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# Faster Parameterized Algorithms for Modification Problems to Minor-Closed Classes

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## Acknowledgements

We would like to thank the reviewers for helpful remarks that improved the presentation of the article.

## Cite As

Laure Morelle, Ignasi Sau, Giannos Stamoulis, and Dimitrios M. Thilikos. Faster Parameterized Algorithms for Modification Problems to Minor-Closed Classes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 93:1-93:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.93

## Abstract

Let G be a minor-closed graph class and let G be an n-vertex graph. We say that G is a k-apex of G if G contains a set S of at most k vertices such that G⧵S belongs to G. Our first result is an algorithm that decides whether G is a k-apex of G in time 2^poly(k)⋅n². This algorithm improves the previous one, given by Sau, Stamoulis, and Thilikos [ICALP 2020, TALG 2022], whose running time was 2^poly(k)⋅n³. The elimination distance of G to G, denoted by ed_G(G), is the minimum number of rounds required to reduce each connected component of G to a graph in G by removing one vertex from each connected component in each round. Bulian and Dawar [Algorithmica 2017] proved the existence of an FPT-algorithm, with parameter k, to decide whether ed_G(G) ≤ k. This algorithm is based on the computability of the minor-obstructions and its dependence on k is not explicit. We extend the techniques used in the first algorithm to decide whether ed_G(G) ≤ k in time 2^{2^{2^poly(k)}}⋅n². This is the first algorithm for this problem with an explicit parametric dependence in k. In the special case where G excludes some apex-graph as a minor, we give two alternative algorithms, one running in time 2^{2^O(k²log k)}⋅n² and one running in time 2^{poly(k)}⋅n³. As a stepping stone for these algorithms, we provide an algorithm that decides whether ed_G(G) ≤ k in time 2^O(tw⋅ k + tw log tw)⋅n, where tw is the treewidth of G. This algorithm combines the dynamic programming framework of Reidl, Rossmanith, Villaamil, and Sikdar [ICALP 2014] for the particular case where G contains only the empty graph (i.e., for treedepth) with the representative-based techniques introduced by Baste, Sau, and Thilikos [SODA 2020]. In all the algorithmic complexities above, poly is a polynomial function whose degree depends on G, while the hidden constants also depend on G. Finally, we provide explicit upper bounds on the size of the graphs in the minor-obstruction set of the class of graphs E_k(G) = {G ∣ ed_G(G) ≤ k}.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Parameterized complexity and exact algorithms
##### Keywords
• Graph minors
• Parameterized algorithms
• Graph modification problems
• Vertex deletion
• Elimination distance
• Irrelevant vertex technique
• Flat Wall Theorem
• Obstructions

