Document Open Access Logo

ETH-Tight Algorithms for Long Path and Cycle on Unit Disk Graphs

Authors Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Meirav Zehavi

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2020.44.pdf
  • Filesize: 0.66 MB
  • 18 pages

Document Identifiers

Related Versions

  • A full version of the paper is available at [Fomin et al., 2020].

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Computational geometry
Keywords
  • Optimality Program
  • ETH
  • Unit Disk Graphs
  • Parameterized Complexity
  • Long Path
  • Long Cycle

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We present an algorithm for the extensively studied Long Path and Long Cycle problems on unit disk graphs that runs in time 2^{𝒪(√k)}(n+m). Under the Exponential Time Hypothesis, Long Path and Long Cycle on unit disk graphs cannot be solved in time 2^{o(√k)}(n+m)^𝒪(1) [de Berg et al., STOC 2018], hence our algorithm is optimal. Besides the 2^{𝒪(√k)}(n+m)^𝒪(1)-time algorithm for the (arguably) much simpler Vertex Cover problem by de Berg et al. [STOC 2018] (which easily follows from the existence of a 2k-vertex kernel for the problem), this is the only known ETH-optimal fixed-parameter tractable algorithm on UDGs. Previously, Long Path and Long Cycle on unit disk graphs were only known to be solvable in time 2^{𝒪(√klog k)}(n+m). This algorithm involved the introduction of a new type of a tree decomposition, entailing the design of a very tedious dynamic programming procedure. Our algorithm is substantially simpler: we completely avoid the use of this new type of tree decomposition. Instead, we use a marking procedure to reduce the problem to (a weighted version of) itself on a standard tree decomposition of width 𝒪(√k).

Cite As Get BibTex

Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. ETH-Tight Algorithms for Long Path and Cycle on Unit Disk Graphs. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.44

Author Details

Fedor V. Fomin
  • University of Bergen, Norway
Daniel Lokshtanov
  • University of California, Santa Barbara, CA, USA
Fahad Panolan
  • Department of Computer Science and Engineering, IIT Hyderabad, India
Saket Saurabh
  • Department of Informatics, University of Bergen, Norway
  • The Institute of Mathematical Sciences, HBNI and IRL 2000 ReLaX, Chennai, India
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

  • Fomin, Fedor V.: Research Council of Norway via the project MULTIVAL.
  • Lokshtanov, Daniel: European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 715744), and United States - Israel Binational Science Foundation grant no. 2018302.
  • Saurabh, Saket: European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 819416), and Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.
  • Zehavi, Meirav: Israel Science Foundation grant no. 1176/18, and United States – Israel Binational Science Foundation grant no. 2018302.

