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Distance from Triviality 2.0: Hybrid Parameterizations

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Combinatorial Algorithms (IWOCA 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13270))

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Abstract

Vertex deletion problems have been at the heart of numerous major advances in Algorithms and Combinatorial Optimization, and especially so in the area of Parameterized Complexity. For a family of graphs \(\mathcal H\), the input to Vertex Deletion to \(\mathcal {H}\) is a graph G and an integer k, and the objective is to decide whether there is a vertex-subset, called a modulator, whose removal from G results in a graph contained in the family \(\mathcal H\), and such that \(|S|\le k\). Traditionally, the majority of the study of Vertex Deletion to \(\mathcal {H}\) problems in Parameterized Complexity has been limited to parameterization by modulator size and structural graph width measures of the input graph such as treewidth. Recent years have seen systematic efforts at: i) quantifying the complexity of modulators in ways other than their size, and ii) studying the complexity landscape of various graph problems under parameterizations that are simultaneously better than both the modulator size and certain width measures of the graph. In this talk we will look at some exciting developments in this direction in relation to two such parameters that are “hybridizations” of the modulator size, and the well-explored graph parameters – treewidth and treedepth.

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Notes

  1. 1.

    For this, we always assume that \(\mathcal {H}\) contains the empty graph and so V(G) is a trivial modulator to \(\mathcal {H}\).

  2. 2.

    Here we make a mild assumption that, checking whether a graph is in \(\mathcal H\) can be done in polynomial time.

  3. 3.

    A family of graphs \(\mathcal H\) is hereditary if for each graph \(G \in \mathcal{H}\), every induced subgraph of G belongs to \(\mathcal H\).

  4. 4.

    When we say CMSO, we refer to the fragment that is sometimes referred to as \({\mathsf{CMSO}}_2\) in the literature. We will only be using a meta-result regarding CMSO definable problems, and thus, we refer to [5, 11, 12] for a an introduction to this topics.

  5. 5.

    We can check if \(G \in \mathcal{H}\) by calling \(\mathscr {X}_\mathsf{mod}\) for the instance (G, 0). We recall that \(\mathcal H\) is closed under disjoint union.

  6. 6.

    Even if \(\mathscr {X}_\mathsf{mod}\) is a decision algorithm, using the self-reducibility like property, we can compute the decomposition itself, see Lemma 3.5 and 3.6 in [1] or [2] for details.

References

  1. Agrawal, A., et al.: Deleting, eliminating and decomposing to hereditary classes are all FPT-equivalent (2021). https://doi.org/10.48550/ARXIV.2104.09950, https://arxiv.org/abs/2104.09950

  2. Agrawal, A., et al.: Deleting, eliminating and decomposing to hereditary classes are all FPT-equivalent. In: ACM-SIAM Symposium on Discrete Algorithms (SODA) SIAM, pp. 1726–1736 (2022)

    Google Scholar 

  3. Agrawal, A., Kanesh, L., Panolan, F., Ramanujan, M.S., Saurabh, S.: An FPT algorithm for elimination distance to bounded degree graphs. In: 38th International Symposium on Theoretical Aspects of Computer Science (STACS), vol. 187, pp. 5:1–5:11 (2021). https://doi.org/10.4230/LIPIcs.STACS.2021.5

  4. Agrawal, A., Ramanujan, M.S.: On the parameterized complexity of clique elimination distance. In: Cao, Y., Pilipczuk, M. (eds.) 15th International Symposium on Parameterized and Exact Computation, (IPEC), vol. 180, pp. 1:1–1:13 (2020). https://doi.org/10.4230/LIPIcs.IPEC.2020.1

  5. Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12(2), 308–340 (1991). https://doi.org/10.1016/0196-6774(91)90006-K

    Article  MathSciNet  MATH  Google Scholar 

  6. van Bevern, R., Komusiewicz, C., Moser, H., Niedermeier, R.: Measuring indifference: unit interval vertex deletion. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 232–243. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-16926-7_22

