a random assignment. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC 2002), pp. 43–52. ACM (2002)
Hermelin, D., Kratsch, S., Soltys, K., Wahlström, M., Wu, X.: Hierarchies of inefficient kernelizability. CoRR, abs/1110.0976 (2011)
Jansen, B.M.P., Bodlaender, H.L.: Vertex cover kernelization revisited: Upper and lower bounds for a refined parameter. In: Proceedings of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). LIPIcs, vol. 9, pp. 177–188. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2011)
Jansen, B.M.P., Kratsch, S.: Data Reduction for Graph Coloring Problems. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 90–101. Springer, Heidelberg (2011)
Kőnig, D.: Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Math. Ann. 77, 453–465 (1916)
Kratsch, S.: Co-nondeterminism in compositions: a kernelization lower bound for a ramsey-type problem. In: SODA, pp. 114–122 (2012)
Kratsch, S., Wahlström, M.: Representative sets and irrelevant vertices: New tools for kernelization. CoRR, abs/1111.2195 (2011)
Kratsch, S., Wahlström, M.: Compression via matroids: a randomized polynomial kernel for odd cycle transversal. In: SODA, pp. 94–103 (2012)
Lokshtanov, D.: Phd thesis, New Methods in Parameterized Algorithms and Complexity (2009)
Mahajan, M., Raman, V.: Parameterizing above guaranteed values: Maxsat and maxcut. J. Algorithms 31(2), 335–354 (1999)
Misra, N., Raman, V., Saurabh, S.: Lower bounds on kernelization. Discrete Optim. 8, 110–128 (2011)
Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford Lecture Series in Mathematics and its Applications, vol. 31. Oxford University Press, Oxford (2006)
Niedermeier, R.: Reflections on multivariate algorithmics and problem parameterization. In: Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS 2010). LIPIcs, vol. 5, pp. 17–32. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2010)
Quine, W.V.: The problem of simplifying truth functions. Amer. Math. Monthly 59, 521–531 (1952)
Stockmeyer, L.J.: The polynomial-time hierarchy. Theor. Comp. Sc. 3, 1–22 (1976)
Thomassé, S.: A quadratic kernel for feedback vertex set. ACM Transactions on Algorithms 6 (2010)
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Lokshtanov, D., Misra, N., Saurabh, S. (2012). Kernelization – Preprocessing with a Guarantee. In: Bodlaender, H.L., Downey, R., Fomin, F.V., Marx, D. (eds) The Multivariate Algorithmic Revolution and Beyond. Lecture Notes in Computer Science, vol 7370. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30891-8_10
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