References

  1. Jochen Alber and Jiří Fiala. Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. In Foundations of Information Technology in the Era of Network and Mobile Computing, pages 26-37. Springer, 2002. Google Scholar
  2. Noga Alon, Phuong Dao, Iman Hajirasouliha, Fereydoun Hormozdiari, and Süleyman Cenk Sahinalp. Biomolecular network motif counting and discovery by color coding. In Proceedings 16th International Conference on Intelligent Systems for Molecular Biology (ISMB), Toronto, Canada, July 19-23, 2008, pages 241-249, 2008. URL: https://doi.org/10.1093/bioinformatics/btn163.
  3. Noga Alon and Shai Gutner. Balanced hashing, color coding and approximate counting. In Parameterized and Exact Computation, 4th International Workshop, IWPEC 2009, Copenhagen, Denmark, September 10-11, 2009, Revised Selected Papers, pages 1-16, 2009. URL: https://doi.org/10.1007/978-3-642-11269-0_1.
  4. Noga Alon and Shai Gutner. Balanced families of perfect hash functions and their applications. ACM Trans. Algorithms, 6(3):54:1-54:12, 2010. URL: https://doi.org/10.1145/1798596.1798607.
  5. Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. Assoc. Comput. Mach., 42(4):844-856, 1995. Google Scholar
  6. Vikraman Arvind and Venkatesh Raman. Approximation algorithms for some parameterized counting problems. In Algorithms and Computation, 13th International Symposium, ISAAC 2002 Vancouver, BC, Canada, November 21-23, 2002, Proceedings, pages 453-464, 2002. URL: https://doi.org/10.1007/3-540-36136-7_40.
  7. Ivona Bezáková, Radu Curticapean, Holger Dell, and Fedor V. Fomin. Finding detours is fixed-parameter tractable. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, pages 54:1-54:14, 2017. Google Scholar
  8. Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Narrow sieves for parameterized paths and packings. J. Comput. Syst. Sci., 87:119-139, 2017. Google Scholar
  9. Andreas Björklund, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Approximate counting of k-paths: Deterministic and in polynomial space. In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, pages 24:1-24:15, 2019. Google Scholar
  10. Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput., 243:86-111, 2015. URL: https://doi.org/10.1016/j.ic.2014.12.008.
  11. Hans L Bodlaender, Rodney G Downey, Michael R Fellows, and Danny Hermelin. On problems without polynomial kernels. Journal of Computer and System Sciences, 75(8):423-434, 2009. Google Scholar
  12. Hans L. Bodlaender, Pål Grønås Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov, and Michal Pilipczuk. A c^k n 5-approximation algorithm for treewidth. SIAM J. Comput., 45(2):317-378, 2016. Google Scholar
  13. Cornelius Brand, Holger Dell, and Thore Husfeldt. Extensor-coding. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 151-164, 2018. URL: https://doi.org/10.1145/3188745.3188902.
  14. Heinz Breu and David G Kirkpatrick. Unit disk graph recognition is np-hard. Computational Geometry, 9(1-2):3-24, 1998. Google Scholar
  15. Timothy M. Chan. Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms, 46(2):178-189, 2003. URL: https://doi.org/10.1016/S0196-6774(02)00294-8.
  16. Jianer Chen, Joachim Kneis, Songjian Lu, Daniel Mölle, Stefan Richter, Peter Rossmanith, Sing-Hoi Sze, and Fenghui Zhang. Randomized divide-and-conquer: Improved path, matching, and packing algorithms. SIAM Journal on Computing, 38(6):2526-2547, 2009. Google Scholar
  17. Brent N. Clark, Charles J. Colbourn, and David S. Johnson. Unit disk graphs. Discrete Mathematics, 86(1-3):165-177, 1990. Google Scholar
  18. 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.
  19. Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 150-159, 2011. Google Scholar
  20. Mark de Berg, Hans L. Bodlaender, Sándor Kisfaludi-Bak, Dániel Marx, and Tom C. van der Zanden. A framework for eth-tight algorithms and lower bounds in geometric intersection graphs. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 574-586, 2018. Google Scholar
  21. Erik D. Demaine, Fedor V. Fomin, Mohammadtaghi Hajiaghayi, and Dimitrios M. Thilikos. Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs. jacm, 52(6):866-893, 2005. Google Scholar
  22. Erik D. Demaine and MohammadTaghi Hajiaghayi. The bidimensionality theory and its algorithmic applications. Comput. J., 51(3):292-302, 2008. URL: https://doi.org/10.1093/comjnl/bxm033.
  23. Frederic Dorn, Eelko Penninkx, Hans L. Bodlaender, and Fedor V. Fomin. Efficient exact algorithms on planar graphs: Exploiting sphere cut decompositions. Algorithmica, 58(3):790-810, 2010. URL: https://doi.org/10.1007/s00453-009-9296-1.
  24. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. Google Scholar
  25. Adrian Dumitrescu and János Pach. Minimum clique partition in unit disk graphs. Graphs and Combinatorics, 27(3):399-411, 2011. Google Scholar
  26. Jörg Flum and Martin Grohe. The parameterized complexity of counting problems. SIAM J. Comput., 33(4):892-922, 2004. Google Scholar
  27. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Going far from degeneracy. In 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, pages 47:1-47:14, 2019. Google Scholar
  28. Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Parameterization above a multiplicative guarantee. In 11th Innovations in Theoretical Computer Science, ITCS 2020 (To Appear), 2020. Google Scholar
  29. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM, 63(4):29:1-29:60, 2016. Google Scholar
  30. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Efficient computation of representative families with applications in parameterized and exact algorithms. jacm, 63(4):29, 2016. URL: https://doi.org/10.1145/2886094.
  31. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Long directed (s, t)-path: FPT algorithm. Inf. Process. Lett., 140:8-12, 2018. Google Scholar