    Chapter  Google Scholar 

  7. Bulian, J., Dawar, A.: Graph isomorphism parameterized by elimination distance to bounded degree. Algorithmica 75(2), 363–382 (2015). https://doi.org/10.1007/s00453-015-0045-3

    Article  MathSciNet  MATH  Google Scholar 

  8. Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996). https://doi.org/10.1016/0020-0190(96)00050-6

    Article  MathSciNet  MATH  Google Scholar 

  9. Cao, Y., Marx, D.: Interval deletion is fixed-parameter tractable. ACM Trans. Algorithms 11(3), 21:1-21:35 (2015). https://doi.org/10.1145/2629595

    Article  MathSciNet  MATH  Google Scholar 

  10. Cao, Y., Marx, D.: Chordal editing is fixed-parameter tractable. Algorithmica 75(1), 118–137 (2015). https://doi.org/10.1007/s00453-015-0014-x

    Article  MathSciNet  MATH  Google Scholar 

  11. Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990). https://doi.org/10.1016/0890-5401(90)90043-H

    Article  MathSciNet  MATH  Google Scholar 

  12. Courcelle, B.: The expression of graph properties and graph transformations in monadic second-order logic. In: Handbook of Graph Grammars and Computing by Graph Transformations, vol. 1: Foundations, pp. 313–400. World Scientific (1997)

    Google Scholar 

  13. Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000). https://doi.org/10.1007/s002249910009

    Article  MathSciNet  MATH  Google Scholar 

  14. Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discret. Appl. Math. 101(1–3), 77–114 (2000). https://doi.org/10.1016/S0166-218X(99)00184-5

    Article  MathSciNet  MATH  Google Scholar 

  15. Courcelle, B., Oum, S.: Vertex-minors, monadic second-order logic, and a conjecture by seese. J. Comb. Theory Ser. B 97(1), 91–126 (2007). https://doi.org/10.1016/j.jctb.2006.04.003

    Article  MathSciNet  MATH  Google Scholar 

  16. Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3

    Book  MATH  Google Scholar 

  17. Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. TCS, Springer, London (2013). https://doi.org/10.1007/978-1-4471-5559-1

    Book  MATH  Google Scholar 

  18. Eiben, E., Ganian, R., Hamm, T., Kwon, O.: Measuring what matters: a hybrid approach to dynamic programming with treewidth. J. Comput. Syst. Sci. 121, 57–75 (2021). https://doi.org/10.1016/j.jcss.2021.04.005

    Article  MathSciNet  MATH  Google Scholar 

  19. Eiben, E., Ganian, R., Kwon, O.: A single-exponential fixed-parameter algorithm for distance-hereditary vertex deletion. J. Comput. Syst. Sci. 97, 121–146 (2018). https://doi.org/10.1016/j.jcss.2018.05.005

    Article  MathSciNet  MATH  Google Scholar 

  20. Flum, J., Grohe, M.: Parameterized Complexity Theory. TTCSAES, Springer, Heidelberg (2006). https://doi.org/10.1007/3-540-29953-X

    Book  MATH  Google Scholar 

  21. Fomin, F.V., Golovach, P.A., Thilikos, D.M.: Parameterized complexity of elimination distance to first-order logic properties. In: 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pp. 1–13 (2021). https://doi.org/10.1109/LICS52264.2021.9470540

  22. Fomin, F.V., Lokshtanov, D., Misra, N., Saurabh, S.: Planar \({F}\)-deletion: approximation, kernelization and optimal FPT algorithms. In: 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 470–479 (2012). https://doi.org/10.1109/FOCS.2012.62

  23. Fomin, F.V., Lokshtanov, D., Panolan, F., Saurabh, S., Zehavi, M.: Hitting topological minors is FPT. In: Proccedings of the 52nd Annual ACM-SIGACT Symposium on Theory of Computing (STOC), pp. 1317–1326 (2020). https://doi.org/10.1145/3357713.3384318

