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Computing Connected-k-Subgraph Cover with Connectivity Requirement

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Theory and Applications of Models of Computation (TAMC 2022)

Abstract

In this paper, we design an \(O(2^{O(\sqrt{t}\log t)}|V|^{O(1)})\) time subexponential FPT algorithm for MinCkSC\(_{con}\) on an H-minor free-graph, where t is an upper bound of solution size.

This research work is supported in part by NSFC (U20A2068, 11771013), and ZJNSFC (LD19A010001).

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References

  1. Alon, N., Seymour, P.D., Thomas, R.: A separator theorem for nonplanar graphs. J. Am. Math. Soc. 3(4), 801–808 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bai, Z., Tu, J., Shi, Y.: An improved algorithm for the vertex cover P\(_3\) problem on graphs of bounded treewidth. Discrete Math. Theor. Comput. Sci. 21 (2019)

    Google Scholar 

  3. Bar-Yehuda, R., Even, S.: A linear time approximation algorithm for the weighted vertex cover problem. J. Algorithms 2, 198–203 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bar-Yehuda, R., Censor-Hillel, K., Schwartzman, G.: A distributed \((2+\epsilon )\)-approximation for vertex cover in \(O(\log \varDelta / \epsilon \log \log \varDelta )\) rounds, J. ACM 64(3), 1–11 (2017)

    Google Scholar 

  5. Bateni, M., Farhadi, A., Hajiaghayi, M.: Polynomial-time approximation scheme for minimum \(k\)-cut in planar and minor-free graphs, In: Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, pp. 1055–1068 (2019)

    Google Scholar 

  6. Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brešar, B., Kardoš, F., Katrenič, J., Semanišin, G.: Minimum \(k\)-path vertex cover. Discret. Appl. Math. 159, 1189–1195 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cabello, S., Gajser, D.: Simple PTAS’s for families of graphs excluding a minor. Discret. Appl. Math. 189, 41–48 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cai, S., Li, Y., Hou, W., Wang, H.: Towards faster local search for minimum weight vertex cover on massive graphs. Inf. Sci. 471, 64–79 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Červenỳ, R., Suchỳ, O.: Faster FPT algorithm for 5-path vertex cover. In: Proceedings of 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019), vol. 32, pp. 1–13 (2019)

    Google Scholar 

  12. Chang, M.S., Chen, L.H., Hung, L.J., Rossmanith, P., Su, P.C.: Fixed-parameter algorithms for vertex cover P\(_3\). Discret. Optim. 19, 12–22 (2016)

    Article  MATH  Google Scholar 

  13. Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theoret. Comput. Sci. 411(40–42), 3736–3756 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cygan, M.: Deterministic parameterized connected vertex cover. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 95–106. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31155-0_9

    Chapter  Google Scholar 

  15. Demaine, E.D., Hajiaghayi, M.: Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica 28(1), 19–36 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Diestel, R.: Graph Theory. GTM, vol. 173. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-53622-3

    Book  MATH  Google Scholar 

  17. Fomin, F.V., Lokshtanov, D., Raman, V., Saurabh, S.: Bidimensionality and EPTAS, In: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, pp. 748–759 (2011)

    Google Scholar 

  18. Frederickson, G.: Fast algorithms for shortest paths in planar graphs with applications. SIAM J. Comput. 16(6), 1004–1022 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guruswami, V., Lee, E.: Inapproximability of \(H\)-transversal/packing. SIAM J. Discret. Math. 31(3), 1552–1571 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hochbaum, D.S.: Approximation algorithms for the set covering and vertex cover problems. SIAM J. Comput. 11(3), 555–556 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kardoš, F., Katrenič, J., Schiermeyer, I.: On computing the minimum \(3\)-path vertex cover and dissociation number of graphs. Theoret. Comput. Sci. 412, 7009–7017 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Katrenič, J.: A faster FPT algorithm for \(3\)-path vertex cover. Inf. Process. Lett. 116(4), 273–278 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Khot, S., Minzer, D., Safra, M.: Pseudorandom sets in grassmann graph have near-perfect expansion. In: 2018 IEEE 59th Annual Symposium on Foundations of Computer Science, FOCS 2018, pp. 592–601 (2018)

    Google Scholar 

  24. Kloks, T. (ed.): Treewidth. LNCS, vol. 842. Springer, Heidelberg (1994). https://doi.org/10.1007/BFb0045375

    Book  MATH  Google Scholar 

  25. Le, H., Zheng, B.: Local search is a PTAS for feedback vertex set in minor-free graphs. Theoret. Comput. Sci. 838, 17–24 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lee, E.:Partitioning a graph into small pieces with applications to path transversal, In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, pp. 1546–1558 (2017)

