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Search-Space Reduction via Essential Vertices

Authors Benjamin Merlin Bumpus , Bart M. P. Jansen , Jari J. H. de Kroon



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Author Details

Benjamin Merlin Bumpus
  • Eindhoven University of Technology, The Netherlands
Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Jari J. H. de Kroon
  • Eindhoven University of Technology, The Netherlands

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Benjamin Merlin Bumpus, Bart M. P. Jansen, and Jari J. H. de Kroon. Search-Space Reduction via Essential Vertices. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 30:1-30:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.30

Abstract

We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Packing and covering problems
  • Theory of computation → Linear programming
  • Theory of computation → Fixed parameter tractability
Keywords
  • fixed-parameter tractability
  • essential vertices
  • covering versus packing

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References

  1. Tobias Achterberg, Robert E. Bixby, Zonghao Gu, Edward Rothberg, and Dieter Weninger. Presolve reductions in mixed integer programming. Technical Report 16-44, ZIB, Takustr.7, 14195 Berlin, 2016. URL: http://nbn-resolving.de/urn:nbn:de:0297-zib-60370.
  2. Takuya Akiba and Yoichi Iwata. Branch-and-reduce exponential/FPT algorithms in practice: A case study of vertex cover. Theor. Comput. Sci., 609:211-225, 2016. URL: https://doi.org/10.1016/j.tcs.2015.09.023.
  3. Jochen Alber, Nadja Betzler, and Rolf Niedermeier. Experiments on data reduction for optimal domination in networks. Annals OR, 146(1):105-117, 2006. URL: https://doi.org/10.1007/s10479-006-0045-4.
  4. Hans L. Bodlaender. Lower bounds for kernelization. In Marek Cygan and Pinar Heggernes, editors, Parameterized and Exact Computation - 9th International Symposium, IPEC 2014, Wroclaw, Poland, September 10-12, 2014. Revised Selected Papers, volume 8894 of Lecture Notes in Computer Science, pages 1-14. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-13524-3_1.
  5. Hans L. Bodlaender, Fedor V. Fomin, Daniel Lokshtanov, Eelko Penninkx, Saket Saurabh, and Dimitrios M. Thilikos. (Meta) Kernelization. In 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, October 25-27, 2009, Atlanta, Georgia, USA, pages 629-638. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/FOCS.2009.46.
  6. Hans L. Bodlaender and Thomas C. van Dijk. A cubic kernel for feedback vertex set and loop cutset. Theory Comput. Syst., 46(3):566-597, 2010. URL: https://doi.org/10.1007/s00224-009-9234-2.
  7. Benjamin Merlin Bumpus, Bart M. P. Jansen, and Jari J. H. de Kroon. Search-space reduction via essential vertices. CoRR, abs/2207.00386, 2022. URL: http://arxiv.org/abs/2207.00386.
  8. Yixin Cao and Dániel Marx. Chordal editing is fixed-parameter tractable. Algorithmica, 75(1):118-137, 2016. URL: https://doi.org/10.1007/s00453-015-0014-x.
  9. Jianer Chen, Iyad A. Kanj, and Ge Xia. Improved upper bounds for vertex cover. Theor. Comput. Sci., 411(40-42):3736-3756, 2010. URL: https://doi.org/10.1016/j.tcs.2010.06.026.
  10. Jianer Chen, Yang Liu, Songjian Lu, Barry O'Sullivan, and Igor Razgon. A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM, 55(5):21:1-21:19, 2008. URL: https://doi.org/10.1145/1411509.1411511.
  11. Maria Chudnovsky, Jim Geelen, Bert Gerards, Luis A. Goddyn, Michael Lohman, and Paul D. Seymour. Packing non-zero A-paths in group-labelled graphs. Comb., 26(5):521-532, 2006. URL: https://doi.org/10.1007/s00493-006-0030-1.
  12. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  13. Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM, 61(4):23:1-23:27, 2014. URL: https://doi.org/10.1145/2629620.
  14. Huib Donkers and Bart M. P. Jansen. Preprocessing to reduce the search space: Antler structures for feedback vertex set. In Lukasz Kowalik, Michal Pilipczuk, and Pawel Rzazewski, editors, Graph-Theoretic Concepts in Computer Science - 47th International Workshop, WG 2021, Warsaw, Poland, June 23-25, 2021, Revised Selected Papers, volume 12911 of Lecture Notes in Computer Science, pages 1-14. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-86838-3_1.
  15. Rodney G. Downey and M. R. Fellows. Parameterized Complexity. Springer Publishing Company, Incorporated, 2012. Google Scholar
  16. Andrew Drucker. New limits to classical and quantum instance compression. SIAM J. Comput., 44(5):1443-1479, 2015. URL: https://doi.org/10.1137/130927115.
  17. Eduard Eiben, Robert Ganian, Thekla Hamm, and O-joung Kwon. Measuring what matters: A hybrid approach to dynamic programming with treewidth. J. Comput. Syst. Sci., 121:57-75, 2021. URL: https://doi.org/10.1016/j.jcss.2021.04.005.
  18. P. Erdös and L. Pósa. On independent circuits contained in a graph. Canadian Journal of Mathematics, 17:347-352, 1965. URL: https://doi.org/10.4153/CJM-1965-035-8.
  19. Andreas Emil Feldmann, Karthik C. S., Euiwoong Lee, and Pasin Manurangsi. A survey on approximation in parameterized complexity: Hardness and algorithms. Algorithms, 13(6):146, 2020. URL: https://doi.org/10.3390/a13060146.
  20. Michael R. Fellows. The lost continent of polynomial time: Preprocessing and kernelization. In Proc. 2nd IWPEC, pages 276-277, 2006. URL: https://doi.org/10.1007/11847250_25.
  21. Fedor Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: theory of parameterized preprocessing. Cambridge University Press, 2019. Google Scholar