  24. Ganian, R., Ordyniak, S., Ramanujan, M.S.: Going beyond primal treewidth for (M)ILP. In: Singh, S.P., Markovitch, S. (eds.) Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, pp. 815–821. AAAI Press (2017)

    Google Scholar 

  25. Ganian, R., Ramanujan, M.S., Szeider, S.: Backdoor treewidth for SAT. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 20–37. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66263-3_2

    Chapter  MATH  Google Scholar 

  26. Ganian, R., Ramanujan, M.S., Szeider, S.: Combining treewidth and backdoors for CSP. In: 34th Symposium on Theoretical Aspects of Computer Science (STACS), vol. 66, pp. 36:1–36:17 (2017). https://doi.org/10.4230/LIPIcs.STACS.2017.36

  27. Ganian, R., Ramanujan, M.S., Szeider, S.: Discovering archipelagos of tractability for constraint satisfaction and counting. ACM Trans. Algorithms 13(2), 291–2932 (2017). https://doi.org/10.1145/3014587

    Article  MathSciNet  MATH  Google Scholar 

  28. Guo, J., Hüffner, F., Niedermeier, R.: A structural view on parameterizing problems: distance from triviality. In: Parameterized and Exact Computation, First International Workshop, (IWPEC), vol. 3162, pp. 162–173 (2004). https://doi.org/10.1007/978-3-540-28639-4_15

  29. Hlinený, P., Oum, S., Seese, D., Gottlob, G.: Width parameters beyond tree-width and their applications. Comput. J. 51(3), 326–362 (2008). https://doi.org/10.1093/comjnl/bxm052

    Article  Google Scholar 

  30. Jacob, A., de Kroon, J.J.H., Majumdar, D., Raman, V.: Parameterized complexity of deletion to scattered graph classes. CoRR abs/2105.04660 (2021). https://arxiv.org/abs/2105.04660

  31. Jacob, A., Majumdar, D., Raman, V.: Parameterized complexity of deletion to scattered graph classes. In: 15th International Symposium on Parameterized and Exact Computation, (IPEC), vol. 180, pp. 18:1–18:17 (2020). https://doi.org/10.4230/LIPIcs.IPEC.2020.18

  32. Jansen, B.M.P., de Kroon, J.J.H.: FPT algorithms to compute the elimination distance to bipartite graphs and more. In: Kowalik, Łukasz, Pilipczuk, Michał, Rzążewski, Paweł (eds.) WG 2021. LNCS, vol. 12911, pp. 80–93. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-86838-3_6

    Chapter  Google Scholar 

  33. Jansen, B.M.P., de Kroon, J.J.H., Wlodarczyk, M.: Vertex deletion parameterized by elimination distance and even less. In: Proceedings of the 53rd Annual ACM-SIGACT Symposium on Theory of Computing (STOC), pp. 1757–1769 (2021). https://doi.org/10.1145/3406325.3451068

  34. Jansen, B.M.P., Lokshtanov, D., Saurabh, S.: A near-optimal planarization algorithm. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1802–1811 (2014). https://doi.org/10.1137/1.9781611973402.130

  35. Kim, E.J., et al.: Linear kernels and single-exponential algorithms via protrusion decompositions. ACM Trans. Algorithms 12(2), 21:1-21:41 (2016). https://doi.org/10.1145/2797140

    Article  MathSciNet  MATH  Google Scholar 

  36. Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci. 20(2), 219–230 (1980). https://doi.org/10.1016/0022-0000(80)90060-4

    Article  MathSciNet  MATH  Google Scholar 

  37. Lindermayr, A., Siebertz, S., Vigny, A.: Elimination distance to bounded degree on planar graphs. In: 45th International Symposium on Mathematical Foundations of Computer Science, (MFCS), vol. 170, pp. 65:1–65:12 (2020). https://doi.org/10.4230/LIPIcs.MFCS.2020.65