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## References

1. Isolde Adler, Martin Grohe, and Stephan Kreutzer. Computing excluded minors. In Proc. of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 641-650, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347153.
2. Akanksha Agrawal, Lawqueen Kanesh, Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. Deleting, Eliminating and Decomposing to Hereditary Classes Are All FPT-Equivalent. In Proc. of the 2022 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1976-2004, 2022. URL: https://doi.org/10.1137/1.9781611977073.79.
3. Akanksha Agrawal, Lawqueen Kanesh, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. An FPT algorithm for elimination distance to bounded degree graphs. In Proc. of the 38th International Symposium on Theoretical Aspects of Computer Science (STACS), volume 187 of LIPIcs, pages 5:1-5:11, 2021. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.5.
4. Akanksha Agrawal and M. S. Ramanujan. On the parameterized complexity of clique elimination distance. In Proc. of the 15th International Symposium on Parameterized and Exact Computation (IPEC), volume 180 of LIPIcs, pages 1:1-1:13, 2020. URL: https://doi.org/10.4230/LIPIcs.IPEC.2020.1.
5. Julien Baste, Ignasi Sau, and Dimitrios M. Thilikos. A complexity dichotomy for hitting connected minors on bounded treewidth graphs: the chair and the banner draw the boundary. In Proc. of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 951-970, 2020. URL: https://doi.org/10.1137/1.9781611975994.57.
6. Julien Baste, Ignasi Sau, and Dimitrios M. Thilikos. Hitting minors on bounded treewidth graphs. I. General upper bounds. SIAM Journal on Discrete Mathematics, 34(3):1623-1648, 2020. URL: https://doi.org/10.1137/19M1287146.
7. Julien Baste, Ignasi Sau, and Dimitrios M. Thilikos. Hitting minors on bounded treewidth graphs. II. Single-exponential algorithms. Theoretical Computer Science, 814:135-152, 2020. URL: https://doi.org/10.1016/j.tcs.2020.01.026.
8. Julien Baste, Ignasi Sau, and Dimitrios M. Thilikos. Hitting minors on bounded treewidth graphs. III. Lower bounds. Journal of Computer and System Sciences, 109:56-77, 2020. URL: https://doi.org/10.1016/j.jcss.2019.11.002.
9. Hans L. Bodlaender, John R. Gilbert, Ton Kloks, and Hjálmtyr Hafsteinsson. Approximating treewidth, pathwidth, and minimum elimination tree height. In Proc. of the 17th International Workshop on Graph-Theoretic Concepts in Computer Science (WG), volume 570 of LNCS, pages 1-12, 1991. URL: https://doi.org/10.1007/3-540-55121-2_1.
10. Jannis Bulian and Anuj Dawar. Graph isomorphism parameterized by elimination distance to bounded degree. Algorithmica, 75(2):363-382, 2016. URL: https://doi.org/10.1007/s00453-015-0045-3.
11. Jannis Bulian and Anuj Dawar. Fixed-parameter tractable distances to sparse graph classes. Algorithmica, 79(1):139-158, 2017. URL: https://doi.org/10.1007/s00453-016-0235-7.
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. Reinhard Diestel. Graph Theory, volume 173. Springer-Verlag, 5th edition, 2017. URL: https://doi.org/10.1007/978-3-662-53622-3.
14. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: https://doi.org/10.1007/978-1-4471-5559-1.
15. Zdeněk Dvořák, Archontia C. Giannopoulou, and Dimitrios M. Thilikos. Forbidden graphs for tree-depth. European Journal of Combinatorics, 33(5):969-979, 2012. URL: https://doi.org/10.1016/j.ejc.2011.09.014.
16. Eduard Eiben, Robert Ganian, Thekla Hamm, and O-joung Kwon. Measuring what matters: A hybrid approach to dynamic programming with treewidth. Journal of Computer and System Sciences, 121:57-75, 2021. URL: https://doi.org/10.1016/j.jcss.2021.04.005.
17. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2006. URL: https://doi.org/10.1007/3-540-29953-X.
18. Fedor V. Fomin, Petr A. Golovach, and Dimitrios M. Thilikos. Parameterized complexity of elimination distance to first-order logic properties. ACM Transactions on Computational Logic, 23(3):17:1-17:35, 2022. URL: https://doi.org/10.1145/3517129.
19. Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar F-deletion: Approximation, kernelization and optimal FPT algorithms. In Proc. of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 470-479, 2012. URL: https://doi.org/10.1109/FOCS.2012.62.
20. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Hitting topological minors is FPT. In Proc. of the 52nd Annual ACM Symposium on Theory of Computing (STOC), pages 1317-1326, 2020. URL: https://doi.org/10.1145/3357713.3384318.
21. Archontia C. Giannopoulou, O-joung Kwon, Jean-Florent Raymond, and Dimitrios M. Thilikos. Lean tree-cut decompositions: Obstructions and algorithms. In Proc. of the 36th International Symposium on Theoretical Aspects of Computer Science (STACS), volume 126 of LIPIcs, pages 32:1-32:14, 2019. URL: https://doi.org/10.4230/LIPIcs.STACS.2019.32.
22. Archontia C. Giannopoulou, Michal Pilipczuk, Jean-Florent Raymond, Dimitrios M. Thilikos, and Marcin Wrochna. Linear kernels for edge deletion problems to immersion-closed graph classes. In Proc. of the 44th International Colloquium on Automata, Languages, and Programming (ICALP), volume 80 of LIPIcs, pages 57:1-57:15, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.57.
23. Archontia C. Giannopoulou, Michal Pilipczuk, Jean-Florent Raymond, Dimitrios M. Thilikos, and Marcin Wrochna. Cutwidth: Obstructions and algorithmic aspects. Algorithmica, 81(2):557-588, 2019. URL: https://doi.org/10.1007/s00453-018-0424-7.
24. Jiong Guo, Falk Hüffner, and Rolf Niedermeier. A structural view on parameterizing problems: Distance from triviality. In Proc. of the 1st International Workshop on Parameterized and Exact Computation (IWPEC), volume 3162 of LNCS, pages 162-173, 2004. URL: https://doi.org/10.1007/978-3-540-28639-4_15.
25. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
26. Bart M. P. Jansen, Jari J. H. de Kroon, and Michał Włodarczyk. Vertex deletion parameterized by elimination distance and even less. In Proc. of the 53rd Annual ACM-SIGACT Symposium on Theory of Computing (STOC), pages 1757-1769, 2021. URL: https://doi.org/10.1145/3406325.3451068.
27. Bart M. P. Jansen, Daniel Lokshtanov, and Saket Saurabh. A near-optimal planarization algorithm. In Proc. of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1802-1811, 2014. URL: https://doi.org/10.1137/1.9781611973402.130.
28. Mamadou Moustapha Kanté and O-joung Kwon. An upper bound on the size of obstructions for bounded linear rank-width, 2014. URL: https://arxiv.org/abs/1412.6201.
29. Mamadou Moustapha Kanté and O-joung Kwon. Linear rank-width of distance-hereditary graphs II. vertex-minor obstructions. European Journal of Combinatorics, 74:110-139, 2018. URL: https://doi.org/10.1016/j.ejc.2018.07.009.
30. Ken-ichi Kawarabayashi. Planarity allowing few error vertices in linear time. In Proc. of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 639-648, 2009. URL: https://doi.org/10.1109/FOCS.2009.45.
31. Ken-ichi Kawarabayashi, Yusuke Kobayashi, and Bruce A. Reed. The disjoint paths problem in quadratic time. Journal of Combinatorial Theory, Series B, 102(2):424-435, 2012. URL: https://doi.org/10.1016/j.jctb.2011.07.004.
32. Ken-ichi Kawarabayashi, Robin Thomas, and Paul Wollan. A new proof of the flat wall theorem. Journal of Combinatorial Theory, Series B, 129:204-238, 2018. URL: https://doi.org/10.1016/j.jctb.2017.09.006.
33. Ken-ichi Kawarabayashi and Paul Wollan. A Shorter Proof of the Graph Minor Algorithm: The Unique Linkage Theorem. In Proc. of the 42nd ACM Symposium on Theory of Computing (STOC), pages 687-694, 2010. URL: https://doi.org/10.1145/1806689.1806784.
34. Eun Jung Kim, Alexander Langer, Christophe Paul, Felix Reidl, Peter Rossmanith, Ignasi Sau, and Somnath Sikdar. Linear kernels and single-exponential algorithms via protrusion decompositions. ACM Transactions on Algorithms, 12(2):21:1-21:41, 2016. URL: https://doi.org/10.1145/2797140.