  32. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Decomposition of map graphs with applications. In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece., pages 60:1-60:15, 2019. Google Scholar
  33. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Finding, hitting and packing cycles in subexponential time on unit disk graphs. Discrete & Computational Geometry, 62(4):879-911, 2019. Google Scholar
  34. Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Eth-tight algorithms for long path and cycle on unit disk graphs. CoRR, abs/2003.00938, 2020. URL: http://arxiv.org/abs/2003.00938.
  35. Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Bidimensionality and geometric graphs. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1563-1575, 2012. Google Scholar
  36. Harold N Gabow and Shuxin Nie. Finding a long directed cycle. ACM Transactions on Algorithms (TALG), 4(1):7, 2008. Google Scholar
  37. William K Hale. Frequency assignment: Theory and applications. Proceedings of the IEEE, 68(12):1497-1514, 1980. Google Scholar
  38. Danny Hermelin, Stefan Kratsch, Karolina Soltys, Magnus Wahlström, and Xi Wu. A completeness theory for polynomial (turing) kernelization. Algorithmica, 71(3):702-730, 2015. URL: https://doi.org/10.1007/s00453-014-9910-8.
  39. Falk Hüffner, Sebastian Wernicke, and Thomas Zichner. Algorithm engineering for color-coding with applications to signaling pathway detection. Algorithmica, 52(2):114-132, 2008. Google Scholar
  40. Harry B. Hunt III, Madhav V. Marathe, Venkatesh Radhakrishnan, S. S. Ravi, Daniel J. Rosenkrantz, and Richard Edwin Stearns. NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs. J. Algorithms, 26(2):238-274, 1998. Google Scholar
  41. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity. Journal of Computer and System Sciences, 63(4):512-530, 2001. Google Scholar
  42. Alon Itai, Christos H. Papadimitriou, and Jayme Luiz Szwarcfiter. Hamilton paths in grid graphs. SIAM J. Comput., 11(4):676-686, 1982. URL: https://doi.org/10.1137/0211056.
  43. Hiro Ito and Masakazu Kadoshita. 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 (ISORA10), pages 120-127, 2010. Google Scholar
  44. Bart M. P. Jansen. Polynomial kernels for hard problems on disk graphs. In Proceedings of the 12th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), volume 6139, pages 310-321. Springer, 2010. Google Scholar
  45. Bart M. P. Jansen, László Kozma, and Jesper Nederlof. Hamiltonicity below dirac’s condition. In Graph-Theoretic Concepts in Computer Science - 45th International Workshop, WG 2019, Vall de Núria, Spain, June 19-21, 2019, Revised Papers, pages 27-39, 2019. Google Scholar
  46. Bart M. P. Jansen, Marcin Pilipczuk, and Marcin Wrochna. Turing kernelization for finding long paths in graph classes excluding a topological minor. Algorithmica, 81(10):3936-3967, 2019. Google Scholar
  47. Karl Kammerlander. C 900-an advanced mobile radio telephone system with optimum frequency utilization. IEEE journal on selected areas in communications, 2(4):589-597, 1984. Google Scholar
  48. Ross J. Kang and Tobias Müller. Sphere and dot product representations of graphs. Discrete & Computational Geometry, 47(3):548-568, 2012. URL: https://doi.org/10.1007/s00454-012-9394-8.
  49. Ioannis Koutis. Faster algebraic algorithms for path and packing problems. In Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP 2008), volume 5125 of Lecture Notes in Computer Science, pages 575-586, 2008. Google Scholar
  50. Ioannis Koutis and Ryan Williams. Algebraic fingerprints for faster algorithms. Commun. ACM, 59(1):98-105, 2016. URL: https://doi.org/10.1145/2742544.
  51. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Slightly superexponential parameterized problems. SIAM J. Comput., 47(3):675-702, 2018. Google Scholar
  52. Burkhard Monien. How to find long paths efficiently. In North-Holland Mathematics Studies, volume 109, pages 239-254. Elsevier, 1985. Google Scholar
  53. George L Nemhauser and Leslie Earl Trotter. Vertex packings: structural properties and algorithms. Mathematical Programming, 8(1):232-248, 1975. Google Scholar
  54. Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Contraction decomposition in unit disk graphs and algorithmic applications in parameterized complexity. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1035-1054, 2019. Google Scholar
  55. Christos H. Papadimitriou and Mihalis Yannakakis. On limited nondeterminism and the complexity of the V.C dimension (extended abstract). In Proceedings of the Eigth Annual Structure in Complexity Theory Conference, San Diego, CA, USA, May 18-21, 1993, pages 12-18, 1993. Google Scholar
  56. Hadas Shachnai and Meirav Zehavi. Representative families: A unified tradeoff-based approach. J. Comput. Syst. Sci., 82(3):488-502, 2016. Google Scholar
  57. Warren D. Smith and Nicholas C. Wormald. Geometric separator theorems & applications. In Proceedings of the 39th Annual Symposium on Foundations of Computer Science (FOCS), pages 232-243. IEEE Computer Society, 1998. Google Scholar
  58. Dekel Tsur. Faster deterministic parameterized algorithm for k-path. Theor. Comput. Sci., 790:96-104, 2019. Google Scholar
  59. DW Wang and Yue-Sun Kuo. A study on two geometric location problems. Information processing letters, 28(6):281-286, 1988. Google Scholar
  60. Ryan Williams. Finding paths of length k in O^*(2^k) time. Inf. Process. Lett., 109(6):315-318, 2009. Google Scholar
  61. Yu-Shuan Yeh, J Wilson, and S Schwartz. Outage probability in mobile telephony with directive antennas and macrodiversity. IEEE journal on selected areas in communications, 2(4):507-511, 1984. Google Scholar
  62. Meirav Zehavi. Mixing color coding-related techniques. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, pages 1037-1049, 2015. URL: https://doi.org/10.1007/978-3-662-48350-3_86.
  63. Meirav Zehavi. A randomized algorithm for long directed cycle. Inf. Process. Lett., 116(6):419-422, 2016. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact