Finding, Hitting and Packing Cycles in Subexponential Time on Unit Disk Graphs

Abstract

We give algorithms with running time \(2^{\mathcal {O}({\sqrt{k}\log {k}})} \cdot n^{\mathcal {O}(1)}\) for the following problems. Given an n-vertex unit disk graph G and an integer k, decide whether G contains

  • a path on exactly/at least k vertices,

  • a cycle on exactly k vertices,

  • a cycle on at least k vertices,

  • a feedback vertex set of size at most k, and

  • a set of k pairwise vertex-disjoint cycles.

For the first three problems, no subexponential time parameterized algorithms were previously known. For the remaining two problems, our algorithms significantly outperform the previously best known parameterized algorithms that run in time \(2^{\mathcal {O}(k^{0.75}\log {k})} \cdot n^{\mathcal {O}(1)}\). Our algorithms are based on a new kind of tree decompositions of unit disk graphs where the separators can have size up to \(k^{\mathcal {O}(1)}\) and there exists a solution that crosses every separator at most \(\mathcal {O}(\sqrt{k})\) times. The running times of our algorithms are optimal up to the \(\log {k}\) factor in the exponent, assuming the exponential time hypothesis.

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Notes

  1. 1.

    The paper [23] does not consider unit square graphs, but the arguments it presents for unit disk graphs can be adapted to handle unit square graphs as well.

References

  1. 1.

    Alber, J., Fiala, J.: Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. In: Baeza-Yates, R., Montanari, U., Santoro, N. (eds.) Foundations of Information Technology in the Era of Network and Mobile Computing, pp. 26–37. Springer, New York (2002)

    Google Scholar 

  2. 2.

    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. Assoc. Comput. Mach. 42(4), 844–856 (1995)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. Assoc. Comput. Mach. 41(1), 153–180 (1994)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Narrow sieves for parameterized paths and packings. J. Comput. Syst. Sci. 87, 119–139 (2017)

    MathSciNet  Article  Google Scholar 

  5. 5.

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

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bodlaender, H.L., Cygan, M., Kratsch, S., Nederlof, J.: Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput. 243, 86–111 (2015)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bodlaender, H.L., Drange, P.G., Dregi, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M.: A $c^k n$ 5-approximation algorithm for treewidth. SIAM J. Comput. 45(2), 317–378 (2016)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Chan, T.M.: Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms 46(2), 178–189 (2003)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Chen, J., Kanj, I.A., Jia, W.: Vertex cover: further observations and further improvements. J. Algorithms 41(2), 280–301 (2001)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86(1–3), 165–177 (1990)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Clarkson, K.L., Varadarajan, K.R.: Improved approximation algorithms for geometric set cover. Discrete Comput. Geom. 37(1), 43–58 (2007)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Cham (2015)

    Google Scholar 

  14. 14.

    Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Subexponential parameterized algorithms on graphs of bounded genus and $H$-minor-free graphs. J. ACM 52(6), 866–893 (2005)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Demaine, E.D., Hajiaghayi, M.: Bidimensionality: new connections between FPT algorithms and PTASs. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), pp. 590–601. ACM-SIAM, New York (2005)

  16. 16.

    Demaine, E.D., Hajiaghayi, M.: The bidimensionality theory and its algorithmic applications. Comput. J. 51(3), 292–302 (2008)

    Article  Google Scholar 

  17. 17.

    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, New York (2012)

    Google Scholar 

  18. 18.

    Dorn, F., Penninkx, E., Bodlaender, H.L., Fomin, F.V.: Efficient exact algorithms on planar graphs: exploiting sphere cut decompositions. Algorithmica 58(3), 790–810 (2010)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Dumitrescu, A., Pach, J.: Minimum clique partition in unit disk graphs. Graphs Combin. 27(3), 399–411 (2011)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Fomin, F.V., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Subexponential parameterized algorithms for planar and apex-minor-free graphs via low treewidth pattern covering. In: Proceedings of the 57th Annual Symposium on Foundations of Computer Science (FOCS 2016), pp. 515–524. IEEE Computer Society, Los Alamitos (2016)

  21. 21.

    Fomin, F.V., Lokshtanov, D., Panolan, F., Saurabh, S.: Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM 63(4), 29 (2016)

    MathSciNet  Article  Google Scholar 

  22. 22.

    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. SIAM, Philadelphia (2011)

  23. 23.

    Fomin, F.V., Lokshtanov, D., Saurabh, S.: Bidimensionality and geometric graphs. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2012), pp. 1563–1575. SIAM, Philadelphia (2012)

  24. 24.

    Fomin, F.V., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Bidimensionality and kernels. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), pp. 503–510. SIAM, Philadelphia (2010)

  25. 25.

    Hale, W.K.: Frequency assignment: theory and applications. Proc. IEEE 68(12), 1497–1514 (1980)

    Article  Google Scholar 

  26. 26.