35. Eun Jung Kim, Maria J. Serna, and Dimitrios M. Thilikos. Data-compression for parametrized counting problems on sparse graphs. In Proc. of the 29th International Symposium on Algorithms and Computation (ISAAC), volume 123 of LIPIcs, pages 20:1-20:13, 2018. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2018.20.
36. Ton Kloks. Treewidth, Computations and Approximations, volume 842 of Lecture Notes in Computer Science. Springer, 1994. URL: https://doi.org/10.1007/BFb0045375.
37. Tomasz Kociumaka and Marcin Pilipczuk. Deleting Vertices to Graphs of Bounded Genus. Algorithmica, 81(9):3655-3691, 2019. URL: https://doi.org/10.1007/s00453-019-00592-7.
38. Jens Lagergren. An upper bound on the size of an obstruction. In Graph Structure Theory, volume 147 of Contemporary Mathematics, pages 601-621. American Mathematical Society, 1991. URL: https://doi.org/10.1090/conm/147/01202.
39. Jens Lagergren. Upper bounds on the size of obstructions and intertwines. Journal of Combinatorial Theory, Series B, 73:7-40, 1998. URL: https://doi.org/10.1006/jctb.1997.1788.
40. Jens Lagergren and Stefan Arnborg. Finding minimal forbidden minors using a finite congruence. In Proc. of the 18th International Colloquium on Automata, Languages and Programming (ICALP), volume 510 of LNCS, pages 532-543, 1991. URL: https://doi.org/10.1007/3-540-54233-7_161.
41. John M. Lewis and Mihalis Yannakakis. The node-deletion problem for hereditary properties is NP-complete. Journal of Computer and System Sciences, 20(2):219-230, 1980. URL: https://doi.org/10.1016/0022-0000(80)90060-4.
42. Dániel Marx and Ildikó Schlotter. Obtaining a planar graph by vertex deletion. Algorithmica, 62(3-4):807-822, 2012. URL: https://doi.org/10.1007/s00453-010-9484-z.
43. Jaroslav Nesetril and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-27875-4.
44. Rolf Niedermeier. Invitation to fixed parameter algorithms, volume 31. Oxford University Press, 2006. URL: https://doi.org/10.1093/ACPROF:OSO/9780198566076.001.0001.
45. Alex Pothen. The complexity of optimal elimination trees. Technical Report. Pennsylvania State University. Dept. of Computer Science, 1988. URL: https://www.cs.purdue.edu/homes/apothen/Papers/shortest-etree1988.pdf.
46. Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, and Somnath Sikdar. A faster parameterized algorithm for treedepth. In Proc. of the 41st International Colloquium on Automata, Languages, and Programming (ICALP), volume 8572 of LNCS, pages 931-942, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_77.
47. Neil Robertson and Paul D. Seymour. Graph Minors. XIII. The Disjoint Paths Problem. Journal of Combinatorial Theory, Series B, 63(1):65-110, 1995. URL: https://doi.org/10.1006/jctb.1995.1006.
48. Neil Robertson and Paul D. Seymour. Graph Minors. XVI. Excluding a non-planar graph. Journal of Combinatorial Theory, Series B, 89(1):43-76, 2003. URL: https://doi.org/10.1016/S0095-8956(03)00042-X.
49. Neil Robertson and Paul D. Seymour. Graph Minors. XX. Wagner’s conjecture. Journal of Combinatorial Theory, Series B, 92(2):325-357, 2004. URL: https://doi.org/10.1016/j.jctb.2004.08.001.
50. Ignasi Sau, Giannos Stamoulis, and Dimitrios M. Thilikos. A more accurate view of the Flat Wall Theorem, 2021. URL: https://arxiv.org/abs/2102.06463.
51. Ignasi Sau, Giannos Stamoulis, and Dimitrios M. Thilikos. k-apices of minor-closed graph classes. II. Parameterized algorithms. ACM Transactions on Algorithms, 18(3):21:1-21:30, 2022. Short version in Proc. of the 47th International Colloquium on Automata, Languages and Programming (ICALP), volume 168 of LIPIcs, pages 95:1-95:20, 2020. URL: https://doi.org/10.1145/3519028.
52. Ignasi Sau, Giannos Stamoulis, and Dimitrios M. Thilikos. k-apices of minor-closed graph classes. I. Bounding the obstructions. Journal of Combinatorial Theory, Series B, 161:180-227, 2023. URL: https://doi.org/10.1016/j.jctb.2023.02.012.
53. Andrew Thomason. The extremal function for complete minors. Journal of Combinatorial Theory, Series B, 81(2):318-338, 2001. URL: https://doi.org/10.1006/jctb.2000.2013.
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