  22. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar F-deletion: Approximation, kernelization and optimal FPT algorithms. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 470-479. IEEE Computer Society, 2012. URL: https://doi.org/10.1109/FOCS.2012.62.
  23. Jim Geelen, Bert Gerards, Bruce A. Reed, Paul D. Seymour, and Adrian Vetta. On the odd-minor variant of Hadwiger’s conjecture. J. Comb. Theory, Ser. B, 99(1):20-29, 2009. URL: https://doi.org/10.1016/j.jctb.2008.03.006.
  24. M.C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. ISSN. Elsevier Science, 2004. Google Scholar
  25. Jiong Guo and Rolf Niedermeier. Invitation to data reduction and problem kernelization. SIGACT News, 38(1):31-45, 2007. URL: https://doi.org/10.1145/1233481.1233493.
  26. Pinar Heggernes, Pim van 't Hof, Bart M. P. Jansen, Stefan Kratsch, and Yngve Villanger. Parameterized complexity of vertex deletion into perfect graph classes. Theor. Comput. Sci., 511:172-180, 2013. URL: https://doi.org/10.1016/j.tcs.2012.03.013.
  27. Bart M. P. Jansen, Jari J. H. de Kroon, and Michal Wlodarczyk. Vertex deletion parameterized by elimination distance and even less. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1757-1769. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451068.
  28. Subhash Khot. On the power of unique 2-prover 1-round games. In John H. Reif, editor, Proceedings on 34th Annual ACM Symposium on Theory of Computing, May 19-21, 2002, Montréal, Québec, Canada, pages 767-775. ACM, 2002. URL: https://doi.org/10.1145/509907.510017.
  29. Stefan Kratsch. Recent developments in kernelization: A survey. Bull. EATCS, 113, 2014. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/285.
  30. Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization. J. ACM, 67(3):16:1-16:50, 2020. URL: https://doi.org/10.1145/3390887.
  31. Jason Li and Jesper Nederlof. Detecting feedback vertex sets of size k in O^*(2.7^k) time. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 971-989. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.58.
  32. Daniel Lokshtanov. Wheel-free deletion is W[2]-hard. In Martin Grohe and Rolf Niedermeier, editors, Parameterized and Exact Computation, Third International Workshop, IWPEC 2008, Victoria, Canada, May 14-16, 2008. Proceedings, volume 5018 of Lecture Notes in Computer Science, pages 141-147. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-79723-4_14.
  33. Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Kernelization - Preprocessing with a guarantee. In Hans L. Bodlaender, Rod Downey, Fedor V. Fomin, and Dániel Marx, editors, The Multivariate Algorithmic Revolution and Beyond - Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, volume 7370 of Lecture Notes in Computer Science, pages 129-161. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-30891-8_10.
  34. Daniel Lokshtanov, N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. Faster parameterized algorithms using linear programming. ACM Trans. Algorithms, 11(2):15:1-15:31, 2014. URL: https://doi.org/10.1145/2566616.
  35. Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. Parameterized complexity and approximability of directed odd cycle transversal. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 2181-2200. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.134.
  36. Konstantin Makarychev and Yury Makarychev. Perturbation resilience. In Tim Roughgarden, editor, Beyond the Worst-Case Analysis of Algorithms, pages 95-119. Cambridge University Press, 2020. URL: https://doi.org/10.1017/9781108637435.008.
  37. George L. Nemhauser and Leslie E. Trotter Jr. Vertex packings: Structural properties and algorithms. Math. Program., 8(1):232-248, 1975. URL: https://doi.org/10.1007/BF01580444.
  38. Bruce A. Reed, Kaleigh Smith, and Adrian Vetta. Finding odd cycle transversals. Oper. Res. Lett., 32(4):299-301, 2004. URL: https://doi.org/10.1016/j.orl.2003.10.009.
  39. Neil Robertson and Paul D. Seymour. Graph minors. XIII. The disjoint paths problem. J. Comb. Theory, Ser. B, 63(1):65-110, 1995. URL: https://doi.org/10.1006/jctb.1995.1006.
  40. Karthik C. S., Bundit Laekhanukit, and Pasin Manurangsi. On the parameterized complexity of approximating dominating set. J. ACM, 66(5):33:1-33:38, 2019. URL: https://doi.org/10.1145/3325116.
  41. A. Schrijver. Combinatorial Optimization - Polyhedra and Efficiency. Springer, 2003. Google Scholar
  42. Leslie G. Valiant and Vijay V. Vazirani. NP is as easy as detecting unique solutions. Theor. Comput. Sci., 47(3):85-93, 1986. URL: https://doi.org/10.1016/0304-3975(86)90135-0.
  43. Karsten Weihe. Covering trains by stations or the power of data reduction. In Algorithms and Experiments (ALEX98), pages 1-8, 1998. URL: https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.57.2173.
  44. Karsten Weihe. On the differences between "Practical" and "Applied". In Stefan Näher and Dorothea Wagner, editors, Algorithm Engineering, 4th International Workshop, WAE 2000, Saarbrücken, Germany, September 5-8, 2000, Proceedings, volume 1982 of Lecture Notes in Computer Science, pages 1-10. Springer, 2000. URL: https://doi.org/10.1007/3-540-44691-5_1.
  45. Sebastian Wernicke. On the algorithmic tractability of single nucleotide polymorphism (SNP) analysis and related problems. diplom.de, 2014. Google Scholar
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