Jiong Guo
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Abstract
To solve NP-hard problems, polynomial-time preprocessing is a natural and promising approach. Preprocessing is based on data reduction techniques that take a problem's input instance and try to perform a reduction to a smaller, equivalent problem kernel. Problem kernelization is a methodology that is rooted in parameterized computational complexity. In this brief survey, we present data reduction and problem kernelization as a promising research field for algorithm and complexity theory.
- F. N. Abu-Khzam, R. L. Collins, M. R. Fellows, M. A. Langston, W. H. Suters, and C. T. Symons. Kernelization algorithms for the Vertex Cover problem: Theory and experiments. In Proc. 6th ACM-SIAM ALENEX, pages 62--69. ACM-SIAM, 2004.Google Scholar
- J. Alber, B. Dorn, and R. Niedermeier. A general data reduction scheme for domination in graphs. In Proc. 32nd SOFSEM, volume 3831 of LNCS, pages 137--147. Springer, 2006. Google ScholarDigital Library
- J. Alber, M. R. Fellows, and R. Niedermeier. Polynomial time data reduction for Dominating Set. Journal of the ACM, 51(3):363--384, 2004. Google ScholarDigital Library
- B. S. Baker. Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM, 41(1):153--180, 1994. Google ScholarDigital Library
- R. E. Bixby. Solving real-world linear programs: A decade and more of progress. Operations Research, 50:3--15, 2002. Google ScholarDigital Library
- H. L. Bodlaender. A cubic kernel for feedback vertex set. In Proc. 24th STACS, LNCS. Springer, 2007. Google ScholarDigital Library
- K. Burrage, V. Estivill-Castro, M. Fellows, M. Langston, S. Mac, and F. Rosamond. The undirected Feedback Vertex Set problem has a poly(k) kernel. In Proc. 2nd IWPEC, volume 4196 of LNCS, pages 192--202. Springer, 2006. Google ScholarDigital Library
- M. Cadoli, F. M. Donini, P. Liberatore, and M. Schaerf. Preprocessing of intractable problems. Information and Computation, 176(2):89--120, 2002. Google ScholarDigital Library
- L. Cai, J. Chen, R. G. Downey, and M. R. Fellows. Advice classes of parameterized tractability. Annals of Pure and Applied Logic, 84:119--138, 1997.Google ScholarCross Ref
- M. Charikar, V. Guruswami, and A. Wirth. Clustering with qualitative information. Journal of Computer and System Sciences, 71(3):360--383, 2005. Google ScholarDigital Library
- J. Chen, H. Fernau, I. A. Kanj, and G. Xia. Parametric duality and kernelization: Lower bounds and upper bounds on kernel size. In Proc. 22nd STACS, volume 3404 of LNCS, pages 269--280. Springer, 2005. Google ScholarDigital Library
- J. Chen, I. A. Kanj, and W. Jia. Vertex Cover: Further observations and further improvements. Journal of Algorithms, 41:280--301, 2001. Google ScholarDigital Library
- B. Chor, M. Fellows, and D. W. Juedes. Linear kernels in linear time, or how to save k colors in O(n
2) steps. In Proc. 30th WG, volume 3353 of LNCS, pages 257--269. Springer, 2004. Google ScholarDigital Library
- P. Damaschke. Parameterized enumeration, transversals, and imperfect phylogeny reconstruction. Theoretical Computer Science, 351(3):337--350, 2006. Google ScholarDigital Library
- I. Dinur and S. Safra. The importance of being biased. In Proc. 34th ACM STOC, pages 33--42. ACM, 2002. Google ScholarDigital Library
- R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer, 1999. Google ScholarDigital Library
- V. Estivill-Castro, M. R. Fellows, M. A. Langston, and F. Rosamond. FPT is P-time extremal structure I. In Proc. 1st ACiD, volume 4 of Texts in Algorithmics, pages 1--41. King's College, 2005.Google Scholar
- J. Flum and M. Grohe. Parameterized Complexity Theory. Springer, 2006. Google ScholarDigital Library
- J. Gramm, J. Guo, F. Hüffner, and R. Niedermeier. Graph-modeled data clustering: Exact algorithms for clique generation. Theory of Computing Systems, 38(4):373--392, 2005. Google ScholarDigital Library
- J. Gramm, J. Guo, F. Hüffner, and R. Niedermeier. Data reduction, exact, and heuristic algorithms for Clique Cover. In Proc. 8th ACM-SIAM ALENEX, pages 86--94. ACM-SIAM, 2006. Long version to appear in The ACM Journal of Experimental Algorithmics. Google ScholarDigital Library
- J. Guo. A more effective linear kernelization for Cluster Editing, November 2006. Submitted.Google Scholar
- J. Guo and R. Niedermeier. Fixed-parameter tractability and data reduction for Multicut in Trees. Networks, 46(3):124--135, 2005. Google ScholarDigital Library
- J. Guo, R. Niedermeier, and S. Wernicke. Parameterized complexity of generalized Vertex Cover problems. In Proc. 9th WADS, volume 3608 of LNCS, pages 36--48. Springer, 2005. Long version to appear under the title "Parameterized complexity of Vertex Cover variants" in Theory of Computing Systems. Google ScholarDigital Library
- J. Guo, R. Niedermeier, and S. Wernicke. Fixed-parameter tractability results for full-degree spanning tree and its dual. In Proc. 2nd IWPEC, volume 4196 of LNCS, pages 203--214. Springer, 2006. Google ScholarDigital Library
- F. Hüffner, R. Niedermeier, and S. Wernicke. Techniques for practical fixed-parameter algorithms. To appear in The Computer Journal, 2007.Google Scholar
- S. Khot and O. Regev. Vertex Cover might be hard to approximate to within 2 - ε. In Proc. 18th IEEE Annual Conference on Computational Complexity, pages 379--386. IEEE, 2003.Google ScholarCross Ref
- P. Liberatore. Monotonic reductions, representative equivalence, and compilation of intractable problems. Journal of the ACM, 48(6):1091--1125, 2001. Google ScholarDigital Library
- D. Marx. Parameterized complexity and approximation algorithms. To appear in The Computer Journal, 2007. Google ScholarDigital Library
- G. L. Nemhauser and L. E. Trotter. Vertex packing: Structural properties and algorithms. Mathematical Programming, 8:232--248, 1975.Google ScholarDigital Library
- R. Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford University Press, 2006.Google Scholar
- R. Niedermeier and P. Rossmanith. A general method to speed up fixed-parameter-tractable algorithms. Information Processing Letters, 73:125--129, 2000. Google ScholarDigital Library
- W. V. Quine. The problem of simplifying truth functions. American Mathematical Monthly, 59:512--531, 1952.Google ScholarCross Ref
- M. Weston. A fixed-parameter tractable algorithm for matrix domination. Information Processing Letters, 90:267--272, 2004. Google ScholarDigital Library
Index Terms
- Invitation to data reduction and problem kernelization
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