  38. Lokshtanov, D., Narayanaswamy, N.S., Raman, V., Ramanujan, M.S., Saurabh, S.: Faster parameterized algorithms using linear programming. ACM Trans. Algorithms 11(2), 15:1–15:31 (2014). https://doi.org/10.1145/2566616. https://doi.org/10.1145/2566616

  39. Lokshtanov, D., Ramanujan, M.S., Saurabh, S., Zehavi, M.: Reducing CMSO model checking to highly connected graphs. In: 45th International Colloquium on Automata, Languages, and Programming, (ICALP), vol. 107, pp. 135:1–135:14 (2018). https://doi.org/10.4230/LIPIcs.ICALP.2018.135

  40. Marx, D.: Chordal deletion is fixed-parameter tractable. Algorithmica 57(4), 747–768 (2010). https://doi.org/10.1007/s00453-008-9233-8

    Article  MathSciNet  MATH  Google Scholar 

  41. Nesetril, J., de Mendez, P.O.: Tree-depth, subgraph coloring and homomorphism bounds. Eur. J. Comb. 27(6), 1022–1041 (2006). https://doi.org/10.1016/j.ejc.2005.01.010

    Article  MathSciNet  MATH  Google Scholar 

  42. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006). https://doi.org/10.1093/acprof:oso/9780198566076.001.0001

    Book  MATH  Google Scholar 

  43. Oum, S.: Rank-width and vertex-minors. J. Comb. Theory Ser. B 95(1), 79–100 (2005). https://doi.org/10.1016/j.jctb.2005.03.003

    Article  MathSciNet  MATH  Google Scholar 

  44. Oum, S.: Approximating rank-width and clique-width quickly. ACM Trans. Algorithms 5(1), 101–1020 (2008). https://doi.org/10.1145/1435375.1435385

    Article  MathSciNet  MATH  Google Scholar 

  45. Oum, S.: Rank-width and well-quasi-ordering. SIAM J. Discret. Math. 22(2), 666–682 (2008). https://doi.org/10.1137/050629616

    Article  MathSciNet  MATH  Google Scholar 

  46. Oum, S.: Rank-width: algorithmic and structural results. Discret. Appl. Math. 231, 15–24 (2017). https://doi.org/10.1016/j.dam.2016.08.006

    Article  MathSciNet  MATH  Google Scholar 

  47. Oum, S., Seymour, P.D.: Approximating clique-width and branch-width. J. Comb. Theory Ser. B 96(4), 514–528 (2006). https://doi.org/10.1016/j.jctb.2005.10.006

    Article  MathSciNet  MATH  Google Scholar 

  48. Reed, B.A., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004). https://doi.org/10.1016/j.orl.2003.10.009

    Article  MathSciNet  MATH  Google Scholar 

  49. Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory Ser. B 63(1), 65–110 (1995). https://doi.org/10.1006/jctb.1995.1006

    Article  MathSciNet  MATH  Google Scholar 

  50. Sau, I., Stamoulis, G., Thilikos, D.M.: An FPT-algorithm for recognizing k-apices of minor-closed graph classes. In: 47th International Colloquium on Automata, Languages, and Programming, (ICALP), vol. 168, pp. 95:1–95:20 (2020). https://doi.org/10.4230/LIPIcs.ICALP.2020.95

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Authors and Affiliations

  1. Indian Institute of Technology Madras, Chennai, India

    Akanksha Agrawal

  2. University of Warwick, Coventry, England

    M. S. Ramanujan

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  1. Akanksha Agrawal
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Correspondence to Akanksha Agrawal .

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  1. Université Paris-Dauphine, Paris, France

    Cristina Bazgan

  2. Universität Trier, Trier, Germany

    Henning Fernau

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Agrawal, A., Ramanujan, M.S. (2022). Distance from Triviality 2.0: Hybrid Parameterizations. In: Bazgan, C., Fernau, H. (eds) Combinatorial Algorithms. IWOCA 2022. Lecture Notes in Computer Science, vol 13270. Springer, Cham. https://doi.org/10.1007/978-3-031-06678-8_1

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