    Har-Peled, S., Lee, M.: Weighted geometric set cover problems revisited. J. Comput. Geom. 3(1), 65–85 (2012)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Har-Peled, S., Quanrud, K.: Approximation algorithms for polynomial-expansion and low-density graphs. In: Proceedings of the 23rd Annual European Symposium (ESA 2015). Lecture Notes in Computer Science, vol. 9294, pp. 717–728. Springer, Heidelberg (2015)

  28. 28.

    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32(1), 130–136 (1985)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Hopcroft, J., Tarjan, R.: Algorithm 447: efficient algorithms for graph manipulation. Commun. ACM 16(6), 372–378 (1973)

    Article  Google Scholar 

  30. 30.

    Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs. J. Algorithms 26(2), 238–274 (1998)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity. J. Comput. Syst. Sci. 63(4), 512–530 (2001)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Ito, H., Kadoshita, M.: Tractability and intractability of problems on unit disk graphs parameterized by domain area. In: Proceedings of the 9th International Symposium on Operations Research and Its Applications (ISORA’10), pp. 120–127 (2010)

  33. 33.

    Jansen, B.: Polynomial kernels for hard problems on disk graphs. In: Proceedings of the 12th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2010). Lecture Notes in Computer Science, vol. 6139, pp. 310–321. Springer, Berlin (2010)

  34. 34.

    Kammerlander, K.: C 900—an advanced mobile radio telephone system with optimum frequency utilization. IEEE J. Sel. Areas Commun. 2(4), 589–597 (1984)

    Article  Google Scholar 

  35. 35.

    Koutis, I.: Faster algebraic algorithms for path and packing problems. In: Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP 2008). Lecture Notes in Computer Science, vol. 5125, pp. 575–586. Springer, Berlin (2008)

  36. 36.

    Koutis, I., Williams, R.: Algebraic fingerprints for faster algorithms. Commun. ACM 59(1), 98–105 (2016)

    Article  Google Scholar 

  37. 37.

    Marx, D.: Efficient approximation schemes for geometric problems? In: Proceedings of the 13th Annual European Symposium on Algorithms (ESA 2005). Lecture Notes in Computer Science, vol. 3669, pp. 448–459. Springer, Berlin (2005)

  38. 38.

    Mustafa, N.H., Raman, R., Ray, S.: Settling the APX-hardness status for geometric set cover. In: Proceedings of the 55th IEEE Annual Symposium on Foundations of Computer Science (FOCS 2014), pp. 541–550. IEEE Computer Society, Philadelphia (2014)

  39. 39.

    Smith, W.D., Wormald, N.C.: Geometric separator theorems & applications. In: Proceedings of the 39th Annual Symposium on Foundations of Computer Science (FOCS 1998), pp. 232–243. IEEE Computer Society, Philadelphia (1998)

  40. 40.

    Thomassé, S.: A $4k^2$ kernel for feedback vertex set. ACM Trans. Algorithms 6(2), 32 (2010)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Wang, D., Kuo, Y.-S.: A study on two geometric location problems. Inf. Process. Lett. 28(6), 281–286 (1988)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Williams, R.: Finding paths of length $k$ in ${O}^*(2^k)$ time. Inf. Process. Lett. 109(6), 315–318 (2009)

    MathSciNet  Article  Google Scholar 

  43. 43.

    Yeh, Y.S., Wilson, J., Schwartz, S.: Outage probability in mobile telephony with directive antennas and macrodiversity. IEEE J. Sel. Areas Commun. 2(4), 507–511 (1984)

    Article  Google Scholar 

  44. 44.

    Zehavi, M.: Mixing color coding-related techniques. In: Proceedings of the 23rd Annual European Symposium on Algorithms (ESA 2015). Lecture Notes in Computer Science, vol. 9294, pp. 1037–1049. Springer, Heidelberg (2013)

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Correspondence to Fahad Panolan.

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The preliminary version of the paper appeared in the proceedings of ICALP 2017. The research leading to the results of the paper is supported by Pareto-Optimal Parameterized Algorithms, ERC Starting Grant 715744, Parameterized Approximation, ERC Starting Grant 306992, Rigorous Theory of Preprocessing, ERC Advanced Investigator Grant 267959, and NFR MULTIVAL project.

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Fomin, F.V., Lokshtanov, D., Panolan, F. et al. Finding, Hitting and Packing Cycles in Subexponential Time on Unit Disk Graphs. Discrete Comput Geom 62, 879–911 (2019). https://doi.org/10.1007/s00454-018-00054-x

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Keywords

  • Longest path
  • Longest cycle
  • Cycle packing
  • Feedback vertex set
  • Unit disk graph
  • Unit square graph
  • Parameterized complexity

Mathematics Subject Classification

  • 68W01
  • 68W40
  • 68Q25