    Google Scholar 

  27. Li, X., Zhang, Z., Huang, X.: Approximation algorithms for minimum (weight) connected \(k\)-path vertex cover. Discret. Appl. Math. 205, 101–108 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, X., Shi, Y., Huang, X.: PTAS for \(\cal{H} \)-free node deletion problems in disk graphs. Discret. Appl. Math. 239, 119–124 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, P., Zhang, Z., Huang, X.: Approximation algorithm for minimum weight connected-\(k\)-subgraph cover. Theoret. Comput. Sci. 838, 160–67 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Novotný, M.: Design and analysis of a generalized canvas protocol. In: Samarati, P., Tunstall, M., Posegga, J., Markantonakis, K., Sauveron, D. (eds.) WISTP 2010. LNCS, vol. 6033, pp. 106–121. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12368-9_8

    Chapter  Google Scholar 

  31. Pourhassan, M., Shi, F., Neumann, F.: Parameterized analysis of multiobjective evolutionary algorithms and the weighted vertex cover problem. Evol. Comput. 27(4), 559–575 (2019)

    Article  Google Scholar 

  32. Ran, Y., Zhang, Z., Huang, X., Li, X., Du, D.: Approximation algorithms for minimum weight connected 3-path vertex cover. Appl. Math. Comput. 347, 723–733 (2019)

    MathSciNet  MATH  Google Scholar 

  33. Tsur, D.: Parameterized algorithm for 3-path vertex vover. Theoret. Comput. Sci. 783, 1–8 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tsur, D.: An \(O^*(2.619^k)\) algorithm for 4-path vertex cover. Discrete Appl. Math. 291, 1–14 (2021)

    Google Scholar 

  35. Tsur, D.: \(l\)-path vertex cover is easier than \(l\)-hitting set for small \(l\), ArXiv Preprint ArXiv:1906.10523 (2019)

  36. Tu, J.: A fixed-parameter algorithm for the vertex cover P\(_3\) problem. Inf. Process. Lett. 115, 96–99 (2015)

    Article  MATH  Google Scholar 

  37. Tu, J., Jin, Z.: An FPT algorithm for the vertex cover P\(_4\) problem. Discret. Appl. Math. 200, 186–190 (2016)

    Article  MATH  Google Scholar 

  38. Tu, J., Shi, Y.: An efficient polynomial time approximation scheme for the vertex cover P\(_3\) problem on planar graphs. Discussiones Math. Graph Theory 39, 55–65 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  39. Tu, J., Zhou, W.: A factor \(2\) approximation algorithm for the vertex cover P\(_3\) problem. Inf. Process. Lett. 111, 683–686 (2011)

    Article  MATH  Google Scholar 

  40. Tu, J., Zhou, W.: A primal-dual approximation algorithm for the vertex cover P\(_3\) problem. Theoret. Comput. Sci. 412, 7044–7048 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Tu, J., Yang, F.: The vertex cover P\(_3\) problem in cubic graphs. Inf. Process. Lett. 113, 481–485 (2013)

    Article  MATH  Google Scholar 

  42. Xiao, M., Kou, S.: Exact algorithms for the maximum dissociation set and minimum \(3\)-path vertex cover problems. Theoret. Comput. Sci. 657, 86–97 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhang, A., Chen, Y., Chen, Z., Lin, G.H.: Improved approximation algorithms for path vertex covers in regular graphs. ArXiv: 1811.01162v1 (2018)

  44. Zhang, Y., Shi, Y.S., Zhang, Z.: Approximation algorithm for the minimum weight connected \(k\)-subgraph cover problem. Theoret. Comput. Sci. 535, 54–58 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang, Z., Li, X., Shi, Y., Nie, H., Zhu, Y.: PTAS for minimum \(k\)-path vertex cover in ball graph. Inf. Process. Lett. 119, 9–13 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgment

This research work is supported in part by NSFC (U20A2068, 11771013), and ZJNSFC (LD19A010001).

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Liu, P., Zhang, Z., Ran, Y., Huang, X. (2022). Computing Connected-k-Subgraph Cover with Connectivity Requirement. In: Du, DZ., Du, D., Wu, C., Xu, D. (eds) Theory and Applications of Models of Computation. TAMC 2022. Lecture Notes in Computer Science, vol 13571. Springer, Cham. https://doi.org/10.1007/978-3-031-20350-3_9

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  • DOI: https://doi.org/10.1007/978-3-031-20350-3_9

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